Robust Optimality and Duality for Nonsmooth Multiobjective Programming Problems with Vanishing Constraints Under Data Uncertainty

Abstract

This article investigates robust optimality conditions and duality results for a class of nonsmooth multiobjective programming problems with vanishing constraints under data uncertainty (UNMPVC). Mathematical programming problems with vanishing constraints constitute a distinctive class of constrained optimization problems because of the presence of complementarity constraints. Moreover, uncertainties are inherent in various real-life problems. The aim of this article is to identify an optimal solution to an uncertain optimization problem with vanishing constraints that remains feasible in every possible future scenario. Stationary conditions are necessary conditions for optimality in mathematical programming problems with vanishing constraints. These conditions can be derived under various constraint qualifications. Employing the properties of convexificators, we introduce generalized standard Abadie constraint qualification (GS-ACQ) for the considered problem, UNMPVC. We introduce a generalized robust version of nonsmooth stationary conditions, namely a weakly stationary point, a Mordukhovich stationary point, and a strong stationary point (RS-stationary) for UNMPVC. By employing GS-ACQ, we establish the necessary conditions for a local weak Pareto solution of UNMPVC. Moreover, under generalized convexity assumptions, we derive sufficient optimality criteria for UNMPVC. Furthermore, we formulate the Wolfe-type and Mond–Weir-type robust dual models corresponding to the primal problem, UNMPVC.

1. Introduction

Multiobjective programming (MOP) is a vital tool for tackling optimization challenges with multiple conflicting objectives commonly encountered in fields like business, economics, and science (see, for instance, [1,2]). By considering multiple objectives simultaneously, MOP aids in making decisions that balance conflicting priorities. Numerous researchers have contributed to the theory and applications of MOP, exploring diverse approaches and settings to address several optimization problems (for instance, see, [3,4]).

Convexity is a cornerstone of optimization theory and algorithms ensuring that stationary points are also global minima. Moreover, the satisfaction of the necessary optimality conditions guarantees global optimality. However, many real-world problems are nonconvex. To address this limitation, the convex optimization theory has been significantly enriched by introducing a variety of generalized convex functions. Mangasarian [5] pioneered the idea of pseudoconvex and pseudoconcave functions as generalizations of convex and concave functions, respectively. De Finetti [6] was one of the first to recognize some of the characteristics of functions having convex level sets. He did not name this class, but he did note that it includes all convex functions and some nonconvex functions. Fenchel [7] was one of the early pioneers in formalizing, naming, and developing the class of quasiconvex functions. For a more comprehensive study on convex functions and their role in optimization theory, see [8,9,10].

Nonsmoothness occurs inevitably in many real-life problems arising in various fields, including engineering, science, and numerous other related fields of modern research (see, [11,12], and the references mentioned therein). For instance, Ştefănescu et al. [13] developed a surrogate-based robust design for a non-smooth radiation source detection problem. Antonelli et al. [14] introduced a total-variation-regularized image segmentation model and discussed the preservation of spatial features in nonsmooth regions. To solve nonlinear complementarity problems, Śmietański [15] devised nonsmooth gauss-newton algorithms. To tackle the challenges posed by nonsmooth functions in optimization problems, the notion of subdifferentials and generalized derivatives has been developed and thoroughly examined. Clarke [16] established various theories for nonsmooth analysis and presented subdifferentials for nonsmooth optimization problems. Michel and Penot [17] introduced a simple way of defining a generalized derivative for arbitrary functions. Mordukhovich and Shao [18] explored the subdifferentials for extended real-valued functions defined based on Banach spaces. Treiman [19] introduced a linear generalized gradient that is smaller than the generalized gradients defined by Clarke and Mordukhovich. Demyanov and Jeyakumar [20] discussed several results of Clarke and Michel–Penot subdifferentials. By extending the concepts of upper convex and lower concave approximations, Demyanov [21] introduced the notion of convexificators. Jeyakumar and Luc [22] presented noncompact convexificators and developed a comprehensive calculus for their computation. Convexificators are weaker versions of well-known subdifferentials, such as Clarke [16], Treiman [19], Morduchovich-Shao [18], and Michel-Penot [17]. In this context, it is important to note that in sharp contrast to various known subdifferentials, convexificators, generally, are not necessarily bounded or convex sets (but are closed sets). In the case of a locally Lipschitz function, the various generalized subdifferentials mentioned above turn out to be convexificators. Furthermore, it is interesting to note that the subdifferentials mentioned above may contain the convex hull of a convexificator (for reference, see, [22]). In the framework of Banach spaces, Luu [23,24] established necessary optimality criteria for MOP involving mixed as well as set constraints via convexificators. Rimpi and Lalitha [25] employed convexificators to establish constraint qualifications (CQ) for nonsmooth optimization problems with mixed constraints.

Over the last few years, mathematical programming problems with vanishing constraints (MPVC) have gained prominence as a significant area of research (see, [26,27,28]). Achtziger and Kanzow [29] are considered the pioneers of MPVC, laying the groundwork for its formulation and development. The nomenclature “vanishing constraints” aptly captures a critical feature of these problems. In many real-world applications, certain constraints become inactive for specific feasible solutions within the decision space (see, [30]). A significant challenge in investigating MPVC is the fact that the feasible set may be non-convex and disconnected, even when all constraint functions are convex (see [30]).

Hoheisel and Kanzow [31] made significant contributions to MPVC by deriving necessary optimality criteria for MPVC. Hoheisel and Kanzow [32] introduced various MPVC-tailored CQs and established interrelationships among them. Later, Hoheisel and Kanzow [33] investigated Guignard and Abadie CQs for MPVC. Kazemi and Kanzi [34] introduced various CQs for MPVC with non-differentiable constraints and established several KKT-type stationarity conditions. For nonsmooth MPVC, Kazemi et al. [35] estimated the Fréchet normal cone of the feasible set and derived corresponding stationarity conditions. Shaker et al. [36] introduced two new Abadie-type CQ and established the necessary optimality criteria for properly efficient solutions for multiobjective MPVC with non-differentiable convex constraints and continuously differentiable objective functions. Sadeghieh et al. [37] established first-order necessary and sufficient optimality criteria for an efficient and weakly efficient solution for nonsmooth multiobjective MPVC. Upadhyay and Ghosh [38] introduced several CQs for MPVC on Hadamard manifolds. Later, Upadhyay et al. [39] introduced several CQs and established Pareto efficiency criteria for multiobjective MPVC based on Hadamard manifolds. Using convexificators, Hu et al. [40] introduced CQs and stationary conditions for MPVC involving non-differentiable functions. By leveraging the concept of convexificators, Lai et al. [41] introduced generalized cottle-type and Guignard-type CQs for nonsmooth semidefinite multiobjective MPVC and derived stationary conditions for the considered problem. Recently, Upadhyay et al. [42] introduced the concept of convexificators within the context of Hadamard manifolds and employed them to investigate multiobjective MPVC on Hadamard manifolds.

Duality is a fundamental concept in optimization that allows us to view the same optimization problem from a different perspective. In many circumstances, tackling the dual problem is often seen to be more advantageous than addressing the primal problem. Wolfe [43] and Mond and Weir [44] introduced two important dual models for differentiable scalar functions, which have subsequently been extended to nonsmooth scalar and multiobjective optimization problems under various generalized convexity and subdifferential assumptions. Mishra et al. [45] presented Mond–Weir- and Wolfe-type dual models for MPVC. Hu et al. [46] formulated new Mond–Weir- and Wolfe-type dual models for MPVC, which eliminate the need for index set calculations. Tung [47] studied Pareto efficiency criteria and various duality results for multiobjective semi-infinite programming problems with vanishing constraints. Recently, in the setting of Hadamard manifolds, for nonsmooth multiobjective semi-infinite programming problems with vanishing constraints, Upadhyay et al. [48] investigated optimality criteria and formulated dual models.

Robust optimization deals with uncertain data in an optimization problem. Mathematical programming problems often incorporate parameters with inherent uncertainty due to estimation errors, rounding discrepancies, or implementation issues. Robust optimization aims to find solutions that remain optimal even when faced with uncertainties in the problem’s parameters. Generally, robust optimization reformulates the original problem into a deterministic equivalent, called the robust counterpart. This allows us to use the standard optimization algorithms to find solutions that are resilient to these uncertainties. Such problems arise frequently in various areas of modern research (see, for instance, [49,50]).

To tackle the barriers posed by the nonsmooth nature of data uncertainty problems, several authors have thoroughly investigated nonsmooth optimization problems with uncertain data under various assumptions. Lee and Lee [51] established nonsmooth optimality criteria for robust MOP. Fakhar et al. [52] developed sufficient optimality criteria and dual models for nonsmooth MOP with uncertain data. Chen et al. [53] introduced two new classes of generalized convex functions, which are not necessarily convex, and established optimality and duality results under the assumptions of generalized convexity for nonsmooth MOP with uncertain data. Recently, Gadhi and Ohda [54] established necessary optimality conditions for robust nonsmooth MOP. Utilizing convexificators, Chen et al. [55] deduced weak Pareto efficiency criteria and duality results for nonconvex MOP with uncertain data.

It is worthwhile to mention that the optimality conditions and duality theorems have so far been studied by several researchers for nonsmooth robust optimization problems defined in Euclidean spaces (see, for instance, [52,53,54]). However, to the best of our knowledge, prior work has not addressed problems involving vanishing constraints and data uncertainty. In this paper, our main focus is to primarily tackle the aforementioned research gap. Motivated by the research work presented in [40,45,53,54], we examine a class of nonsmooth multiobjective programming problems with vanishing constraints under data uncertainty via convexificators. We introduce GS-ACQ for UNMPVC and introduce generalized robust versions of various stationary points, namely RW-stationary, RM-stationary, and RS-stationary points, for UNMPVC. Subsequently, under generalized convexity assumptions, we deduce sufficient optimality conditions for UNMPVC. We further enrich the analysis of the primal problem, UNMPVC, by formulating WRD and MWRD-type dual models. This enables us to derive weak, strong, and strict converse duality relationships between the original problem and its dual model. The importance of results established in this paper is demonstrated through suitable illustrative examples.

The fundamental contributions and novelty of this paper are twofold. At first, we introduce GS-ACQ and robust stationary conditions for UNMPVC, which is employed to extend the optimality conditions derived in [40] for nonsmooth MPVC to a more general category of optimization problems dealing with data uncertainty. Moreover, we extend the duality theorems established in [45] for MPVC to a wider class of mathematical programming problems dealing with data uncertainty. Secondly, we generalize the robust optimality and duality results established in [54,55] to a broader class of general robust mathematical programming problems incorporating vanishing constraints. To the best of our knowledge, optimality and duality results in terms of convexificators presented in this paper have not been given before.

The present article is structured as follows: we review some fundamental definitions and mathematical concepts in Section 2. In Section 3, we introduce GS-ACQ constraint qualification and various stationary conditions for UNMPVC. Moreover, we deduce sufficient optimality conditions for UNMPVC. In Section 4 and Section 5, we formulate two dual models, namely WRD and MWRD, related to the problem of UNMPVC. We deduce various duality relationships between the UNMPVC and its corresponding dual models. In Section 6, we discuss the applicability of the results derived in this article in truss topology optimization. In Section 7, we conclude the article by providing a summary of the key findings and various possible future research avenues.

2. Preliminaries and Notations

The following abbreviations are used in this article:

$$\begin{array}{cc}\hfill \mathrm{MOP}& :\mathrm{Multiobjective}\mathrm{programming}\mathrm{problems}\hfill \\ \hfill \mathrm{CQ}& :\mathrm{Constraint}\mathrm{qualifications}\hfill \\ \hfill \mathrm{UNMPVC}& :\mathrm{Nonsmooth}\mathrm{multiobjective}\mathrm{programming}\mathrm{problems}\mathrm{with}\mathrm{vanishing}\hfill \\ \hfill & \mathrm{constraints}\mathrm{under}\mathrm{data}\mathrm{uncertainty}\hfill \\ \hfill \mathrm{GS}\text{-}\mathrm{ACQ}& :\mathrm{Generalized}\mathrm{standard}\mathrm{Abadie}3.\mathrm{constraint}\mathrm{qualification}\hfill \\ \hfill \mathrm{RW}\text{-}\mathrm{staionary}& :\mathrm{Robust}\mathrm{weakly}\mathrm{stationary}\mathrm{point}\hfill \\ \hfill \mathrm{RM}\text{-}\mathrm{staionary}& :\mathrm{Robust}\mathrm{Mordukhovich}\mathrm{stationary}\mathrm{point}\hfill \\ \hfill \mathrm{RS}\text{-}\mathrm{staionary}& :\mathrm{Robust}\mathrm{strong}\mathrm{stationary}\mathrm{point}\hfill \\ \hfill \mathrm{WRD}& :\mathrm{Wolfe}\text{-}\mathrm{type}\mathrm{robust}\mathrm{dual}\mathrm{model}\hfill \\ \hfill \mathrm{MWRD}& :\mathrm{Mond}\text{–}\mathrm{Weir}\text{-}\mathrm{type}\mathrm{robust}\mathrm{dual}\mathrm{model}\hfill \end{array}$$

Throughout this article, we use the notation ${\mathbb{R}}^n$ to denote the n-dimensional Euclidean space and $\mathbb{N}$ to denote the set of all natural numbers. Let $\mathtt{b},\mathtt{d}\in {\mathbb{R}}^n$. The notations given below will be employed:

$$\begin{array}{cc}\hfill \mathtt{b}\prec \mathtt{d}⟺& {\mathtt{b}}_{\ell }<{\mathtt{d}}_{\ell },\forall \ell \in \{1,\cdots ,n\},\hfill \\ \hfill \mathtt{b}⪯\mathtt{d}⟺& \left\{\begin{array}{c}{\mathtt{b}}_{\ell }\le {\mathtt{d}}_{\ell },\forall \ell \in \{1,\cdots ,n\},\hfill \\ {\mathtt{b}}_k<{\mathtt{d}}_k,\mathrm{for}\mathrm{at}\mathrm{least}\mathrm{one}k\in \{1,\cdots ,n\}.\hfill \end{array}\right.\hfill \end{array}$$

Let $\mathcal{D}$ be a nonempty subset of ${\mathbb{R}}^n$. We denote the closure of $\mathcal{D}$ as $cl\mathcal{D}$, its convex hull as $co\mathcal{D}$, and the convex cone generated by $\mathcal{D}$ (including the origin) as $cone\mathcal{D}$.

Let $\mathcal{G}:{\mathbb{R}}^n\to \overline{\mathbb{R}}$ be an extended real-valued function. Then, the domain of function $\mathcal{G}$ is denoted by $dom\left(\mathcal{G}\right)$ and is defined as follows:

$$\mathrm{dom}\left(\mathcal{G}\right):=\{\mathfrak{a}\in {\mathbb{R}}^n:\mathcal{G}\left(\mathfrak{a}\right)\ne +\infty \}.$$

We now define the following sets:

$${\mathcal{D}}^{-}:=\left\{\mathfrak{a}\in {\mathbb{R}}^n:⟨\mathfrak{a},\mathfrak{b}⟩\le 0,\forall \mathfrak{b}\in \mathcal{D}\right\},$$

$${\mathcal{D}}^s:=\left\{\mathfrak{a}\in {\mathbb{R}}^n:⟨\mathfrak{a},\mathfrak{b}⟩<0,\forall \mathfrak{b}\in \mathcal{D}\right\}.$$

The following definitions are used in this article:

Definition 1 

([56]). The contingent cone $\mathcal{T}(\mathcal{D},\mathfrak{a})$ of a nonempty subset $\mathcal{D}$ of ${\mathbb{R}}^n$ at $\mathfrak{a}\in cl\mathcal{D}$ is defined as

$$\mathcal{T}(\mathcal{D},\mathfrak{a}):=\left\{\mathtt{d}\in {\mathbb{R}}^n:\exists {\alpha }_n\downarrow 0\&{\mathtt{d}}_n\to \mathtt{d}\mathit{such}\mathit{that}\mathfrak{a}+{\alpha }_n{\mathtt{d}}_n\in \mathcal{D},\forall n\in \mathbb{N}\right\}.$$

Definition 2 

([56]). The lower and upper Dini derivatives of function $\mathcal{G}$ at $\mathfrak{a}$ in some direction $\mathtt{d}\in {\mathbb{R}}^n$ are defined, respectively, as

$${\mathcal{G}}^{-}(\mathfrak{a};\mathtt{d}):=\underset{\mu \downarrow 0}{lim\; inf}{\displaystyle \frac{\mathcal{G}(\mathfrak{a}+\mu \mathtt{d})-\mathcal{G}\left(\mathfrak{a}\right)}{\mu }},$$

$${\mathcal{G}}^{+}(\mathfrak{a};\mathtt{d}):=\underset{\mu \downarrow 0}{lim\; sup}{\displaystyle \frac{\mathcal{G}(\mathfrak{a}+\mu \mathtt{d})-\mathcal{G}\left(\mathfrak{a}\right)}{\mu }}.$$

Definition 3 

([56]). $\mathcal{G}:{\mathbb{R}}^n\to \overline{\mathbb{R}}$ is said to have an upper semi-regular convexificator (USRC) at $\mathfrak{a}\in \mathit{dom}\left(\mathcal{G}\right)$, denoted as ${\partial }^{*}\mathcal{G}\left(\mathfrak{a}\right)$, provided that the set ${\partial }^{*}\mathcal{G}\left(\mathfrak{a}\right)$ is a closed set and for every $\mathtt{d}\in {\mathbb{R}}^n$, one has

$${\mathcal{G}}^{+}(\mathfrak{a};\mathtt{d})\le \underset{\kappa \in {\partial }^{*}\mathcal{G}\left(\mathfrak{a}\right)}{sup}⟨\kappa ,\mathtt{d}⟩.$$

Definition 4 

([57]). Let $\mathcal{G}:{\mathbb{R}}^n\to \overline{\mathbb{R}}$ and admit an bounded USRC at $\mathfrak{a}\in \mathit{dom}\left(\mathcal{G}\right)$, denoted by ${\partial }^{*}\mathcal{G}\left(\mathfrak{a}\right)$. Then,

• $\mathcal{G}$ is ${\partial }^{*}$-convex at $\mathfrak{a}$ if, for each $\mathfrak{b}\in {\mathbb{R}}^n$,

  $$\mathcal{G}\left(\mathfrak{b}\right)-\mathcal{G}\left(\mathfrak{a}\right)\ge ⟨\zeta ,\mathfrak{b}-\mathfrak{a}⟩,\forall \zeta \in {\partial }^{*}\mathcal{G}\left(\mathfrak{a}\right).$$
• $\mathcal{G}$ is strictly ${\partial }^{*}$-convex at $\mathfrak{a}$ if, for each $\mathfrak{b}(\ne \mathfrak{a})\in {\mathbb{R}}^n$,

  $$\mathcal{G}\left(\mathfrak{b}\right)-\mathcal{G}\left(\mathfrak{a}\right)>⟨\zeta ,\mathfrak{b}-\mathfrak{a}⟩,\forall \zeta \in {\partial }^{*}\mathcal{G}\left(\mathfrak{a}\right).$$
• $\mathcal{G}$ is ${\partial }^{*}$-pseudoconvex at $\mathfrak{a}$ if, for each $\mathfrak{b}\in {\mathbb{R}}^n$,

  $$\mathcal{G}\left(\mathfrak{b}\right)<\mathcal{G}\left(\mathfrak{a}\right)\Rightarrow ⟨\zeta ,\mathfrak{b}-\mathfrak{a}⟩<0,\forall \zeta \in {\partial }^{*}\mathcal{G}\left(\mathfrak{a}\right).$$
• $\mathcal{G}$ is strictly ${\partial }^{*}$-pseudoconvex at $\mathfrak{a}$ if, for each $\mathfrak{b}(\ne \mathfrak{a})\in {\mathbb{R}}^n$,

  $$\mathcal{G}\left(\mathfrak{b}\right)\le \mathcal{G}\left(\mathfrak{a}\right)\Rightarrow ⟨\zeta ,\mathfrak{b}-\mathfrak{a}⟩<0,\forall \zeta \in {\partial }^{*}\mathcal{G}\left(\mathfrak{a}\right).$$
• $\mathcal{G}$ is ${\partial }^{*}$-quasiconvex at $\mathfrak{a}$ if, for each $\mathfrak{b}\in {\mathbb{R}}^n$,

  $$\mathcal{G}\left(\mathfrak{b}\right)\le \mathcal{G}\left(\mathfrak{a}\right)\Rightarrow ⟨\zeta ,\mathfrak{b}-\mathfrak{a}⟩\le 0,\forall \zeta \in {\partial }^{*}\mathcal{G}\left(\mathfrak{a}\right).$$

Definition 5 

([58]). $\mathcal{G}:{\mathbb{R}}^n\to \mathbb{R}$ is locally Lipschitz if every point $d\in {\mathbb{R}}^n$ possesses a neighbourhood ${\mathcal{N}}_d\subset {\mathbb{R}}^n$ such that

$$|\mathcal{G}\left(d_1\right)-\mathcal{G}\left(d_2\right)|\le M\Vert d_1-d_2\Vert ,\forall d_1,d_2\in {\mathcal{N}}_d,$$

for a constant $M>0$ depending on ${\mathcal{N}}_d$.

The following theorems are utilized to establish various results in this article. Moreover, we use the notation $\mathbb{T}$ to represent the set defined as $\mathbb{T}:=\{1,\dots ,r\}$.

Theorem 1 

([55]). Let $f_t:{\mathbb{R}}^n\to \mathbb{R}$, $t\in \mathbb{T}$ be a locally Lipschitz function. Moreover, we assume that $f_t$ admits USRC ${\partial }^{*}{f}_t\left(\mathfrak{a}\right)$ at $\mathfrak{a}\in {\mathbb{R}}^n$ for all $t\in \mathbb{T}$. Then $\Theta \left(\mathfrak{a}\right):=max\{f_1\left(\mathfrak{a}\right),\cdots ,f_r\left(\mathfrak{a}\right)\}$ admits an USRC which is convex and is given as

$${\partial }^{*}\Theta \left(\mathfrak{a}\right)=co\left\{\bigcup _{t\in \mathbb{T}\left(\mathfrak{a}\right)}{\partial }^{*}{f}_t\left(\mathfrak{a}\right)\right\},$$

where $\mathbb{T}\left(\mathfrak{a}\right):=\{t\in \mathbb{T}:f_t\left(\mathfrak{a}\right)=\Theta \left(\mathfrak{a}\right)\}.$

Theorem 2 

([55]). Let $f_t:{\mathbb{R}}^n\to \mathbb{R}$, $t\in \mathbb{T}$ and $\overline{\mathfrak{a}}\in {\mathbb{R}}^n$. Define

$${\Theta }_t\left(\overline{\mathfrak{a}}\right):=\underset{{\mathfrak{u}}_t\in {\Omega }_t}{max}{f}_t(\overline{\mathfrak{a}},{\mathfrak{u}}_t),t\in \mathbb{T},$$

and

$${\Omega }_t\left(\overline{\mathfrak{a}}\right):=\{{\overline{\mathfrak{u}}}_t\in {\Omega }_t:F_t\left(\overline{\mathfrak{a}}\right)=f_t(\overline{\mathfrak{a}},{\overline{\mathfrak{u}}}_t)\}.$$

Then

$${\partial }^{*}{\Theta }_t\left(\overline{\mathfrak{a}}\right)=\left\{{\partial }^{*}{f}_t(\overline{\mathfrak{a}},{\overline{\mathfrak{u}}}_t):{\overline{\mathfrak{u}}}_t\in {\Omega }_t\left(\overline{\mathfrak{a}}\right)\right\}=\bigcup _{{\overline{\mathfrak{u}}}_t\in {\Omega }_t\left(\overline{\mathfrak{a}}\right)}{\partial }^{*}{f}_t(\overline{\mathfrak{a}},{\overline{\mathfrak{u}}}_t),$$

where for each $t\in \mathbb{T}$, ${\Omega }_t$ is nonempty convex subset of ${\mathbb{R}}^{n_t}$.

3. Optimality Criteria

In this section, we consider a nonsmooth MOP with vanishing constraints under data uncertainty. We introduce GS-ACQ at a feasible point of UNMPVC. Moreover, we introduce RW-stationary, RM-stationary and RS-stationary points for UNMPVC. By employing GS-ACQ, we establish that the RS-stationary is a necessary optimality criterion for UNMPVC. Subsequently, we establish sufficient optimality criteria for UNMPVC.

In the remaining portion of this article, we examine the following UNMPVC:

$$\begin{array}{cc}\hfill \left(\mathrm{UNMPVC}\right)\mathrm{Minimize}& \mathcal{G}\left(\mathfrak{a}\right):=({\mathcal{G}}_1\left(\mathfrak{a}\right),\cdots ,{\mathcal{G}}_r\left(\mathfrak{a}\right)),\hfill \\ \hfill \mathrm{subject}\mathrm{to}& {\Psi }_j(\mathfrak{a},{\mathfrak{u}}_j)\le 0,j=1,\cdots ,p,\hfill \\ \hfill & {\Theta }_{\mathfrak{s}}(\mathfrak{a},{\mathfrak{u}}_{\mathfrak{s}})=0,\mathfrak{s}=1,\dots ,\ell ,\hfill \\ \hfill & {\mathtt{K}}_{\mathfrak{i}}(\mathfrak{a},{\mathfrak{u}}_{\mathfrak{i}})\ge 0,i=1,\dots ,m,\hfill \\ \hfill & {\mathtt{L}}_{\mathfrak{i}}(\mathfrak{a},{\mathfrak{u}}_{\mathfrak{i}}){\mathtt{K}}_{\mathfrak{i}}(\mathfrak{a},{\mathfrak{u}}_{\mathfrak{i}})\le 0,i=1,\dots ,m,\hfill \end{array}$$

where ${\mathcal{G}}_t:{\mathbb{R}}^n\to \mathbb{R}$, $t\in \mathbb{T}:=\{1,\cdots ,r\}$, ${\Psi }_j:{\mathbb{R}}^n\times {\mathbb{R}}^{n_j}\to \mathbb{R}$, $j\in \mathbb{J}:=\{1,\cdots ,p\}$, ${\Theta }_{\mathfrak{s}}:{\mathbb{R}}^n\times {\mathbb{R}}^{n_{\mathfrak{s}}}\to \mathbb{R}$, $\mathfrak{s}\in \mathbb{S}:=\{1,\cdots ,\ell \}$, ${\mathtt{K}}_{\mathfrak{i}}:{\mathbb{R}}^n\times {\mathbb{R}}^{n_{\mathfrak{i}}}\to \mathbb{R}$, $\mathfrak{i}\in \mathtt{I}:=\{1,\cdots ,m\}$, and ${\mathtt{L}}_{\mathfrak{i}}:{\mathbb{R}}^n\times {\mathbb{R}}^{n_{\mathfrak{i}}}\to \mathbb{R}$, $\mathfrak{i}\in \mathtt{I}:=\{1,\cdots ,m\}$ are extended real-valued functions and admit bounded USRC; ${\mathfrak{u}}_j\in {\Omega }_j,j\in \mathbb{J}$, ${\mathfrak{u}}_{\mathfrak{s}}\in {\Omega }_{\mathfrak{s}},\mathfrak{s}\in \mathbb{S}$ and ${\mathfrak{u}}_{\mathfrak{i}}\in {\Omega }_{\mathfrak{i}},\mathfrak{i}\in \mathtt{I}$ are uncertain parameters; and ${\Omega }_j$, ${\Omega }_{\mathfrak{s}}$, ${\Omega }_{\mathfrak{i}}$ are nonempty convex subsets of ${\mathbb{R}}^{n_j}$, ${\mathbb{R}}^{n_{\mathfrak{s}}}$ and ${\mathbb{R}}^{n_{\mathfrak{i}}}$, respectively.

Associated with UNMPVC, the robust multiobjective programming problem (RNMPVC) is defined as

$$\begin{array}{cc}\hfill \left(\mathrm{RNMPVC}\right)\mathrm{Minimize}& \mathcal{G}\left(\mathfrak{a}\right):=({\mathcal{G}}_1\left(\mathfrak{a}\right),\cdots ,{\mathcal{G}}_r\left(\mathfrak{a}\right)),\hfill \\ \hfill \mathrm{subject}\mathrm{to}& {\Psi }_j(\mathfrak{a},{\mathfrak{u}}_j)\le 0,\forall {\mathfrak{u}}_j\in {\Omega }_j,j\in \mathbb{J},\hfill \\ \hfill & {\Theta }_{\mathfrak{s}}(\mathfrak{a},{\mathfrak{u}}_{\mathfrak{s}})=0,\forall {\mathfrak{u}}_{\mathfrak{s}}\in {\Omega }_{\mathfrak{s}},\mathfrak{s}\in \mathbb{S},\hfill \\ \hfill & {\mathtt{K}}_{\mathfrak{i}}(\mathfrak{a},{\mathfrak{u}}_{\mathfrak{i}})\ge 0,\forall {\mathfrak{u}}_{\mathfrak{i}}\in {\Omega }_{\mathfrak{i}},\mathfrak{i}\in \mathtt{I},\hfill \\ \hfill & {\mathtt{L}}_{\mathfrak{i}}(\mathfrak{a},{\mathfrak{u}}_{\mathfrak{i}}){\mathtt{K}}_{\mathfrak{i}}(\mathfrak{a},{\mathfrak{u}}_{\mathfrak{i}})\le 0,\forall {\mathfrak{u}}_{\mathfrak{i}}\in {\Omega }_{\mathfrak{i}},\mathfrak{i}\in \mathtt{I}.\hfill \end{array}$$

The feasible set ${\mathbb{F}}_{\mathcal{G}}$ of RNMPVC is

$$\begin{array}{cc}\hfill {\mathbb{F}}_{\mathcal{G}}=\{\mathfrak{a}\in {\mathbb{R}}^n:& {\Psi }_j(\mathfrak{a},{\mathfrak{u}}_j)\le 0,\forall {\mathfrak{u}}_j\in {\Omega }_j,j\in \mathbb{J},{\Theta }_{\mathfrak{s}}(\mathfrak{a},{\mathfrak{u}}_{\mathfrak{s}})=0,\hfill \\ & \forall {\mathfrak{u}}_{\mathfrak{s}}\in {\Omega }_{\mathfrak{s}},\mathfrak{s}\in \mathbb{S},{\mathtt{K}}_{\mathfrak{i}}(\mathfrak{a},{\mathfrak{u}}_{\mathfrak{i}})\ge 0,\forall {\mathfrak{u}}_{\mathfrak{i}}\in {\Omega }_{\mathfrak{i}},\mathfrak{i}\in \mathtt{I},\hfill \\ & {\mathtt{L}}_{\mathfrak{i}}(\mathfrak{a},{\mathfrak{u}}_{\mathfrak{i}}){\mathtt{K}}_{\mathfrak{i}}(\mathfrak{a},{\mathfrak{u}}_{\mathfrak{i}})\le 0,\forall {\mathfrak{u}}_{\mathfrak{i}}\in {\Omega }_{\mathfrak{i}},\mathfrak{i}\in \mathtt{I}\}.\hfill \end{array}$$

Let us define

$${\Psi }_j^{*}\left(\mathfrak{a}\right):=\underset{{\mathfrak{u}}_j\in {\Omega }_j}{max}{\Psi }_j(\mathfrak{a},{\mathfrak{u}}_j),j\in \mathbb{J},$$

$${\Theta }_{\mathfrak{s}}^{*}\left(\mathfrak{a}\right):=\underset{{\mathfrak{u}}_{\mathfrak{s}}\in {\Omega }_{\mathfrak{s}}}{max}{\Theta }_{\mathfrak{s}}(\mathfrak{a},{\mathfrak{u}}_{\mathfrak{s}}),\mathfrak{s}\in \mathbb{S},$$

$${\mathtt{K}}_{\mathfrak{i}}^{*}\left(\mathfrak{a}\right):=\underset{{\mathfrak{u}}_{\mathfrak{i}}\in {\Omega }_{\mathfrak{i}}}{max}{\mathtt{K}}_{\mathfrak{i}}(\mathfrak{a},{\mathfrak{u}}_{\mathfrak{i}}),\mathfrak{i}\in \mathtt{I},$$

$${\mathtt{L}}_{\mathfrak{i}}^{*}\left(\mathfrak{a}\right):=\underset{{\mathfrak{u}}_{\mathfrak{i}}\in {\Omega }_{\mathfrak{i}}}{max}{\mathtt{L}}_{\mathfrak{i}}(\mathfrak{a},{\mathfrak{u}}_{\mathfrak{i}}),\mathfrak{i}\in \mathtt{I},$$

and

$$\begin{array}{cc}\hfill {\Omega }_j\left(\mathfrak{a}\right):=\{{\mathfrak{u}}_j\in {\Omega }_j:& {\Psi }_j^{*}\left(\mathfrak{a}\right)={\Psi }_j(\mathfrak{a},{\mathfrak{u}}_j),j\in \mathbb{J}\},\hfill \\ \hfill {\Omega }_{\mathfrak{s}}\left(\mathfrak{a}\right):=\{{\mathfrak{u}}_{\mathfrak{s}}\in {\Omega }_{\mathfrak{s}}:& {\Theta }_{\mathfrak{s}}^{*}\left(\mathfrak{a}\right)={\Theta }_{\mathfrak{s}}(\mathfrak{a},{\mathfrak{u}}_{\mathfrak{s}}),\mathfrak{s}\in \mathbb{S}\},\hfill \\ \hfill {\Omega }_{\mathfrak{i}}\left(\mathfrak{a}\right):=\{{\mathfrak{u}}_{\mathfrak{i}}\in {\Omega }_{\mathfrak{i}}:& {\mathtt{K}}_{\mathfrak{i}}^{*}\left(\mathfrak{a}\right)={\mathtt{K}}_{\mathfrak{i}}(\mathfrak{a},{\mathfrak{u}}_{\mathfrak{i}}),{\mathtt{L}}_{\mathfrak{i}}^{*}\left(\mathfrak{a}\right)={\mathtt{L}}_{\mathfrak{i}}(\mathfrak{a},{\mathfrak{u}}_{\mathfrak{i}})\hfill \\ & {\mathtt{L}}_{\mathfrak{i}}^{*}\left(\mathfrak{a}\right){\mathtt{K}}_{\mathfrak{i}}^{*}\left(\mathfrak{a}\right)={\mathtt{L}}_{\mathfrak{i}}(\mathfrak{a},{\mathfrak{u}}_{\mathfrak{i}}){\mathtt{K}}_{\mathfrak{i}}(\mathfrak{a},{\mathfrak{u}}_{\mathfrak{i}}),\mathfrak{i}\in \mathtt{I}\}.\hfill \end{array}$$

Therefore, the problem RNMPVC can be transformed into the following nonsmooth multiobjective programming problems with vanishing constraints (NMPVC*):

$$\begin{array}{cc}\hfill (\mathrm{NMPVC}*)\mathrm{Minimize}& \mathcal{G}\left(\mathfrak{a}\right):=({\mathcal{G}}_1\left(\mathfrak{a}\right),\cdots ,{\mathcal{G}}_r\left(\mathfrak{a}\right)),\hfill \\ \hfill \mathrm{subject}\mathrm{to}& {\Psi }_j^{*}\left(\mathfrak{a}\right)\le 0,j\in \mathbb{J},\hfill \\ \hfill & {\Theta }_{\mathfrak{s}}^{*}\left(\mathfrak{a}\right)=0,\mathfrak{s}\in \mathbb{S},\hfill \\ \hfill & {\mathtt{K}}_{\mathfrak{i}}^{*}\left(\mathfrak{a}\right)\ge 0,\mathfrak{i}\in \mathtt{I},\hfill \\ \hfill & {\mathtt{L}}_{\mathfrak{i}}^{*}\left(\mathfrak{a}\right){\mathtt{K}}_{\mathfrak{i}}^{*}\left(\mathfrak{a}\right)\le 0,\mathfrak{i}\in \mathtt{I}.\hfill \end{array}$$

The feasible set ${\mathbb{F}}_{\mathcal{G}}^{*}$ of NMPVC* is:

$$\begin{array}{cc}\hfill {\mathbb{F}}_{\mathcal{G}}^{*}=\{\mathfrak{a}\in {\mathbb{R}}^n:& {\Psi }_j^{*}\left(\mathfrak{a}\right)\le 0,j\in \mathbb{J},{\Theta }_{\mathfrak{s}}^{*}\left(\mathfrak{a}\right)=0,\mathfrak{s}\in \mathbb{S},\hfill \\ & {\mathtt{K}}_{\mathfrak{i}}^{*}\left(\mathfrak{a}\right)\ge 0,\mathfrak{i}\in \mathtt{I},{\mathtt{L}}_{\mathfrak{i}}^{*}\left(\mathfrak{a}\right){\mathtt{K}}_{\mathfrak{i}}^{*}\left(\mathfrak{a}\right)\le 0,\mathfrak{i}\in \mathtt{I}\}.\hfill \end{array}$$

In the sequel, the following definitions are used.

Definition 6 

([55]). A feasible element $\overline{\mathfrak{a}}\in {\mathbb{F}}_{\mathcal{G}}$ is a robust Pareto (robust weak Pareto, respectively) solution of UNMPVC if there is no other feasible point $\mathfrak{a}\in {\mathbb{F}}_{\mathcal{G}}$, such that $\mathcal{G}\left(\mathfrak{a}\right)⪯\mathcal{G}\left(\overline{\mathfrak{a}}\right)$ $(\mathcal{G}\left(\mathfrak{a}\right)⪯\mathcal{G}\left(\overline{\mathfrak{a}}\right),$ $\mathrm{respectively})$.

Definition 7 

([55]). A feasible element $\overline{\mathfrak{a}}\in {\mathbb{F}}_{\mathcal{G}}$ is a robust local Pareto (robust local weak Pareto, respectively) solution of UNMPVC if for any neighbourhood $\mathcal{N}$ of $\overline{\mathfrak{a}}$ there is no other feasible point $\mathfrak{a}\in {\mathbb{F}}_{\mathcal{G}}$, such that $\mathcal{G}\left(\mathfrak{a}\right)⪯\mathcal{G}\left(\overline{\mathfrak{a}}\right)(\mathcal{G}\left(\mathfrak{a}\right)\prec \mathcal{G}\left(\overline{\mathfrak{a}}\right),\mathrm{respectively}).$

To facilitate the development of our main results, we define the subsequent index sets for a given $\overline{\mathfrak{a}}\in {\mathbb{F}}_{\mathcal{G}}^{*}$.

$$\begin{array}{cccc}& {\mathfrak{T}}_{\Psi }=\{j:{\Psi }_j^{*}\left(\overline{\mathfrak{a}}\right)=0\},\hfill & & {\mathfrak{T}}_{+0}=\{i:{\mathtt{K}}_{\mathfrak{i}}^{*}\left(\overline{\mathfrak{a}}\right)>0,{\mathtt{L}}_{\mathfrak{i}}^{*}\left(\overline{\mathfrak{a}}\right)=0\},\hfill \\ & {\mathfrak{T}}_{+-}=\{i:{\mathtt{K}}_{\mathfrak{i}}^{*}\left(\overline{\mathfrak{a}}\right)>0,{\mathtt{L}}_{\mathfrak{i}}^{*}\left(\overline{\mathfrak{a}}\right)<0\},\hfill & & {\mathfrak{T}}_{0+}=\{i:{\mathtt{K}}_{\mathfrak{i}}^{*}\left(\overline{\mathfrak{a}}\right)=0,{\mathtt{L}}_{\mathfrak{i}}^{*}\left(\overline{\mathfrak{a}}\right)>0\},\hfill \\ & {\mathfrak{T}}_{0-}=\{i:{\mathtt{K}}_{\mathfrak{i}}^{*}\left(\overline{\mathfrak{a}}\right)=0,{\mathtt{L}}_{\mathfrak{i}}^{*}\left(\overline{\mathfrak{a}}\right)<0\},\hfill & & {\mathfrak{T}}_{00}=\{i:{\mathtt{K}}_{\mathfrak{i}}^{*}\left(\overline{\mathfrak{a}}\right)=0,{\mathtt{L}}_{\mathfrak{i}}^{*}\left(\overline{\mathfrak{a}}\right)=0\}.\hfill \end{array}$$

We define the subsequent notations as follows:

$$\begin{array}{cc}& \mathcal{F}:=\bigcup _{t\in \mathbb{T}}co{\partial }^{*}{\mathcal{G}}_t\left(\overline{\mathfrak{a}}\right),{\mathcal{F}}^t:=\bigcup _{q\in \mathbb{T}\setminus \left\{t\right\}}co{\partial }^{*}{\mathcal{G}}_q\left(\overline{\mathfrak{a}}\right),\hfill \\ & \mathcal{G}:=\bigcup _{j\in {\mathfrak{T}}_{\Psi }}co{\partial }^{*}{\Psi }_j^{*}\left(\overline{\mathfrak{a}}\right),\mathcal{H}:=\bigcup _{\mathfrak{s}\in \mathbb{S}}co{\partial }^{*}{\Theta }_{\mathfrak{s}}^{*}\left(\overline{\mathfrak{a}}\right)\cup co{\partial }^{*}(-{\Theta }_{\mathfrak{s}}^{*})\left(\overline{\mathfrak{a}}\right),\hfill \\ & {\mathcal{L}}_{+0}:=\bigcup _{\mathfrak{i}\in {\mathfrak{T}}_{+0}}co{\partial }^{*}{\mathtt{L}}_{\mathfrak{i}}^{*}\left(\overline{\mathfrak{a}}\right),\hfill \\ & {\mathcal{K}}_{0+}:=\bigcup _{\mathfrak{i}\in {\mathfrak{T}}_{0+}}co{\partial }^{*}{\mathtt{K}}_{\mathfrak{i}}^{*}\left(\overline{\mathfrak{a}}\right)\cup co{\partial }^{*}(-{\mathtt{K}}_{\mathfrak{i}}^{*})\left(\overline{\mathfrak{a}}\right),\hfill \\ & {\mathcal{K}}_{0-}:=\bigcup _{\mathfrak{i}\in {\mathfrak{T}}_{0-}}co{\partial }^{*}(-{\mathtt{K}}_{\mathfrak{i}}^{*})\left(\overline{\mathfrak{a}}\right),\hfill \\ & {\mathcal{K}}_{00}:=\bigcup _{\mathfrak{i}\in {\mathfrak{T}}_{00}}co{\partial }^{*}(-{\mathtt{K}}_{\mathfrak{i}}^{*})\left(\overline{\mathfrak{a}}\right),\hfill \\ & \Gamma \left(\overline{\mathfrak{a}}\right):={\left({\mathcal{F}}^{t_0}\right)}^{-}\cap {\mathcal{G}}^{-}\cap {\mathcal{H}}^{-}\cap {\mathcal{L}}_{+0}^{-}\cap {\mathcal{K}}_{0+}^{-}\cap {\mathcal{K}}_{0-}^{-}\cap {\mathcal{K}}_{00}^{-},\hfill \\ & \mathcal{S}:=\{\mathfrak{a}\in {\mathbb{R}}^n:{\Psi }_j^{*}\left(\mathfrak{a}\right)\le 0,j\in \mathbb{J},{\Theta }_{\mathfrak{s}}^{*}\left(\mathfrak{a}\right)=0,\mathfrak{s}\in \mathbb{S},\hfill \\ & {\mathtt{K}}_{\mathfrak{i}}^{*}\left(\mathfrak{a}\right)\ge 0,\mathfrak{i}\in \mathtt{I},{\mathtt{L}}_{\mathfrak{i}}^{*}\left(\mathfrak{a}\right){\mathtt{K}}_{\mathfrak{i}}^{*}\left(\mathfrak{a}\right)\le 0,\mathfrak{i}\in \mathtt{I}\},\hfill \\ & {\mathcal{S}}^t:=\{\mathfrak{a}\in {\mathbb{R}}^n:{\mathcal{G}}_q\left(\mathfrak{a}\right)\le {\mathcal{G}}_q\left(\overline{\mathfrak{a}}\right),\forall q\in \mathbb{T}\setminus \left\{t\right\},{\Psi }_j^{*}\left(\mathfrak{a}\right)\le 0,j\in \mathbb{J},\hfill \\ & {\Theta }_{\mathfrak{s}}^{*}\left(\mathfrak{a}\right)=0,\mathfrak{s}\in \mathbb{S},{\mathtt{K}}_{\mathfrak{i}}^{*}\left(\mathfrak{a}\right)\ge 0,\mathfrak{i}\in \mathtt{I},{\mathtt{L}}_{\mathfrak{i}}^{*}\left(\mathfrak{a}\right){\mathtt{K}}_{\mathfrak{i}}^{*}\left(\mathfrak{a}\right)\le 0,\mathfrak{i}\in \mathtt{I}\}.\hfill \end{array}$$

Generalized standard Abadie constraint qualification (GS-ACQ) for UNMPVC is introduced in the following definition, which will be useful in establishing first-order necessary local weak Pareto efficiency conditions for UNMPVC.

Definition 8. 

For $\overline{\mathfrak{a}}\in \mathcal{S}$, if at least one of the dual sets used to define $\Gamma \left(\overline{\mathfrak{a}}\right)$ is non-zero and

$$\Gamma \left(\overline{\mathfrak{a}}\right)\subset \mathcal{T}({\mathcal{S}}^{t_0},\overline{\mathfrak{a}}),$$

then the generalized standard Abadie constraint qualification (GS-ACQ) is satisfied at $\overline{\mathfrak{a}}\in \mathcal{S}$.

In the subsequent definitions, we introduce various stationary conditions for UNMPVC employing convexificators.

Definition 9. 

Let $\overline{\mathfrak{a}}\in {\mathbb{F}}_{\mathcal{G}}$. $\overline{\mathfrak{a}}$ is a generalized robust weakly stationary (RW-stationary) point if there exist $\sigma =({\sigma }^{\mathcal{G}},{\sigma }^{\Psi },{\sigma }^{\Theta },{\sigma }^{\mathtt{K}})\in {\mathbb{R}}^{r+p+\ell +m},\tau =({\tau }^{\Theta },{\tau }^{\mathtt{K}},{\tau }^{\mathtt{L}})\in {\mathbb{R}}^{\ell +2m}$ and ${\overline{\mathfrak{u}}}_j\in {\Omega }_j\left(\overline{\mathfrak{a}}\right),$ $j\in \mathbb{J},{\overline{\mathfrak{u}}}_{\mathfrak{s}}\in {\Omega }_{\mathfrak{s}}\left(\overline{\mathfrak{a}}\right),\mathfrak{s}\in \mathbb{S},{\overline{\mathfrak{u}}}_{\mathfrak{i}}\in {\Omega }_{\mathfrak{i}}\left(\overline{\mathfrak{a}}\right),\mathfrak{i}\in \mathtt{I},$ such that

$$\begin{array}{cc}\hfill 0\in \sum _{t\in \mathbb{T}}{\sigma }_t^{\mathcal{G}}co{\partial }^{*}{\mathcal{G}}_t\left(\overline{\mathfrak{a}}\right)& +\sum _{j\in {\mathfrak{T}}_{\Psi }}{\sigma }_j^{\Psi }co{\partial }^{*}{\Psi }_j(\overline{\mathfrak{a}},{\overline{\mathfrak{u}}}_j)\hfill \\ & +\sum _{\mathfrak{s}\in \mathbb{S}}[{\sigma }_{\mathfrak{s}}^{\Theta }co{\partial }^{*}{\Theta }_{\mathfrak{s}}(\overline{\mathfrak{a}},{\overline{\mathfrak{u}}}_{\mathfrak{s}})+{\tau }_{\mathfrak{s}}^{\Theta }co{\partial }^{*}(-{\Theta }_{\mathfrak{s}})(\overline{\mathfrak{a}},{\overline{\mathfrak{u}}}_{\mathfrak{s}})]\hfill \\ \hfill & +\sum _{\mathfrak{i}\in \mathtt{I}}{\sigma }_{\mathfrak{i}}^{\mathtt{K}}co{\partial }^{*}(-{\mathtt{K}}_{\mathfrak{i}})(\overline{\mathfrak{a}},{\overline{\mathfrak{u}}}_{\mathfrak{i}})\hfill \\ \hfill & +\sum _{\mathfrak{i}\in \mathtt{I}}[{\tau }_{\mathfrak{i}}^{\mathtt{K}}co{\partial }^{*}{\mathtt{K}}_{\mathfrak{i}}(\overline{\mathfrak{a}},{\overline{\mathfrak{u}}}_{\mathfrak{i}})+{\tau }_{\mathfrak{i}}^{\mathtt{L}}co{\partial }^{*}{\mathtt{L}}_{\mathfrak{i}}(\overline{\mathfrak{a}},{\overline{\mathfrak{u}}}_{\mathfrak{i}})],\hfill \end{array}$$

(1)

$${\sigma }_t^{\mathcal{G}}>0,t\in \mathbb{T},{\sigma }_j^{\Psi }\ge 0,j\in {\mathfrak{T}}_{\Psi },{\sigma }_{\mathfrak{s}}^{\Theta }\ge 0,{\tau }_{\mathfrak{s}}^{\Theta }\ge 0,\mathfrak{s}\in \mathbb{S},{\sigma }_{\mathfrak{i}}^{\mathtt{K}}\ge 0,{\tau }_{\mathfrak{i}}^{\mathtt{K}},\ge 0{\tau }_{\mathfrak{i}}^{\mathtt{L}},\ge 0,\mathfrak{i}\in \mathtt{I},$$

(2)

$${\sigma }_{{\mathfrak{T}}_{+0}\cup {\mathfrak{T}}_{+-}}^{\mathtt{K}}={\tau }_{{\mathfrak{T}}_{+0}\cup {\mathfrak{T}}_{+-}}^{\mathtt{K}}={\tau }_{{\mathfrak{T}}_{0+}\cup {\mathfrak{T}}_{+-}\cup {\mathfrak{T}}_{0-}}^{\mathtt{L}}=0,{\sigma }_i^{\mathtt{K}}-{\tau }_i^{\mathtt{K}}\ge 0,\mathfrak{i}\in {\mathfrak{T}}_{0-}.$$

(3)

Definition 10. 

Let $\overline{\mathfrak{a}}\in {\mathbb{F}}_{\mathcal{G}}$. $\overline{\mathfrak{a}}$ is a generalized robust Mordukhovich stationary (RM-stationary) point if there exist $\sigma =({\sigma }^{\mathcal{G}},{\sigma }^{\Psi },{\sigma }^{\Theta },{\sigma }^{\mathtt{K}})\in {\mathbb{R}}^{r+p+\ell +m},\tau =({\tau }^{\Theta },{\tau }^{\mathtt{K}},{\tau }^{\mathtt{L}})\in {\mathbb{R}}^{\ell +2m}$ and ${\overline{\mathfrak{u}}}_j\in {\Omega }_j\left(\overline{\mathfrak{a}}\right),$ $j\in \mathbb{J},{\overline{\mathfrak{u}}}_{\mathfrak{s}}\in {\Omega }_{\mathfrak{s}}\left(\overline{\mathfrak{a}}\right),\mathfrak{s}\in \mathbb{S},{\overline{\mathfrak{u}}}_{\mathfrak{i}}\in {\Omega }_{\mathfrak{i}}\left(\overline{\mathfrak{a}}\right),\mathfrak{i}\in \mathtt{I},$ such that they satisfy conditions (1)–(3) along with the subsequent condition:

$$\forall \mathfrak{i}\in {\mathfrak{T}}_{00},{\tau }_{\mathfrak{i}}^{\mathtt{L}}({\sigma }_{\mathfrak{i}}^{\mathtt{K}}-{\tau }_{\mathfrak{i}}^{\mathtt{K}})=0.$$

Definition 11. 

Let $\overline{\mathfrak{a}}\in {\mathbb{F}}_{\mathcal{G}}$. $\overline{\mathfrak{a}}$ is a generalized robust strong stationary (RS-stationary) point if there exist $\sigma =({\sigma }^{\mathcal{G}},{\sigma }^{\Psi },{\sigma }^{\Theta },{\sigma }^{\mathtt{K}})\in {\mathbb{R}}^{r+p+\ell +m},\tau =({\tau }^{\Theta },{\tau }^{\mathtt{K}},{\tau }^{\mathtt{L}})\in {\mathbb{R}}^{\ell +2m}$ and ${\overline{\mathfrak{u}}}_j\in {\Omega }_j\left(\overline{\mathfrak{a}}\right),$ $j\in \mathbb{J},{\overline{\mathfrak{u}}}_{\mathfrak{s}}\in {\Omega }_{\mathfrak{s}}\left(\overline{\mathfrak{a}}\right),\mathfrak{s}\in \mathbb{S},{\overline{\mathfrak{u}}}_{\mathfrak{i}}\in {\Omega }_{\mathfrak{i}}\left(\overline{\mathfrak{a}}\right),\mathfrak{i}\in \mathtt{I},$ such that they satisfy conditions (1)–(3) along with the subsequent condition:

$$\forall \mathfrak{i}\in {\mathfrak{T}}_{00},{\tau }_{\mathfrak{i}}^{\mathtt{L}}=0,({\sigma }_{\mathfrak{i}}^{\mathtt{K}}-{\tau }_{\mathfrak{i}}^{\mathtt{K}})\ge 0.$$

From the above definitions, we conclude that

$$\mathrm{RS}\text{-}\mathrm{stationary}\Rightarrow \mathrm{RM}\text{-}\mathrm{stationary}\Rightarrow \mathrm{RW}\text{-}\mathrm{stationary}.$$

In the following theorem, by employing GS-ACQ, we establish that the RS-stationary conditions are necessary for a local weak Pareto efficiency of UNMPVC.

Theorem 3. 

Let $\overline{\mathfrak{a}}\in {\mathbb{F}}_{\mathcal{G}}$ be a robust local weak Pareto solution of UNMPVC. Assume that ${\mathcal{G}}_t,t\in \mathbb{T},$ ${\Psi }_j,j\in \mathbb{J}$, ${\Theta }_{\mathfrak{s}},\mathfrak{s}\in \mathbb{S}$, ${\mathtt{K}}_{\mathfrak{i}},{\mathtt{L}}_{\mathfrak{i}},\mathfrak{i}\in \mathtt{I}$ are locally Lipschitz functions and admit bounded USRC. If GS-ACQ holds at $\overline{\mathfrak{a}}$ and the cone

$$\begin{array}{cc}\hfill D^{t_0}=cone(co{\mathcal{F}}^{t_0})& +cone(co\mathcal{G})+cone(co\mathcal{H})+cone(co{\mathcal{K}}_{0+})\hfill \\ & +cone(co{\mathcal{K}}_{0-})+cone(co{\mathcal{L}}_{0+})+cone(co{\mathcal{K}}_{00}),\hfill \end{array}$$

is closed, then there exist ${\tilde{\mathfrak{u}}}_j\in {\Omega }_j\left(\overline{\mathfrak{a}}\right)$, ${\tilde{\mathfrak{u}}}_{\mathfrak{s}}\in {\Omega }_{\mathfrak{s}}\left(\overline{\mathfrak{a}}\right)$, and ${\tilde{\mathfrak{u}}}_{\mathfrak{i}}\in {\Omega }_{\mathfrak{i}}\left(\overline{\mathfrak{a}}\right)$, such that $\overline{\mathfrak{a}}$ is a RS-stationary point of UNMPVC.

Proof. 

Since $\overline{\mathfrak{a}}\in {\mathbb{F}}_{\mathcal{G}}$ is a robust local weak Pareto solution of UNMPVC, then $\overline{\mathfrak{a}}$ is a robust local weak Pareto solution of NMPVC*. To derive the above result, it is enough to demonstrate that for every $t\in \mathbb{T}$

$$0\in co{\partial }^{*}{\mathcal{G}}_t\left(\overline{\mathfrak{a}}\right)+D^t.$$

(4)

Let us consider an alternative scenario where there exists an index $t_0\in \mathbb{T}$ such that the following condition holds:

$$0\notin co{\partial }^{*}{\mathcal{G}}_{t_0}\left(\overline{\mathfrak{a}}\right)+D^{t_0}.$$

Since ${\mathcal{G}}_{t_0}$ admits bounded USRC, $co{\partial }^{*}{\mathcal{G}}_{t_0}\left(\overline{\mathfrak{a}}\right)$ and $D^{t_0}$ are compact convex set and closed convex set, respectively; therefore, $co{\partial }^{*}{\mathcal{G}}_{t_0}\left(\overline{\mathfrak{a}}\right)+D^{t_0}$ is a closed convex set in ${\mathbb{R}}^n$. According to the convex separation theorem, there exists $\mathtt{d}\in {\mathbb{R}}^n$, such that the following inequality holds:

$$⟨\xi +\beta ,\mathtt{d}⟩<0,\forall \xi \in co{\partial }^{*}{\mathcal{G}}_{t_0}\left(\overline{\mathfrak{a}}\right),\forall \beta \in D^{t_0}.$$

Given that every cone includes the zero vector, we obtain

$$⟨\xi ,\mathtt{d}⟩<0,\forall \xi \in co{\partial }^{*}{\mathcal{G}}_{t_0}\left(\overline{\mathfrak{a}}\right).$$

Thus,

$${\mathcal{G}}_{t_0}^{+}(\overline{\mathfrak{a}},\mathtt{d})<0.$$

Hence $\exists \delta >0$, such that

$${\mathcal{G}}_{t_0}(\overline{\mathfrak{a}}+\mu \mathtt{d})<{\mathcal{G}}_{t_0}\left(\overline{\mathfrak{a}}\right),\forall \mu \in (0,\delta ).$$

Also, we deduce that $⟨\beta ,\mathtt{d}⟩\le 0,\forall \beta \in D^{t_0}.$ Consequently,

$$⟨{\beta }_t^1,\mathtt{d}⟩\le 0,\forall {\beta }_t^1\in co{\partial }^{*}{\mathcal{G}}_t\left(\overline{\mathfrak{a}}\right),\forall t\in \mathbb{T}\setminus \left\{t_0\right\},$$

(5)

$$⟨{\beta }_j^2,\mathtt{d}⟩\le 0,\forall {\beta }_j^2\in co{\partial }^{*}{\Psi }_j^{*}\left(\overline{\mathfrak{a}}\right),\forall j\in {\mathfrak{T}}_{\Psi },$$

(6)

$$⟨{\beta }_{\mathfrak{s}}^3,\mathtt{d}⟩\le 0,\forall {\beta }_{\mathfrak{s}}^3\in co{\partial }^{*}{\Theta }_{\mathfrak{s}}^{*}\left(\overline{\mathfrak{a}}\right)\cup co{\partial }^{*}(-{\Theta }_{\mathfrak{s}}^{*})\left(\overline{\mathfrak{a}}\right),\forall \mathfrak{s}\in \mathbb{S},$$

(7)

$$⟨{\beta }_{\mathfrak{i}}^4,\mathtt{d}⟩\le 0,\forall {\beta }_{\mathfrak{i}}^4\in co{\partial }^{*}{\mathtt{K}}_{\mathfrak{i}}^{*}\left(\overline{\mathfrak{a}}\right)\cup co{\partial }^{*}(-{\mathtt{K}}_{\mathfrak{i}}^{*})\left(\overline{\mathfrak{a}}\right),\forall \mathfrak{i}\in {\mathfrak{T}}_{0+},$$

(8)

$$⟨{\beta }_{\mathfrak{i}}^5,\mathtt{d}⟩\le 0,\forall {\beta }_{\mathfrak{i}}^5\in co{\partial }^{*}(-{\mathtt{K}}_{\mathfrak{i}}^{*})\left(\overline{\mathfrak{a}}\right),\forall \mathfrak{i}\in {\mathfrak{T}}_{0-},$$

(9)

$$⟨{\beta }_{\mathfrak{i}}^6,\mathtt{d}⟩\le 0,\forall {\beta }_{\mathfrak{i}}^6\in co{\partial }^{*}{\mathtt{L}}_{\mathfrak{i}}^{*}\left(\overline{\mathfrak{a}}\right),\forall \mathfrak{i}\in {\mathfrak{T}}_{+0},$$

(10)

$$⟨{\beta }_{\mathfrak{i}}^7,\mathtt{d}⟩\le 0,\forall {\beta }_{\mathfrak{i}}^7\in co{\partial }^{*}(-{\mathtt{K}}_{\mathfrak{i}}^{*})\left(\overline{\mathfrak{a}}\right),\forall \mathfrak{i}\in {\mathfrak{T}}_{00}.$$

(11)

From (5)–(11), we obtain $\mathtt{d}\in \Gamma \left(\overline{\mathfrak{a}}\right).$ According to GS-ACQ, we obtain

$$\mathtt{d}\in \mathcal{T}({\mathcal{S}}^{t_0},\overline{\mathfrak{a}}).$$

Therefore, there exist ${\mu }_n\downarrow 0$ and ${\mathtt{d}}_n\to \mathtt{d}$, such that

$$\overline{\mathfrak{a}}+{\mu }_n{\mathtt{d}}_n\in {\mathcal{S}}^{t_0}.$$

(12)

Let $C_{t_0}>0$ denote the Lipschitz constant for ${\mathcal{G}}_{t_0}$ in the vicinity of $\overline{\mathfrak{a}}$. Then for sufficiently large n we have

$${\mathcal{G}}_{t_0}({\mathfrak{a}}_n+{\mu }_n{\mathtt{d}}_n)\le {\mathcal{G}}_{t_0}(\overline{\mathfrak{a}}+{\mu }_n\mathtt{d})+C_{t_0}{\mu }_n\parallel {\mathtt{d}}_n-\mathtt{d}\parallel .$$

Hence, for a sufficiently large n, we have

$${\mathcal{G}}_{t_0}(\overline{\mathfrak{a}}+{\mu }_n{\mathtt{d}}_n)<{\mathcal{G}}_{t_0}\left(\overline{\mathfrak{a}}\right).$$

(13)

From (12) and (13), we derive a contradiction with the local weak Pareto efficiency at $\overline{\mathfrak{a}}.$ Consequently, (4) holds. As a result, there exist non-negative multipliers ${\sigma }_t^{\mathcal{G}}>0,t\in \mathbb{T},$ ${\sigma }_j^{\Psi }\ge 0,j\in {\mathfrak{T}}_{\Psi },{\sigma }_{\mathfrak{s}}^{\Theta }\ge 0,{\tau }_{\mathfrak{s}}^{\Theta }\ge 0,\mathfrak{s}\in \mathbb{S},{\tau }_{\mathfrak{i}}^{\mathtt{K}}\ge 0,\mathfrak{i}\in {\mathfrak{T}}_{0+}{\sigma }_{\mathfrak{i}}^{\mathtt{K}}\ge 0,$ $\mathfrak{i}\in {\mathfrak{T}}_{0+}\cup {\mathfrak{T}}_{0-}\cup {\mathfrak{T}}_{00},$ ${\tau }_{\mathfrak{i}}^{\mathtt{L}}\ge 0,\mathfrak{i}\in {\mathfrak{T}}_{+0}$, such that

$$\begin{array}{cc}\hfill 0\in \sum _{t\in \mathbb{T}}{\sigma }_t^{\mathcal{G}}co{\partial }^{*}{\mathcal{G}}_t\left(\overline{\mathfrak{a}}\right)& +\sum _{j\in {\mathfrak{T}}_{\Psi }}{\sigma }_j^{\Psi }co{\partial }^{*}{\Psi }_j^{*}\left(\overline{\mathfrak{a}}\right)+\sum _{\mathfrak{s}\in \mathbb{S}}{\sigma }_{\mathfrak{s}}^{\Theta }co{\partial }^{*}{\Theta }_{\mathfrak{s}}^{*}\left(\overline{\mathfrak{a}}\right)\hfill \\ & +\sum _{\mathfrak{s}\in \mathbb{S}}{\tau }_{\mathfrak{s}}^{\Theta }co{\partial }^{*}(-{\Theta }_{\mathfrak{s}}^{*})\left(\overline{\mathfrak{a}}\right)+\sum _{\mathfrak{i}\in {\mathfrak{T}}_{0+}\cup {\mathfrak{T}}_{0-}\cup {\mathfrak{T}}_{00}}{\sigma }_{\mathfrak{i}}^{\mathtt{K}}co{\partial }^{*}(-{\mathtt{K}}_{\mathfrak{i}}^{*})\left(\overline{\mathfrak{a}}\right)\hfill \\ \hfill & +\sum _{\mathfrak{i}\in {\mathfrak{T}}_{0+}}{\tau }_{\mathfrak{i}}^{\mathtt{K}}co{\partial }^{*}{\mathtt{K}}_{\mathfrak{i}}^{*}\left(\overline{\mathfrak{a}}\right)+\sum _{\mathfrak{i}\in {\mathfrak{T}}_{+0}}{\tau }_{\mathfrak{i}}^{\mathtt{L}}co{\partial }^{*}{\mathtt{L}}_{\mathfrak{i}}^{*}\left(\overline{\mathfrak{a}}\right).\hfill \end{array}$$

(14)

Taking ${\sigma }_{{\mathfrak{T}}_{+0}\cup {\mathfrak{T}}_{+-}}^{\mathtt{K}}=0,{\tau }_{{\mathfrak{T}}_{00}\cup {\mathfrak{T}}_{0-}\cup {\mathfrak{T}}_{+-}\cup {\mathfrak{T}}_{+0}}^{\mathtt{K}}=0,{\tau }_{{\mathfrak{T}}_{+-}\cup {\mathfrak{T}}_{0+}\cup {\mathfrak{T}}_{0-}\cup {\mathfrak{T}}_{00}}^{\mathtt{L}}=0,$ from (14), we obtain

$$\begin{array}{cc}\hfill 0\in \sum _{t\in \mathbb{T}}{\sigma }_t^{\mathcal{G}}co{\partial }^{*}{\mathcal{G}}_t\left(\overline{\mathfrak{a}}\right)& +\sum _{j\in {\mathfrak{T}}_{\Psi }}{\sigma }_j^{\Psi }co{\partial }^{*}{\Psi }_j^{*}\left(\overline{\mathfrak{a}}\right)\hfill \\ & +\sum _{\mathfrak{s}\in \mathbb{S}}[{\sigma }_{\mathfrak{s}}^{\Theta }co{\partial }^{*}{\Theta }_{\mathfrak{s}}^{*}\left(\overline{\mathfrak{a}}\right)+{\tau }_{\mathfrak{s}}^{\Theta }co{\partial }^{*}(-{\Theta }_{\mathfrak{s}}^{*})\left(\overline{\mathfrak{a}}\right)]\hfill \\ \hfill & +\sum _{\mathfrak{i}\in \mathtt{I}}{\sigma }_{\mathfrak{i}}^{\mathtt{K}}co{\partial }^{*}(-{\mathtt{K}}_{\mathfrak{i}}^{*})\left(\overline{\mathfrak{a}}\right)\hfill \\ \hfill & +\sum _{\mathfrak{i}\in \mathtt{I}}[{\tau }_{\mathfrak{i}}^{\mathtt{K}}co{\partial }^{*}{\mathtt{K}}_{\mathfrak{i}}^{*}\left(\overline{\mathfrak{a}}\right)+{\tau }_{\mathfrak{i}}^{\mathtt{L}}co{\partial }^{*}{\mathtt{L}}_{\mathfrak{i}}^{*}\left(\overline{\mathfrak{a}}\right)],\hfill \end{array}$$

$${\sigma }_t^{\mathcal{G}}>0,t\in \mathbb{T},{\sigma }_j^{\Psi }\ge 0,j\in {\mathfrak{T}}_{\Psi },{\sigma }_{\mathfrak{s}}^{\Theta }\ge 0,{\tau }_{\mathfrak{s}}^{\Theta }\ge 0,\mathfrak{s}\in \mathbb{S},{\tau }_{\mathfrak{i}}^{\mathtt{K}}\ge 0,{\sigma }_{\mathfrak{i}}^{\mathtt{K}}\ge 0,{\tau }_{\mathfrak{i}}^{\mathtt{L}}\ge 0,\mathfrak{i}\in \mathtt{I}$$

$${\sigma }_{{\mathfrak{T}}_{+0}\cup {\mathfrak{T}}_{+-}}^{\mathtt{K}}=0,{\tau }_{{\mathfrak{T}}_{00}\cup {\mathfrak{T}}_{0-}\cup {\mathfrak{T}}_{+-}\cup {\mathfrak{T}}_{+0}}^{\mathtt{K}}=0,{\tau }_{{\mathfrak{T}}_{+-}\cup {\mathfrak{T}}_{0+}\cup {\mathfrak{T}}_{0-}}^{\mathtt{L}}=0,$$

$$\forall \mathfrak{i}\in {\mathfrak{T}}_{00},{\tau }_{\mathfrak{i}}^{\mathtt{L}}=0,({\sigma }_{\mathfrak{i}}^{\mathtt{K}}-{\tau }_{\mathfrak{i}}^{\mathtt{K}})={\sigma }_{\mathfrak{i}}^{\mathtt{K}}\ge 0.$$

Hence, $\overline{\mathfrak{a}}$ is an RS-stationary point of NMPVC*. From Theorem 2, there exist ${\tilde{\mathfrak{u}}}_j\in {\Omega }_j\left(\overline{\mathfrak{a}}\right)$, ${\tilde{\mathfrak{u}}}_{\mathfrak{s}}\in {\Omega }_{\mathfrak{s}}\left(\overline{\mathfrak{a}}\right)$, and ${\tilde{\mathfrak{u}}}_{\mathfrak{i}}\in {\Omega }_{\mathfrak{i}}\left(\overline{\mathfrak{a}}\right)$, such that $\overline{\mathfrak{a}}$ is a RS-stationary point of UNMPVC. □

Remark 1. 

1 

It is worthwhile to note that Theorem 3 extends the scope of Theorem 3.1 established by Hu et al. [40] for nonsmooth MPVC to a broader class of programming problems involving data uncertainty. If ${\Omega }_j,j\in \mathbb{J},$ ${\Omega }_{\mathfrak{s}},\mathfrak{s}\in \mathbb{S}$, and ${\Omega }_{\mathfrak{i}},\mathfrak{i}\in \mathtt{I}$ are singleton sets, then Theorem 3 reduces to Theorem 3.1 established in [40].

2 

In the domain of robust optimization, Theorem 3 extends the robust first-order optimality conditions established by Chen et al. [55] for nonsmooth MOP with uncertain data to encompass a more general class of programming problems, UNMPVC. In particular, if $\mathbb{S}=\varnothing$ and $\mathtt{I}=\varnothing$, Theorem 3 reduces to Theorem 3.3 established by Chen et al. [55].

We consider the following non-trivial example in the Euclidean space to highlight the significance of the necessary optimality conditions derived in Theorem 3.

Example 1. 

Let us examine the following problem given by

$$\begin{array}{cc}\hfill \left(P1\right)\mathit{Minimize}& \mathcal{G}\left(\mathfrak{a}\right)=({\mathcal{G}}_1\left(\mathfrak{a}\right),{\mathcal{G}}_2\left(\mathfrak{a}\right)):=\left(\right|{\mathfrak{a}}_1|,|{\mathfrak{a}}_2\left|\right),\hfill \\ \hfill \mathit{subject}\mathit{to}& \Psi (\mathfrak{a},{\mathfrak{u}}_1):={\mathfrak{a}}_{1}sin{\mathfrak{u}}_1\le 0,\forall {\mathfrak{u}}_1\in {\Omega }_1:=\left[0,{\displaystyle \frac{\pi }{2}}\right],\hfill \\ & \mathtt{K}(\mathfrak{a},{\mathfrak{u}}_2):={\mathfrak{a}}_2{\mathfrak{u}}_2\le 0,\forall {\mathfrak{u}}_2\in {\Omega }_2:=[0,1],\hfill \\ & \mathtt{L}(\mathfrak{a},{\mathfrak{u}}_2)\mathtt{K}(\mathfrak{a},{\mathfrak{u}}_2):={\mathfrak{a}}_1{\mathfrak{a}}_2{\mathfrak{u}}_2\le 0,\forall {\mathfrak{u}}_2\in {\Omega }_2:=[0,1],\hfill \end{array}$$

where ${\mathcal{G}}_t:{\mathbb{R}}^2\to \overline{\mathbb{R}},t\in \{1,2\}$, $\Psi :{\mathbb{R}}^2\times {\Omega }_1\to \overline{\mathbb{R}}$, $\mathtt{K}:{\mathbb{R}}^2\times {\Omega }_2\to \overline{\mathbb{R}}$, $\mathtt{L}:{\mathbb{R}}^2\times {\Omega }_2\to \overline{\mathbb{R}}$ and $z=({\mathfrak{a}}_1,{\mathfrak{a}}_2)\in {\mathbb{R}}^2.$ It is evident that

$$\begin{array}{cccc}& {\Psi }^{*}\left(\mathfrak{a}\right)=\underset{{\mathfrak{u}}_1\in {\Omega }_1}{max}{\mathfrak{a}}_{1}sin{\mathfrak{u}}_1={\mathfrak{a}}_1,\hfill & & {\mathtt{K}}^{*}\left(\mathfrak{a}\right)=\underset{{\mathfrak{u}}_2\in {\Omega }_2}{max}{\mathfrak{a}}_2{\mathfrak{u}}_2={\mathfrak{a}}_2,\hfill \\ & {\mathtt{L}}^{*}\left(\mathfrak{a}\right){\mathtt{K}}^{*}\left(\mathfrak{a}\right)={\mathfrak{a}}_1{\mathfrak{a}}_2,\hfill \end{array}$$

and $\overline{\mathfrak{a}}=(0,0)$ is a robust local weak Pareto solution in (P1). Furthermore, in view of Definition 3, the upper semi-regular convexificators at $\overline{\mathfrak{a}}$ of each function constituting (P1) are given by

$$\begin{array}{cccc}& {\partial }^{*}{\mathcal{G}}_1\left(\overline{\mathfrak{a}}\right)=\{(-1,0),(1,0)\},\hfill & & {\partial }^{*}{\mathcal{G}}_2\left(\overline{\mathfrak{a}}\right)=\{(0,-1),(0,1)\},\hfill \\ & {\partial }^{*}{\Psi }^{*}\left(\overline{\mathfrak{a}}\right)=\left\{(1,0)\right\},\hfill & & {\partial }^{*}{\mathtt{K}}^{*}\left(\overline{\mathfrak{a}}\right)=\left\{(0,1)\right\},\hfill \\ & {\partial }^{*}(-{\mathtt{K}}^{*})\left(\overline{\mathfrak{a}}\right)=\left\{(0,-1)\right\},\hfill & & {\partial }^{*}{\mathtt{L}}^{*}\left(\overline{\mathfrak{a}}\right)=\left\{(1,0)\right\}.\hfill \end{array}$$

We have

$${\mathcal{G}}^{-}=\{\mathfrak{a}:{\mathfrak{a}}_1\le 0\},$$

$${\mathcal{K}}_{00}^{-}={\mathcal{K}}_{0-}^{-}=\{\mathfrak{a}:{\mathfrak{a}}_2\ge 0\}.$$

Hence, GS-ACQ is satisfied at $\overline{\mathfrak{a}}=(0,0)$ and $D^t$ is a closed set. Moreover, there exist ${\sigma }_1^{\mathcal{G}}=1,$ ${\sigma }_2^{\mathcal{G}}=1,{\sigma }^{\Psi }=1,{\sigma }^{\mathtt{K}}=1,{\tau }^{\mathtt{K}}=0,{\tau }^{\mathtt{L}}=0$ and ${\xi }_1^{\mathcal{G}}=(-1,0)\in {\partial }^{*}{\mathcal{G}}_1^{*}\left(\overline{\mathfrak{a}}\right),{\xi }_2^{\mathcal{G}}=(0,1)\in {\partial }^{*}{\mathcal{G}}_2^{*}\left(\overline{\mathfrak{a}}\right),{\eta }^{\Psi }=(1,0)\in {\partial }^{*}{\Psi }^{*}\left(\overline{\mathfrak{a}}\right),{\eta }^{\mathtt{K}}=(0,1)\in {\partial }^{*}{\mathtt{K}}^{*}\left(\overline{\mathfrak{a}}\right),{\eta }^{-\mathtt{K}}=(0,-1)\in {\partial }^{*}(-{\mathtt{K}}^{*})\left(\overline{\mathfrak{a}}\right),$ ${\eta }^{\mathtt{L}}=(1,0)\in {\partial }^{*}{\mathtt{L}}^{*}\left(\overline{\mathfrak{a}}\right)$, such that $\overline{\mathfrak{a}}=(0,0)$ is a RS-stationary point of (P1).

Next, we derive the sufficient optimality conditions, under the assumptions of generalized convexity, for the robust local weak Pareto solution and robust local Pareto solution of UNMPVC.

Theorem 4. 

Let us assume that $\overline{\mathfrak{a}}\in {\mathbb{F}}_{\mathcal{G}}$ be an RS-stationary point of UNMPVC. Let us consider the following index sets:

$${\mathfrak{T}}_{0-}^{\mathtt{K}}:=\{\mathfrak{i}\in {\mathfrak{T}}_{0-}:{\tau }_{\mathfrak{i}}^{\mathtt{K}}>0\},$$

$${\mathfrak{T}}_{0+}^{\mathtt{K}}:=\{\mathfrak{i}\in {\mathfrak{T}}_{0+}:{\tau }_{\mathfrak{i}}^{\mathtt{K}}>0\},$$

$${\mathfrak{T}}_{+0}^{\mathtt{L}}:=\{\mathfrak{i}\in {\mathfrak{T}}_{+0}:{\tau }_{\mathfrak{i}}^{\mathtt{L}}>0\}.$$

Assume that ${\Psi }_j,j\in {\mathfrak{T}}_{\Psi }$, $\pm {\Theta }_{\mathfrak{s}},\mathfrak{s}\in \mathbb{S}$, $-{\mathtt{K}}_{\mathfrak{i}},\mathfrak{i}\in {\mathfrak{T}}_{0+}\cup {\mathfrak{T}}_{0-}\cup {\mathfrak{T}}_{00}$, ${\mathtt{K}}_{\mathfrak{i}},\mathfrak{i}\in {\mathfrak{T}}_{0+}$, and ${\mathtt{L}}_{\mathfrak{i}},\mathfrak{i}\in {\mathfrak{T}}_{+0}$, are ${\partial }^{*}$-quasiconvex at $\overline{\mathfrak{a}}$. Then,

1 

$\overline{\mathfrak{a}}$ is a robust weak Pareto solution of UNMPVC, if ${\mathcal{G}}_t,t\in \mathbb{T}$ are ${\partial }^{*}$-pseudoconvex at $\overline{\mathfrak{a}}$ and ${\mathfrak{T}}_{0-}^{\mathtt{K}}\cup {\mathfrak{T}}_{0+}^{\mathtt{K}}\cup {\mathfrak{T}}_{+0}^{\mathtt{L}}=\varnothing$;

2 

$\overline{\mathfrak{a}}$ is a robust Pareto solution of UNMPVC, if ${\mathcal{G}}_t,t\in \mathbb{T}$ are strictly ${\partial }^{*}$-pseudoconvex at $\overline{\mathfrak{a}}$ and ${\mathfrak{T}}_{0-}^{\mathtt{K}}\cup {\mathfrak{T}}_{0+}^{\mathtt{K}}\cup {\mathfrak{T}}_{+0}^{\mathtt{L}}=\varnothing$.

Proof. 

• Suppose that $\overline{\mathfrak{a}}$ is not a local robust weak Pareto solution of UNMPVC. Thus, there exists $\mathfrak{b}\in {\mathbb{F}}_{\mathcal{G}}$ such that $\mathcal{G}\left(\mathfrak{b}\right)\prec \mathcal{G}\left(\overline{\mathfrak{a}}\right)$. Since ${\mathcal{G}}_t,t\in \mathbb{T}$ are ${\partial }^{*}$-pseudoconvex at $\overline{\mathfrak{a}}$, and therefore we have

  $$⟨{\xi }_t^1,\mathfrak{b}-\overline{\mathfrak{a}}⟩<0,\forall {\xi }_t^1\in {\partial }^{*}{\mathcal{G}}_t\left(\overline{\mathfrak{a}}\right),\forall t\in \mathbb{T}.$$

  (15)
  From the feasibility conditions of UNMPVC, we have

  $${\Psi }_j(\mathfrak{b},{\mathfrak{u}}_j)\le {\Psi }_j(\overline{\mathfrak{a}},{\overline{\mathfrak{u}}}_j),{\overline{\mathfrak{u}}}_j\in {\Omega }_j\left(\overline{\mathfrak{a}}\right),\forall j\in {\mathfrak{T}}_{\Psi }.$$
  Since ${\Psi }_j,j\in \mathbb{J}$ are ${\partial }^{*}$-quasiconvex at $\overline{\mathfrak{a}}$, we therefore have

  $$⟨{\xi }_j^2,\mathfrak{b}-\overline{\mathfrak{a}}⟩\le 0,\forall {\xi }_j^2\in {\partial }^{*}{\Psi }_j(\overline{\mathfrak{a}},{\overline{\mathfrak{u}}}_j),\forall j\in {\mathfrak{T}}_{\Psi }.$$

  (16)
  In a similar manner, we obtain

  $$⟨{\xi }_{\mathfrak{s}}^3,\mathfrak{b}-\overline{\mathfrak{a}}⟩\le 0,\forall {\xi }_{\mathfrak{s}}^3\in {\partial }^{*}{\Theta }_{\mathfrak{s}}(\overline{\mathfrak{a}},{\overline{\mathfrak{u}}}_{\mathfrak{s}}),{\overline{\mathfrak{u}}}_{\mathfrak{s}}\in {\Omega }_{\mathfrak{s}}\left(\overline{\mathfrak{a}}\right),\forall \mathfrak{s}\in \mathbb{S},$$

  (17)

  $$⟨{\xi }_{\mathfrak{s}}^4,\mathfrak{b}-\overline{\mathfrak{a}}⟩\le 0,\forall {\xi }_{\mathfrak{s}}^4\in {\partial }^{*}(-{\Theta }_{\mathfrak{s}})(\overline{\mathfrak{a}},{\overline{\mathfrak{u}}}_{\mathfrak{s}}),{\overline{\mathfrak{u}}}_{\mathfrak{s}}\in {\Omega }_{\mathfrak{s}}\left(\overline{\mathfrak{a}}\right),\forall \mathfrak{s}\in \mathbb{S}$$

  (18)

  $$\begin{array}{c}\hfill ⟨{\xi }_{\mathfrak{i}}^5,\mathfrak{b}-\overline{\mathfrak{a}}⟩\le 0,\forall {\xi }_{\mathfrak{i}}^5\in {\partial }^{*}(-{\mathtt{K}}_{\mathfrak{i}})(\overline{\mathfrak{a}},{\overline{\mathfrak{u}}}_{\mathfrak{i}}),{\overline{\mathfrak{u}}}_{\mathfrak{i}}\in {\Omega }_{\mathfrak{i}}\left(\overline{\mathfrak{a}}\right),\forall \mathfrak{i}\in {\mathfrak{T}}_{0+}\cup {\mathfrak{T}}_{0-}\cup {\mathfrak{T}}_{00},\end{array}$$

  (19)

  $$⟨{\xi }_{\mathfrak{i}}^6,\mathfrak{b}-\overline{\mathfrak{a}}⟩\le 0,\forall {\xi }_{\mathfrak{i}}^6\in {\partial }^{*}{\mathtt{L}}_{\mathfrak{i}}(\overline{\mathfrak{a}},{\overline{\mathfrak{u}}}_{\mathfrak{i}}),{\overline{\mathfrak{u}}}_{\mathfrak{i}}\in {\Omega }_{\mathfrak{i}}\left(\overline{\mathfrak{a}}\right),\forall \mathfrak{i}\in {\mathfrak{T}}_{+0},$$

  (20)

  $$⟨{\xi }_{\mathfrak{i}}^7,\mathfrak{b}-\overline{\mathfrak{a}}⟩\le 0,\forall {\xi }_{\mathfrak{i}}^7\in {\partial }^{*}{\mathtt{K}}_{\mathfrak{i}}(\overline{\mathfrak{a}},{\overline{\mathfrak{u}}}_{\mathfrak{i}}),{\overline{\mathfrak{u}}}_{\mathfrak{i}}\in {\Omega }_{\mathfrak{i}}\left(\overline{\mathfrak{a}}\right),\forall \mathfrak{i}\in {\mathfrak{T}}_{0+}.$$

  (21)
  From (15)–(21) and from the assumption ${\mathfrak{T}}_{0-}^{\mathtt{K}}\cup {\mathfrak{T}}_{0+}^{\mathtt{K}}\cup {\mathfrak{T}}_{+0}^{\mathtt{L}}=\varnothing$, there exist multipliers ${\sigma }_t^{\mathcal{G}}>0,t\in \mathbb{T},{\sigma }_j^{\Psi }\ge 0,j\in {\mathfrak{T}}_{\Psi },{\sigma }_{\mathfrak{s}}^{\Theta }\ge 0,{\tau }_{\mathfrak{s}}^{\Theta }\ge 0,\mathfrak{s}\in \mathbb{S},{\sigma }_{\mathfrak{i}}^{\mathtt{K}}\ge 0,$ $\mathfrak{i}\in {\mathfrak{T}}_{0+}\cup {\mathfrak{T}}_{0-}\cup {\mathfrak{T}}_{00},$ ${\tau }_{\mathfrak{i}}^{\mathtt{L}}\ge 0,\mathfrak{i}\in {\mathfrak{T}}_{+0},{\tau }_{\mathfrak{i}}^{\mathtt{K}}\ge 0,\mathfrak{i}\in {\mathfrak{T}}_{0+}$, such that

  $$\begin{array}{cc}\hfill ⟨\sum _{t\in \mathbb{T}}{\sigma }_t^{\mathcal{G}}{\xi }_t^1+\sum _{j\in {\mathfrak{T}}_{\Psi }}{\sigma }_j^{\Psi }{\xi }_j^2& +\sum _{\mathfrak{s}\in \mathbb{S}}[{\sigma }_{\mathfrak{i}}^{\Theta }{\xi }_{\mathfrak{i}}^3+{\tau }_{\mathfrak{i}}^{\Theta }{\xi }_{\mathfrak{i}}^4]\hfill \\ & +\sum _{\mathfrak{i}\in \mathtt{I}}{\sigma }_{\mathfrak{i}}^{\mathtt{K}}{\xi }_{\mathfrak{i}}^5+\sum _{\mathfrak{i}\in \mathtt{I}}[{\tau }_{\mathfrak{i}}^{\mathtt{K}}{\xi }_{\mathfrak{i}}^6+{\tau }_{\mathfrak{i}}^{\mathtt{L}}{\xi }_{\mathfrak{i}}^7],\mathfrak{b}-\overline{\mathfrak{a}}⟩<0,\hfill \end{array}$$

  (22)

  from which we obtain a contradiction to the fact that $\overline{\mathfrak{a}}$ is a RS-stationary point of UNMPVC.
• Using the definition of a strictly ${\partial }^{*}$-pseudoconvex function, the proof is similar to the first part. Hence, we omit it for brevity.

□

Remark 2. 

1 

Theorem 4 extends Theorem 3.4 derived in [55] from nonsmooth uncertain programming problems to a broader category of mathematical programming problems, namely UNMPVC. For $\mathtt{I}=\varnothing$ and $\mathbb{S}=\varnothing$ Theorem 4 simplifies to Theorem 3.4 derived in [55].

2 

Theorem 4 extends Theorem 3.3 derived in [40] from nonsmooth MPVC to a wider category of optimization problems, UNMPVC. If ${\Omega }_j,j\in \mathbb{J},$ ${\Omega }_{\mathfrak{s}},\mathfrak{s}\in \mathbb{S}$ and ${\Omega }_{\mathfrak{i}},\mathfrak{i}\in \mathtt{I}$ are singleton sets, then Theorem 4 is similar to the sufficient optimality criteria derived in [40].

To illustrate the practical relevance of the results established in Theorem 4, we furnish the subsequent example:

Example 2. 

Let us consider the following nonsmooth multiobjective programming problem incorporating vanishing constraints under data uncertainty given by

$$\begin{array}{cc}\hfill \left(P2\right)\mathit{Minimize}& \mathcal{G}\left(\mathfrak{a}\right)=({\mathcal{G}}_1\left(\mathfrak{a}\right),{\mathcal{G}}_2\left(\mathfrak{a}\right)):=(z_1^2,{\mathfrak{a}}_2),\hfill \\ \hfill \mathit{subject}\mathit{to}& \Psi (\mathfrak{a},{\mathfrak{u}}_1):=-{\mathfrak{a}}_2{e}^{-{\mathfrak{u}}_1}\le 0,\forall {\mathfrak{u}}_1\in {\Omega }_1:=[0,+\infty ),\hfill \\ & \mathtt{K}(\mathfrak{a},{\mathfrak{u}}_2):={\mathfrak{a}}_1{\mathfrak{u}}_2\ge 0,\forall {\mathfrak{u}}_2\in {\Omega }_2:=[0,1],\hfill \\ & \mathtt{L}(\mathfrak{a},{\mathfrak{u}}_2)\mathtt{K}(\mathfrak{a},{\mathfrak{u}}_2):=-\left|{\mathfrak{a}}_2\right|{\mathfrak{a}}_1{\mathfrak{u}}_2\le 0,\forall {\mathfrak{u}}_2\in {\Omega }_2:=[0,1],\hfill \end{array}$$

where ${\mathcal{G}}_t:{\mathbb{R}}^2\to \overline{\mathbb{R}},t\in \{1,2\}$, $\Psi :{\mathbb{R}}^2\times {\Omega }_1\to \overline{\mathbb{R}}$, $\mathtt{K}:{\mathbb{R}}^2\times {\Omega }_2\to \overline{\mathbb{R}}$, $\mathtt{L}:{\mathbb{R}}^2\times {\Omega }_2\to \overline{\mathbb{R}}$, and $z=({\mathfrak{a}}_1,{\mathfrak{a}}_2)\in {\mathbb{R}}^2.$ It is evident that

$$\begin{array}{cc}& {\Psi }^{*}\left(\mathfrak{a}\right)=\underset{{\mathfrak{u}}_1\in {\Omega }_1}{max}-{\mathfrak{a}}_2{e}^{-{\mathfrak{u}}_1}=-{\mathfrak{a}}_2,\hfill \\ & {\mathtt{K}}^{*}\left(\mathfrak{a}\right)=\underset{{\mathfrak{u}}_2\in {\Omega }_2}{max}{\mathfrak{a}}_1{\mathfrak{u}}_2={\mathfrak{a}}_1,\hfill \\ & {\mathtt{L}}^{*}\left(\mathfrak{a}\right){\mathtt{K}}^{*}\left(\mathfrak{a}\right)=-{\mathfrak{a}}_1\left|{\mathfrak{a}}_2\right|,\hfill \end{array}$$

and $\overline{\mathfrak{a}}=(0,0)$ is a robust local weak Pareto solution of (P2).

Furthermore, according to Definition 3, the upper semi-regular convexificators at $\overline{\mathfrak{a}}$ of each function constituting (P2) are given by

$$\begin{array}{cccc}& {\partial }^{*}{\mathcal{G}}_1\left(\overline{\mathfrak{a}}\right)=\left\{(0,0)\right\},\hfill & & {\partial }^{*}{\mathcal{G}}_2\left(\overline{\mathfrak{a}}\right)=\left\{(1,0)\right\},\hfill \\ & {\partial }^{*}{\Psi }^{*}\left(\overline{\mathfrak{a}}\right)=\left\{(0,-1)\right\},\hfill & & {\partial }^{*}{\mathtt{K}}^{*}\left(\overline{\mathfrak{a}}\right)=\left\{(1,0)\right\},\hfill \\ & {\partial }^{*}(-{\mathtt{K}}^{*})\left(\overline{\mathfrak{a}}\right)=\left\{(-1,0)\right\},\hfill & & {\partial }^{*}{\mathtt{L}}^{*}\left(\overline{\mathfrak{a}}\right)=\{(0,-1),(0,1)\}.\hfill \end{array}$$

We know that $\overline{\mathfrak{a}}=(0,0)$ is a RS-stationary point and ${\mathfrak{T}}_{0-}^{\mathtt{K}}\cup {\mathfrak{T}}_{0+}^{\mathtt{K}}\cup {\mathfrak{T}}_{+0}^{\mathtt{L}}=\varnothing$. Moreover, at $\overline{\mathfrak{a}}$, ${\mathcal{G}}_t,$ $t\in \{1,2\}$ are ${\partial }^{*}$-pseudoconvex and Ψ, $\pm {\mathtt{K}}_{\mathfrak{i}}$, and ${\mathtt{L}}_{\mathfrak{i}}$ are ${\partial }^{*}$-quasiconvex. Therefore, the requirements of Theorem 4 are fulfilled at $\overline{\mathfrak{a}}$.

In the next example, we investigate a UNMPVC with the objective function having three components. Employing MATLAB R2024a, we generate the Pareto front of the considered problem.

Example 3. 

Let us consider the following nonsmooth MPVC under data uncertainty given by

$$\begin{array}{cc}\hfill \left(P3\right)\mathit{Minimize}& \mathcal{G}\left(\mathfrak{a}\right)=({\mathcal{G}}_1\left(\mathfrak{a}\right),{\mathcal{G}}_2\left(\mathfrak{a}\right),{\mathcal{G}}_3\left(\mathfrak{a}\right)):=({({\mathfrak{a}}_1-1)}^2,{({\mathfrak{a}}_2-3)}^2,e^{-3{\mathfrak{a}}_3}),\hfill \\ \hfill \mathit{subject}\mathit{to}& \Psi (\mathfrak{a},{\mathfrak{u}}_1):=-{\mathfrak{a}}_2{\mathfrak{u}}_1\le 0,{\mathfrak{u}}_1\in {\Omega }_1:=\left[0.5,1.5\right],\hfill \\ & \mathtt{K}(\mathfrak{a},{\mathfrak{u}}_2):={\mathfrak{a}}_1{\mathfrak{u}}_2\ge 0,{\mathfrak{u}}_2\in {\Omega }_2:=\left[50,100\right],\hfill \\ & \mathtt{L}(\mathfrak{a},{\mathfrak{u}}_2)\mathtt{K}(\mathfrak{a},{\mathfrak{u}}_3):=-{\mathfrak{a}}_1{\mathfrak{a}}_2{\mathfrak{u}}_3\le 0,\forall {\mathfrak{u}}_3\in {\Omega }_3:=\left[50,100\right],\hfill \end{array}$$

where ${\mathcal{G}}_t:{\mathbb{R}}^2\to \overline{\mathbb{R}},t\in \{1,2\}$, $\Psi :{\mathbb{R}}^2\times {\Omega }_1\to \overline{\mathbb{R}}$, $\mathtt{K}:{\mathbb{R}}^2\times {\Omega }_2\to \overline{\mathbb{R}}$, $\mathtt{L}:{\mathbb{R}}^2\times {\Omega }_2\to \overline{\mathbb{R}}$, and $z=({\mathfrak{a}}_1,{\mathfrak{a}}_2)\in {\mathbb{R}}^2.$ Then, the robust counterpart of (P3) is given by

$$\begin{array}{cc}\hfill \left(\mathit{RC}\right)\mathit{Minimize}& \mathcal{G}\left(\mathfrak{a}\right)=({\mathcal{G}}_1\left(\mathfrak{a}\right),{\mathcal{G}}_2\left(\mathfrak{a}\right),{\mathcal{G}}_3\left(\mathfrak{a}\right)):=({({\mathfrak{a}}_1-1)}^2,{({\mathfrak{a}}_2-3)}^2,e^{-3{\mathfrak{a}}_3}),\hfill \\ \hfill \mathit{subject}\mathit{to}& {\Psi }^{*}\left(\mathfrak{a}\right):=0.5{\mathfrak{a}}_2\le 0,\hfill \\ \hfill & {\mathtt{K}}^{*}\left(\mathfrak{a}\right):=100{\mathfrak{a}}_1\ge 0,\hfill \\ \hfill & {\mathtt{L}}^{*}\left(\mathfrak{a}\right){\mathtt{K}}^{*}\left(\mathfrak{a}\right):=-50{\mathfrak{a}}_1{\mathfrak{a}}_2\le 0.\hfill \end{array}$$

Now, employing MATLAB R2024a, we generate the Pareto front of the problem (P3) and the robust counterpart (RC), as shown in the following figures:

Employing MATLAB R2024a, we generate 5000 random values of ${\mathfrak{a}}_{\mathfrak{i}}\in [-5,5]$ for each $\mathfrak{i}\in \{1,2,3\}$ and verify whether it is feasible or not. Then, from the set of feasible solutions, we find the Pareto front of the considered problem (P3) and the RC. In Figure 1 and Figure 2, the feasible solutions are represented by a blue colour, and the Pareto front solutions are represented by a red colour.

Figure 1. Pareto front of the uncertain problem (P3).

Figure 2. Pareto front of the robust counterpart (RC).

Since the robust counterpart is formed by maximizing the constraints over the uncertain parameters, some of the feasible points of the problem (P3) may no longer be feasible in the robust counterpart.

4. Wolfe-Type Robust Dual Model

This section deals with formulating a Wolfe-type robust dual model (WRD) relating to the problem, UNMPVC. We further extend our analysis by establishing weak, strong, and strict converse duality results. These theorems illuminate the profound interrelationships between UNMPVC and its corresponding dual model, WRD.

We formulate WRD corresponding to the problem UNMPVC as follows:

$$\begin{array}{cc}\hfill \underset{(\mathfrak{b},\sigma )}{\mathrm{Maximize}}\{\mathcal{G}\left(\mathfrak{b}\right)& +\sum _{j\in {\mathfrak{T}}_{\Psi }}{\sigma }_j^{\Psi }{\Psi }_j(\mathfrak{b},{\mathfrak{u}}_j)e+\sum _{\mathfrak{s}\in \mathbb{S}}{\sigma }_{\mathfrak{s}}^{\Theta }{\Theta }_{\mathfrak{s}}(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{s}})e\hfill \\ \hfill & +\sum _{\mathfrak{s}\in \mathbb{S}}{\tau }_{\mathfrak{s}}^{\Theta }(-{\Theta }_{\mathfrak{s}})(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{s}})e+\sum _{\mathfrak{i}\in \mathtt{I}}{\sigma }_{\mathfrak{i}}^{\mathtt{K}}(-{\mathtt{K}}_{\mathfrak{i}})(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{i}})e\hfill \\ \hfill & +\sum _{\mathfrak{i}\in \mathtt{I}}{\tau }_{\mathfrak{i}}^{\mathtt{K}}{\mathtt{K}}_{\mathfrak{i}}(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{i}})e+\sum _{\mathfrak{i}\in \mathtt{I}}{\tau }_{\mathfrak{i}}^{\mathtt{L}}{\mathtt{L}}_{\mathfrak{i}}(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{i}})e,e=(1,1,\cdots ,1)\in {\mathbb{R}}^r\},\hfill \end{array}$$

$$\mathrm{subject}\mathrm{to}(\mathfrak{b},\sigma )\in {\mathbb{F}}_{\mathcal{G}}^{WRD},$$

where $\mathcal{G}\left(\mathfrak{b}\right)=({\mathcal{G}}_1\left(\mathfrak{b}\right),\cdots ,{\mathcal{G}}_r\left(\mathfrak{b}\right))$ and ${\mathbb{F}}_{\mathcal{G}}^{WRD}$ is feasible set of WRD given by

$$\begin{array}{cc}\hfill {\mathbb{F}}_{\mathcal{G}}^{WRD}:=\{\sigma =& ({\sigma }^{\mathcal{G}},{\sigma }^{\Psi },{\sigma }^{\Theta },{\tau }^{\Theta },{\sigma }^{\mathtt{K}},{\tau }^{\mathtt{K}},{\tau }^{\mathtt{L}})\in {\mathbb{R}}^r\times {\mathbb{R}}^p\times {\mathbb{R}}^{2\ell }\times {\mathbb{R}}^{3m},{\left({\sigma }^{\mathcal{G}}\right)}^{T}e=1,\hfill \\ & \mathfrak{b}\in {\mathbb{R}}^n:0\in \sum _{t\in \mathbb{T}}{\sigma }_t^{\mathcal{G}}co{\partial }^{*}{\mathcal{G}}_t\left(\mathfrak{b}\right)+\sum _{j\in {\mathfrak{T}}_{\Psi }}{\sigma }_j^{\Psi }co{\partial }^{*}{\Psi }_j(\mathfrak{b},{\mathfrak{u}}_j)\hfill \\ \hfill & +\sum _{\mathfrak{s}\in \mathbb{S}}[{\sigma }_{\mathfrak{s}}^{\Theta }co{\partial }^{*}{\Theta }_{\mathfrak{s}}(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{s}})+{\tau }_{\mathfrak{s}}^{\Theta }co{\partial }^{*}(-{\Theta }_{\mathfrak{s}})(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{s}})]\hfill \\ \hfill & +\sum _{\mathfrak{i}\in \mathtt{I}}{\sigma }_{\mathfrak{i}}^{\mathtt{K}}co{\partial }^{*}(-{\mathtt{K}}_{\mathfrak{i}})(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{i}})\hfill \\ \hfill & +\sum _{\mathfrak{i}\in \mathtt{I}}[{\tau }_{\mathfrak{i}}^{\mathtt{K}}co{\partial }^{*}{\mathtt{K}}_{\mathfrak{i}}(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{i}})+{\tau }_{\mathfrak{i}}^{\mathtt{L}}co{\partial }^{*}{\mathtt{L}}_{\mathfrak{i}}(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{i}})],\hfill \\ \hfill & {\sigma }_t^{\mathcal{G}}>0,t\in \mathbb{T},{\sigma }_j^{\Psi }\ge 0,j\in {\mathfrak{T}}_{\Psi },{\sigma }_{\mathfrak{s}}^{\Theta }\ge 0,{\tau }_{\mathfrak{s}}^{\Theta }\ge 0,\mathfrak{s}\in \mathbb{S},\hfill \\ \hfill & {\sigma }_{\mathfrak{i}}^{\mathtt{K}}\ge 0,{\tau }_{\mathfrak{i}}^{\mathtt{K}}\ge 0,{\tau }_{\mathfrak{i}}^{\mathtt{L}}\ge 0,\mathfrak{i}\in \mathtt{I},\hfill \\ & {\sigma }_{{\mathfrak{T}}_{+0}\cup {\mathfrak{T}}_{+-}}^{\mathtt{K}}={\tau }_{{\mathfrak{T}}_{+0}\cup {\mathfrak{T}}_{+-}}^{\mathtt{K}}={\tau }_{{\mathfrak{T}}_{0+}\cup {\mathfrak{T}}_{+-}\cup {\mathfrak{T}}_{0-}}^{\mathtt{L}}=0,\hfill \\ \hfill & {\sigma }_i^{\mathtt{K}}-{\tau }_i^{\mathtt{K}}\ge 0,\mathfrak{i}\in {\mathfrak{T}}_{0-},{\tau }_{\mathfrak{i}}^{\mathtt{L}}=0,({\sigma }_{\mathfrak{i}}^{\mathtt{K}}-{\tau }_{\mathfrak{i}}^{\mathtt{K}})\ge 0,\forall \mathfrak{i}\in {\mathfrak{T}}_{00}\}.\hfill \end{array}$$

The subsequent theorem delves into the weak duality relation between UNMPVC and WRD.

Theorem 5. 

Let $\mathfrak{a}$ $\in {\mathbb{F}}_{\mathcal{G}}$ and $(\mathfrak{b},\sigma )\in {\mathbb{F}}_{\mathcal{G}}^{WRD}$. Suppose that ${\mathcal{G}}_t,t\in \mathbb{T},{\Psi }_j,j\in {\mathfrak{T}}_{\Psi },\pm {\Theta }_{\mathfrak{s}},\mathfrak{s}\in \mathbb{S}$, $-{\mathtt{K}}_{\mathfrak{i}},\mathfrak{i}\in {\mathfrak{T}}_{0+}\cup {\mathfrak{T}}_{0-}\cup {\mathfrak{T}}_{00},$ ${\mathtt{K}}_{\mathfrak{i}},\mathfrak{i}\in {\mathfrak{T}}_{0+}$, and ${\mathtt{L}}_{\mathfrak{i}},\mathfrak{i}\in {\mathfrak{T}}_{+0}$ are ${\partial }^{*}$-convex functions at $\mathfrak{b}$ and admit bounded USRC. Moreover, we suppose that ${\mathfrak{T}}_{0-}^{\mathtt{K}}\cup {\mathfrak{T}}_{0+}^{\mathtt{K}}\cup {\mathfrak{T}}_{+0}^{\mathtt{L}}=\varnothing$. Then, we have

$$\begin{array}{cc}\hfill \mathcal{G}\left(\mathfrak{a}\right)\nprec \mathcal{G}\left(\mathfrak{b}\right)& +\sum _{j\in {\mathfrak{T}}_{\Psi }}{\sigma }_j^{\Psi }{\Psi }_j(\mathfrak{b},{\mathfrak{u}}_j)e+\sum _{\mathfrak{s}\in \mathbb{S}}{\sigma }_{\mathfrak{s}}^{\Theta }{\Theta }_{\mathfrak{s}}(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{s}})e+\sum _{\mathfrak{s}\in \mathbb{S}}{\tau }_{\mathfrak{s}}^{\Theta }(-{\Theta }_{\mathfrak{s}})(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{s}})e\hfill \\ \hfill & +\sum _{\mathfrak{i}\in \mathtt{I}}{\sigma }_{\mathfrak{i}}^{\mathtt{K}}(-{\mathtt{K}}_{\mathfrak{i}})(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{i}})e+\sum _{\mathfrak{i}\in \mathtt{I}}{\tau }_{\mathfrak{i}}^{\mathtt{K}}{\mathtt{K}}_{\mathfrak{i}}(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{i}})e+\sum _{\mathfrak{i}\in \mathtt{I}}{\tau }_{\mathfrak{i}}^{\mathtt{L}}{\mathtt{L}}_{\mathfrak{i}}(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{i}})e.\hfill \end{array}$$

(23)

Proof. 

On the other hand, suppose that

$$\begin{array}{cc}\hfill \mathcal{G}\left(\mathfrak{a}\right)\prec \mathcal{G}\left(\mathfrak{b}\right)& +\sum _{j\in {\mathfrak{T}}_{\Psi }}{\sigma }_j^{\Psi }{\Psi }_j(\mathfrak{b},{\mathfrak{u}}_j)e+\sum _{\mathfrak{s}\in \mathbb{S}}{\sigma }_{\mathfrak{s}}^{\Theta }{\Theta }_{\mathfrak{s}}(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{s}})e+\sum _{\mathfrak{s}\in \mathbb{S}}{\tau }_{\mathfrak{s}}^{\Theta }(-{\Theta }_{\mathfrak{s}})(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{s}})e\hfill \\ \hfill & +\sum _{\mathfrak{i}\in \mathtt{I}}{\sigma }_{\mathfrak{i}}^{\mathtt{K}}(-{\mathtt{K}}_{\mathfrak{i}})(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{i}})e+\sum _{\mathfrak{i}\in \mathtt{I}}{\tau }_{\mathfrak{i}}^{\mathtt{K}}{\mathtt{K}}_{\mathfrak{i}}(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{i}})e+\sum _{\mathfrak{i}\in \mathtt{I}}{\tau }_{\mathfrak{i}}^{\mathtt{L}}{\mathtt{L}}_{\mathfrak{i}}(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{i}})e.\hfill \end{array}$$

(24)

Multiplying (24) by ${\sigma }_t^{\mathcal{G}}>0,t\in \mathbb{T}$, and since ${\left({\sigma }^{\mathcal{G}}\right)}^{T}e=1$, we obtain

$$\begin{array}{cc}\hfill \sum _{t\in \mathbb{T}}{\sigma }_t^{\mathcal{G}}{\mathcal{G}}_t\left(\mathfrak{a}\right)& <\sum _{t\in \mathbb{T}}{\sigma }_t^{\mathcal{G}}{\mathcal{G}}_t\left(\mathfrak{b}\right)+\sum _{j\in {\mathfrak{T}}_{\Psi }}{\sigma }_j^{\Psi }{\Psi }_j(\mathfrak{b},{\mathfrak{u}}_j)\hfill \\ & +\sum _{\mathfrak{s}\in \mathbb{S}}{\sigma }_{\mathfrak{s}}^{\Theta }{\Theta }_{\mathfrak{s}}(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{s}})+\sum _{\mathfrak{s}\in \mathbb{S}}{\tau }_{\mathfrak{s}}^{\Theta }(-{\Theta }_{\mathfrak{s}})(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{s}})\hfill \\ & +\sum _{\mathfrak{i}\in \mathtt{I}}{\sigma }_{\mathfrak{i}}^{\mathtt{K}}(-{\mathtt{K}}_{\mathfrak{i}})(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{i}})\hfill \\ & +\sum _{\mathfrak{i}\in \mathtt{I}}[{\sigma }_{\mathfrak{i}}^{\mathtt{K}}{\mathtt{K}}_{\mathfrak{i}}(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{i}})+{\sigma }_{\mathfrak{i}}^{\mathtt{L}}{\mathtt{L}}_{\mathfrak{i}}(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{i}})].\hfill \end{array}$$

(25)

Let $\mathfrak{a}\in {\mathbb{F}}_{\mathcal{G}}$. In light of the fact that ${\mathcal{G}}_t,t\in \mathbb{T}$ are ${\partial }^{*}$-convex at $\mathfrak{b}$, and we obtain the subsequent inequalities:

$${\mathcal{G}}_t\left(\mathfrak{a}\right)-{\mathcal{G}}_t\left(\mathfrak{b}\right)\ge ⟨{\xi }_t^1,\mathfrak{a}-\mathfrak{b}⟩,\forall {\xi }_t^1\in {\partial }^{*}{\mathcal{G}}_t\left(\mathfrak{b}\right),\forall t\in \mathbb{T}.$$

(26)

In a similar manner, by employing the ${\partial }^{*}$-convexity assumptions of ${\Psi }_j(·,{\mathfrak{u}}_j),$ $j\in {\mathfrak{T}}_{\Psi },$ $\pm {\Theta }_{\mathfrak{s}}(·,{\mathfrak{u}}_{\mathfrak{s}}),$ $\mathfrak{s}\in \mathbb{S},$ $-{\mathtt{K}}_{\mathfrak{i}}(·,{\mathfrak{u}}_{\mathfrak{i}}),$ $\mathfrak{i}\in {\mathfrak{T}}_{0+}\cup {\mathfrak{T}}_{0-}\cup {\mathfrak{T}}_{00}$, ${\mathtt{K}}_{\mathfrak{i}}(·,{\mathfrak{u}}_{\mathfrak{i}}),\mathfrak{i}\in {\mathfrak{T}}_{0+}$, and ${\mathtt{L}}_{\mathfrak{i}}(·,{\mathfrak{u}}_{\mathfrak{i}}),\mathfrak{i}\in {\mathfrak{T}}_{+0}$ at $\mathfrak{b}$, we obtain the subsequent inequalities:

$$\begin{array}{c}\hfill {\Psi }_j(\mathfrak{a},{\mathfrak{u}}_j)-{\Psi }_j(\mathfrak{b},{\mathfrak{u}}_j)\ge ⟨{\xi }_j^2,\mathfrak{a}-\mathfrak{b}⟩,\forall {\xi }_j^2\in {\partial }^{*}{\Psi }_j(\mathfrak{b},{\mathfrak{u}}_j),\forall j\in {\mathfrak{T}}_{\Psi },\end{array}$$

(27)

$${\Theta }_{\mathfrak{s}}(\mathfrak{a},{\mathfrak{u}}_{\mathfrak{s}})-{\Theta }_{\mathfrak{s}}(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{s}})\ge ⟨{\xi }_{\mathfrak{s}}^3,\mathfrak{a}-\mathfrak{b}⟩,\forall {\xi }_{\mathfrak{s}}^3\in {\partial }^{*}{\Theta }_{\mathfrak{s}}(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{s}}),\forall \mathfrak{s}\in \mathbb{S},$$

(28)

$$\begin{array}{cc}\hfill (-{\Theta }_{\mathfrak{s}})(\mathfrak{a},{\mathfrak{u}}_{\mathfrak{s}})-(-{\Theta }_{\mathfrak{s}})(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{s}})\ge ⟨{\xi }_{\mathfrak{s}}^4,\mathfrak{a}-\mathfrak{b}⟩,\forall {\xi }_{\mathfrak{s}}^4\in {\partial }^{*}(-{\Theta }_{\mathfrak{s}})& (\mathfrak{b},{\mathfrak{u}}_{\mathfrak{s}}),\hfill \\ & \forall \mathfrak{s}\in \mathbb{S},\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill (-{\mathtt{K}}_{\mathfrak{i}})(\mathfrak{a},{\mathfrak{u}}_{\mathfrak{i}})-(-{\mathtt{K}}_{\mathfrak{i}})(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{i}})\ge ⟨{\xi }_{\mathfrak{i}}^5,\mathfrak{a}-\mathfrak{b}⟩,& \forall {\xi }_{\mathfrak{i}}^5\in {\partial }^{*}(-{\mathtt{K}}_{\mathfrak{i}})(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{i}}),\hfill \\ & \forall \mathfrak{i}\in {\mathfrak{T}}_{0+}\cup {\mathfrak{T}}_{0-}\cup {\mathfrak{T}}_{00},\hfill \end{array}$$

(30)

$${\mathtt{K}}_{\mathfrak{i}}(\mathfrak{a},{\mathfrak{u}}_{\mathfrak{i}})-{\mathtt{K}}_{\mathfrak{i}}(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{i}})\ge ⟨{\xi }_{\mathfrak{i}}^6,\mathfrak{a}-\mathfrak{b}⟩,\forall {\xi }_{\mathfrak{i}}^6\in {\partial }^{*}{\mathtt{K}}_{\mathfrak{i}}(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{i}}),\forall \mathfrak{i}\in {\mathfrak{T}}_{0+},$$

(31)

$${\mathtt{L}}_{\mathfrak{i}}(\mathfrak{a},{\mathfrak{u}}_{\mathfrak{i}})-{\mathtt{L}}_{\mathfrak{i}}(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{i}})\ge ⟨{\xi }_{\mathfrak{i}}^7,\mathfrak{a}-\mathfrak{b}⟩,\forall {\xi }_{\mathfrak{i}}^7\in {\partial }^{*}{\mathtt{K}}_{\mathfrak{i}}(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{i}}),\forall \mathfrak{i}\in {\mathfrak{T}}_{+0},$$

(32)

If ${\mathfrak{T}}_{0-}^{\mathtt{K}}\cup {\mathfrak{T}}_{0+}^{\mathtt{K}}\cup {\mathfrak{T}}_{+0}^{\mathtt{L}}=\varnothing$, then multiplying (26)–(32) by ${\sigma }_t^{\mathcal{G}}>0,t\in \mathbb{T}$, ${\sigma }_j^{\Psi }\ge 0,j\in {\mathfrak{T}}_{\Psi }$, ${\sigma }_{\mathfrak{s}}^{\Theta }\ge 0,\mathfrak{s}\in \mathbb{S}$, ${\tau }_{\mathfrak{s}}^{\Theta }\ge 0,\mathfrak{s}\in \mathbb{S}$, ${\sigma }_{\mathfrak{i}}^{\mathtt{K}}\ge 0,\mathfrak{i}\in {\mathfrak{T}}_{0+}\cup {\mathfrak{T}}_{0-}\cup {\mathfrak{T}}_{00}$, ${\tau }_{\mathfrak{i}}^{\mathtt{K}}\ge 0,\mathfrak{i}\in {\mathfrak{T}}_{0+},{\tau }_{\mathfrak{i}}^{\mathtt{L}}\ge 0,$ $\mathfrak{i}\in {\mathfrak{T}}_{0+}$, respectively, and then by summing these inequalities, we obtain the following:

$$\begin{array}{cc}\hfill \sum _{t\in \mathbb{T}}{\sigma }_t^{\mathcal{G}}{\mathcal{G}}_t\left(\mathfrak{a}\right)& -\sum _{t\in \mathbb{T}}{\sigma }_t^{\mathcal{G}}{\mathcal{G}}_t\left(\mathfrak{b}\right)+\sum _{j\in {\mathfrak{T}}_{\Psi }}{\sigma }_j^{\Psi }{\Psi }_j(\mathfrak{a},{\mathfrak{u}}_j)-\sum _{j\in {\mathfrak{T}}_{\Psi }}{\sigma }_j^{\Psi }{\Psi }_j(\mathfrak{b},{\mathfrak{u}}_j)\hfill \\ & +\sum _{\mathfrak{s}\in \mathbb{S}}{\sigma }_{\mathfrak{s}}^{\Theta }{\Theta }_{\mathfrak{s}}(\mathfrak{a},{\mathfrak{u}}_{\mathfrak{s}})-\sum _{\mathfrak{s}\in \mathbb{S}}{\sigma }_{\mathfrak{s}}^{\Theta }{\Theta }_{\mathfrak{s}}(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{s}})+\sum _{\mathfrak{s}\in \mathbb{S}}{\tau }_{\mathfrak{s}}^{\Theta }(-{\Theta }_{\mathfrak{s}})(\mathfrak{a},{\mathfrak{u}}_{\mathfrak{s}})\hfill \\ & -\sum _{\mathfrak{s}\in \mathbb{S}}{\tau }_{\mathfrak{s}}^{\Theta }(-{\Theta }_{\mathfrak{s}})(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{s}})+\sum _{\mathfrak{i}\in \mathtt{I}}{\sigma }_{\mathfrak{i}}^{\mathtt{K}}(-{\mathtt{K}}_{\mathfrak{i}})(\mathfrak{a},{\mathfrak{u}}_{\mathfrak{i}})-\sum _{\mathfrak{i}\in \mathtt{I}}{\sigma }_{\mathfrak{i}}^{\mathtt{K}}(-{\mathtt{K}}_{\mathfrak{i}})(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{i}})\hfill \\ & +\sum _{\mathfrak{i}\in \mathtt{I}}{\tau }_{\mathfrak{i}}^{\mathtt{K}}{\mathtt{K}}_{\mathfrak{i}}(\mathfrak{a},{\mathfrak{u}}_{\mathfrak{i}})-\sum _{\mathfrak{i}\in \mathtt{I}}{\tau }_{\mathfrak{i}}^{\mathtt{K}}{\mathtt{K}}_{\mathfrak{i}}(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{i}})+\sum _{\mathfrak{i}\in \mathtt{I}}{\tau }_{\mathfrak{i}}^{\mathtt{L}}{\mathtt{L}}_{\mathfrak{i}}(\mathfrak{a},{\mathfrak{u}}_{\mathfrak{i}})\hfill \\ & -\sum _{\mathfrak{i}\in \mathtt{I}}{\tau }_{\mathfrak{i}}^{\mathtt{L}}{\mathtt{L}}_{\mathfrak{i}}(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{i}})\ge ⟨\sum _{t\in \mathbb{T}}{\sigma }_t^{\mathcal{G}}{\xi }_t^1+\sum _{j\in {\mathfrak{T}}_{\Psi }}{\sigma }_j^{\Psi }{\xi }_j^2+\sum _{\mathfrak{s}\in \mathbb{S}}[{\sigma }_{\mathfrak{s}}^{\Theta }{\xi }_{\mathfrak{s}}^3+{\tau }_{\mathfrak{s}}^{\Theta }{\xi }_{\mathfrak{s}}^4]\hfill \\ & +\sum _{\mathfrak{i}\in \mathtt{I}}{\sigma }_{\mathfrak{i}}^{\mathtt{K}}{\xi }_{\mathfrak{i}}^5+\sum _{\mathfrak{i}\in \mathtt{I}}[{\tau }_{\mathfrak{i}}^{\mathtt{K}}{\xi }_{\mathfrak{i}}^6+{\tau }_{\mathfrak{i}}^{\mathtt{L}}{\xi }_{\mathfrak{i}}^7],\mathfrak{a}-\mathfrak{b}⟩.\hfill \end{array}$$

Given the feasibility conditions of $\mathfrak{b}\in {\mathbb{F}}_{\mathcal{G}}^{WRD}$, there exist ${\overline{\xi }}_t^1\in co{\partial }^{*}{\mathcal{G}}_t\left(\mathfrak{b}\right),$ $t\in \mathbb{T}$, ${\overline{\xi }}_j^2\in co{\partial }^{*}{\Psi }_j(\mathfrak{b},{\mathfrak{u}}_j),$ $j\in {\mathfrak{T}}_{\Psi }$, ${\overline{\xi }}_{\mathfrak{s}}^3\in co{\partial }^{*}{\Theta }_{\mathfrak{s}}(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{s}}),$ $\mathfrak{s}\in \mathbb{S}$, ${\overline{\xi }}_{\mathfrak{s}}^4\in co{\partial }^{*}(-{\Theta }_{\mathfrak{s}})(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{s}}),$ $\mathfrak{s}\in \mathbb{S}$, ${\overline{\xi }}_{\mathfrak{i}}^5\in co{\partial }^{*}(-{\mathtt{K}}_{\mathfrak{i}})(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{i}}),$ $\mathfrak{i}\in \mathtt{I}$, ${\overline{\xi }}_{\mathfrak{i}}^6\in co{\partial }^{*}{\mathtt{K}}_{\mathfrak{i}}(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{i}}),$ $\mathfrak{i}\in \mathtt{I}$, and ${\overline{\xi }}_{\mathfrak{i}}^7\in co{\partial }^{*}{\mathtt{L}}_{\mathfrak{i}}(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{i}}),$ $\mathfrak{i}\in \mathtt{I}$, such that

$$\sum _{t\in \mathbb{T}}{\sigma }_t^{\mathcal{G}}{\overline{\xi }}_t^1+\sum _{j\in {\mathfrak{T}}_{\Psi }}{\sigma }_j^{\Psi }{\overline{\xi }}_j^2+\sum _{\mathfrak{s}\in \mathbb{S}}[{\sigma }_{\mathfrak{s}}^{\Theta }{\overline{\xi }}_{\mathfrak{s}}^3+{\tau }_{\mathfrak{s}}^{\Theta }{\overline{\xi }}_{\mathfrak{s}}^4]+\sum _{\mathfrak{i}\in \mathtt{I}}{\sigma }_{\mathfrak{i}}^{\mathtt{K}}{\overline{\xi }}_{\mathfrak{i}}^5+\sum _{\mathfrak{i}\in \mathtt{I}}[{\tau }_{\mathfrak{i}}^{\mathtt{K}}{\overline{\xi }}_{\mathfrak{i}}^5+{\tau }_{\mathfrak{i}}^{\mathtt{L}}{\overline{\xi }}_{\mathfrak{i}}^6]=0.$$

Therefore,

$$\begin{array}{cc}\hfill \sum _{t\in \mathbb{T}}{\sigma }_t^{\mathcal{G}}{\mathcal{G}}_t\left(\mathfrak{a}\right)& -\sum _{t\in \mathbb{T}}{\sigma }_t^{\mathcal{G}}{\mathcal{G}}_t\left(\mathfrak{b}\right)+\sum _{j\in {\mathfrak{T}}_{\Psi }}{\sigma }_j^{\Psi }{\Psi }_j(\mathfrak{a},{\mathfrak{u}}_j)-\sum _{j\in {\mathfrak{T}}_{\Psi }}{\sigma }_j^{\Psi }{\Psi }_j(\mathfrak{b},{\mathfrak{u}}_j)\hfill \\ & +\sum _{\mathfrak{s}\in \mathbb{S}}{\sigma }_{\mathfrak{s}}^{\Theta }{\Theta }_{\mathfrak{s}}(\mathfrak{a},{\mathfrak{u}}_{\mathfrak{s}})-\sum _{\mathfrak{s}\in \mathbb{S}}{\sigma }_{\mathfrak{s}}^{\Theta }{\Theta }_{\mathfrak{s}}(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{s}})+\sum _{\mathfrak{s}\in \mathbb{S}}{\tau }_{\mathfrak{s}}^{\Theta }(-{\Theta }_{\mathfrak{s}})(\mathfrak{a},{\mathfrak{u}}_{\mathfrak{s}})\hfill \\ & -\sum _{\mathfrak{s}\in \mathbb{S}}{\tau }_{\mathfrak{s}}^{\Theta }(-{\Theta }_{\mathfrak{s}})(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{s}})+\sum _{\mathfrak{i}\in \mathtt{I}}{\sigma }_{\mathfrak{i}}^{\mathtt{K}}(-{\mathtt{K}}_{\mathfrak{i}})(\mathfrak{a},{\mathfrak{u}}_{\mathfrak{i}})-\sum _{\mathfrak{i}\in \mathtt{I}}{\sigma }_{\mathfrak{i}}^{\mathtt{K}}(-{\mathtt{K}}_{\mathfrak{i}})(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{i}})\hfill \\ & +\sum _{\mathfrak{i}\in \mathtt{I}}{\tau }_{\mathfrak{i}}^{\mathtt{K}}{\mathtt{K}}_{\mathfrak{i}}(\mathfrak{a},{\mathfrak{u}}_{\mathfrak{i}})-\sum _{\mathfrak{i}\in \mathtt{I}}{\tau }_{\mathfrak{i}}^{\mathtt{K}}{\mathtt{K}}_{\mathfrak{i}}(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{i}})+\sum _{\mathfrak{i}\in \mathtt{I}}{\tau }_{\mathfrak{i}}^{\mathtt{L}}{\mathtt{L}}_{\mathfrak{i}}(\mathfrak{a},{\mathfrak{u}}_{\mathfrak{i}})\hfill \\ & -\sum _{\mathfrak{i}\in \mathtt{I}}{\tau }_{\mathfrak{i}}^{\mathtt{L}}{\mathtt{L}}_{\mathfrak{i}}(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{i}})\ge 0.\hfill \end{array}$$

Since $\mathfrak{a}\in {\mathbb{F}}_{\mathcal{G}}$, we have ${\Psi }_j(\mathfrak{a},{\mathfrak{u}}_j)\le 0,j\in {\mathfrak{T}}_{\Psi },{\Theta }_{\mathfrak{s}}(\mathfrak{a},{\mathfrak{u}}_{\mathfrak{s}})=0,\mathfrak{s}\in \mathbb{S},-{\mathtt{K}}_{\mathfrak{i}}(\mathfrak{a},{\mathfrak{u}}_{\mathfrak{i}})<0,$ $\mathfrak{i}\in {\mathfrak{T}}_{+},$ ${\mathtt{K}}_{\mathfrak{i}}(\mathfrak{a},{\mathfrak{u}}_{\mathfrak{i}})=0,\mathfrak{i}\in {\mathfrak{T}}_0,{\mathtt{L}}_{\mathfrak{i}}(\mathfrak{a},{\mathfrak{u}}_{\mathfrak{i}})>0,\mathfrak{i}\in {\mathfrak{T}}_{0+},{\mathtt{L}}_{\mathfrak{i}}(\mathfrak{a},{\mathfrak{u}}_{\mathfrak{i}})=0,\mathfrak{i}\in {\mathfrak{T}}_{00}\cup {\mathfrak{T}}_{+0},$ ${\mathtt{L}}_{\mathfrak{i}}(\mathfrak{a},{\mathfrak{u}}_{\mathfrak{i}})=0,$ $\mathfrak{i}\in {\mathfrak{T}}_{0-}\cup {\mathfrak{T}}_{+-}.$ Therefore,

$$\begin{array}{cc}\hfill \sum _{t\in \mathbb{T}}{\sigma }_t^{\mathcal{G}}{\mathcal{G}}_t\left(\mathfrak{a}\right)& \ge \sum _{t\in \mathbb{T}}{\sigma }_t^{\mathcal{G}}{\mathcal{G}}_t\left(\mathfrak{b}\right)+\sum _{j\in {\mathfrak{T}}_{\Psi }}{\sigma }_j^{\Psi }{\Psi }_j(\mathfrak{b},{\mathfrak{u}}_j)\hfill \\ & +\sum _{\mathfrak{s}\in \mathbb{S}}[{\sigma }_{\mathfrak{s}}^{\Theta }{\Theta }_{\mathfrak{s}}(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{s}})+{\tau }_{\mathfrak{s}}^{\Theta }(-{\Theta }_{\mathfrak{s}})(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{s}})]+\sum _{\mathfrak{i}\in \mathtt{I}}{\sigma }_{\mathfrak{i}}^{\mathtt{K}}(-{\mathtt{K}}_{\mathfrak{i}})(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{i}})\hfill \\ & +\sum _{\mathfrak{i}\in \mathtt{I}}[{\sigma }_{\mathfrak{i}}^{\mathtt{K}}{\mathtt{K}}_{\mathfrak{i}}(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{i}})+{\sigma }_{\mathfrak{i}}^{\mathtt{L}}{\mathtt{L}}_{\mathfrak{i}}(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{i}})].\hfill \end{array}$$

Hence, from (25), a contradiction is reached, which concludes the proof of the weak duality theorem. □

In the subsequent theorem, a strong duality result is established, which relates UNMPVC with the associated WRD dual model.

Theorem 6. 

Let $\overline{\mathfrak{a}}\in {\mathbb{F}}_{\mathcal{G}}$ be a robust local weak Pareto solution of UNMPVC. Suppose that ${\mathcal{G}}_t,t\in \mathbb{T},{\Psi }_j,j\in {\mathfrak{T}}_{\Psi },\pm {\Theta }_{\mathfrak{s}},\mathfrak{s}\in \mathbb{S},-{\mathtt{K}}_{\mathfrak{i}},\mathfrak{i}\in {\mathfrak{T}}_{0+}\cup {\mathfrak{T}}_{0-}\cup {\mathfrak{T}}_{00},$ ${\mathtt{K}}_{\mathfrak{i}},\mathfrak{i}\in {\mathfrak{T}}_{0+}$, and ${\mathtt{L}}_{\mathfrak{i}},\mathfrak{i}\in {\mathfrak{T}}_{+0}$ are ${\partial }^{*}$-convex functions at $\overline{\mathfrak{a}}$ and admit bounded USRC. Assume also that at $\overline{\mathfrak{a}}$, GS-ACQ holds. Then, there exists $\overline{\sigma }=({\overline{\sigma }}^{\mathcal{G}},{\overline{\sigma }}^{\Psi },{\overline{\sigma }}^{\Theta },{\overline{\tau }}^{\Theta },{\overline{\sigma }}^{\mathtt{K}},{\overline{\tau }}^{\mathtt{K}},{\overline{\tau }}^{\mathtt{L}})\in {\mathbb{R}}^r\times {\mathbb{R}}^p\times {\mathbb{R}}^{2\ell }\times {\mathbb{R}}^{3m}$, such that $(\overline{\mathfrak{a}},\overline{\sigma })$ becomes a robust local weak Pareto solution of the WRD. Moreover, the corresponding objective function values are equal.

Proof. 

Given that GS-ACQ holds as a robust local weak Pareto solution $\overline{\mathfrak{a}}$ of UNMPVC, then, in light of Theorem 3, there exists

$$\overline{\sigma }=({\overline{\sigma }}^{\mathcal{G}},{\overline{\sigma }}^{\Psi },{\overline{\sigma }}^{\Theta },{\overline{\tau }}^{\Theta },{\overline{\sigma }}^{\mathtt{K}},{\overline{\tau }}^{\mathtt{K}},{\overline{\tau }}^{\mathtt{L}})\in {\mathbb{R}}^r\times {\mathbb{R}}^p\times {\mathbb{R}}^{2\ell }\times {\mathbb{R}}^{3m},$$

such that $\overline{\mathfrak{a}}$ satisfies RS-stationary conditions of UNMPVC. Thus, there exist ${\overline{\xi }}_t^1\in co{\partial }^{*}{\mathcal{G}}_t\left(\overline{\mathfrak{a}}\right),$ $t\in \mathbb{T}$, ${\overline{\xi }}_j^2\in co{\partial }^{*}{\Psi }_j(\overline{\mathfrak{a}},$ ${\overline{\mathfrak{u}}}_j),$ $j\in {\mathfrak{T}}_{\Psi },$ ${\overline{\mathfrak{u}}}_j\in {\Omega }_j\left(\overline{\mathfrak{a}}\right)$, ${\overline{\xi }}_{\mathfrak{s}}^3\in co{\partial }^{*}{\Theta }_{\mathfrak{s}}(\overline{\mathfrak{a}},$ ${\overline{\mathfrak{u}}}_{\mathfrak{s}}),$ ${\overline{\xi }}_{\mathfrak{s}}^4\in co{\partial }^{*}(-{\Theta }_{\mathfrak{s}})(\overline{\mathfrak{a}},{\overline{\mathfrak{u}}}_{\mathfrak{s}}),$ $\mathfrak{s}\in \mathbb{S},{\overline{\mathfrak{u}}}_{\mathfrak{s}}\in {\Omega }_{\mathfrak{s}}\left(\overline{\mathfrak{a}}\right)$,${\overline{\xi }}_{\mathfrak{i}}^5\in co{\partial }^{*}(-{\mathtt{K}}_{\mathfrak{i}})(\overline{\mathfrak{a}},{\overline{\mathfrak{u}}}_{\mathfrak{i}}),$ $\mathfrak{i}\in \mathtt{I}$,${\overline{\xi }}_{\mathfrak{i}}^6\in co{\partial }^{*}{\mathtt{K}}_{\mathfrak{i}}(\overline{\mathfrak{a}},{\overline{\mathfrak{u}}}_{\mathfrak{i}}),\mathfrak{i}\in \mathtt{I},$ and ${\overline{\xi }}_{\mathfrak{i}}^7\in co{\partial }^{*}{\mathtt{L}}_{\mathfrak{i}}(\overline{\mathfrak{a}},{\overline{\mathfrak{u}}}_{\mathfrak{i}}),\mathfrak{i}\in \mathtt{I},{\overline{\mathfrak{u}}}_{\mathfrak{i}}\in {\Omega }_{\mathfrak{i}}\left(\overline{\mathfrak{a}}\right)$, such that

$$\sum _{t\in \mathbb{T}}{\overline{\sigma }}_t^{\mathcal{G}}{\overline{\xi }}_t^1+\sum _{j\in {\mathfrak{T}}_{\Psi }}{\overline{\sigma }}_j^{\Psi }{\overline{\xi }}_j^2+\sum _{\mathfrak{s}\in \mathbb{S}}[{\overline{\sigma }}_{\mathfrak{s}}^{\Theta }{\overline{\xi }}_{\mathfrak{s}}^3+{\overline{\tau }}_{\mathfrak{s}}^{\Theta }{\overline{\xi }}_{\mathfrak{s}}^4]+\sum _{\mathfrak{i}\in \mathtt{I}}{\overline{\sigma }}_{\mathfrak{i}}^{\mathtt{K}}{\overline{\xi }}_{\mathfrak{i}}^5+\sum _{\mathfrak{i}\in \mathtt{I}}[{\overline{\tau }}_{\mathfrak{i}}^{\mathtt{K}}{\overline{\xi }}_{\mathfrak{i}}^6+{\overline{\tau }}_{\mathfrak{i}}^{\mathtt{L}}{\overline{\xi }}_{\mathfrak{i}}^7]=0,$$

$$\begin{array}{cc}& {\overline{\sigma }}_t^{\mathcal{G}}>0,t\in \mathbb{T},{\overline{\sigma }}_j^{\Psi }\ge 0,j\in {\mathfrak{T}}_{\Psi },{\overline{\sigma }}_{\mathfrak{s}}^{\Theta }\ge 0,{\overline{\tau }}_{\mathfrak{s}}^{\Theta }\ge 0,\mathfrak{s}\in \mathbb{S},{\overline{\sigma }}_{\mathfrak{i}}^{\mathtt{K}}\ge 0,{\overline{\tau }}_{\mathfrak{i}}^{\mathtt{K}}\ge 0,{\overline{\tau }}_{\mathfrak{i}}^{\mathtt{L}}\ge 0,\mathfrak{i}\in \mathtt{I}.\hfill \end{array}$$

Therefore, $(\overline{\mathfrak{a}},\overline{\sigma })\in {\mathbb{F}}_{\mathcal{G}}^{WRD}$. Theorem 5 then implies that for any $(\mathfrak{b},\sigma )\in {\mathbb{F}}_{\mathcal{G}}^{WRD}$, we have

$$\begin{array}{cc}\hfill \mathcal{G}\left(\overline{\mathfrak{a}}\right)\nprec \mathcal{G}\left(\mathfrak{b}\right)& +\sum _{j\in {\mathfrak{T}}_{\Psi }}{\sigma }_j^{\Psi }{\Psi }_j(\mathfrak{b},{\mathfrak{u}}_j)\hfill \\ & +\sum _{\mathfrak{s}\in \mathbb{S}}[{\sigma }_{\mathfrak{s}}^{\Theta }{\Theta }_{\mathfrak{s}}(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{s}})+{\tau }_{\mathfrak{s}}^{\Theta }(-{\Theta }_{\mathfrak{s}})(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{s}})]\hfill \\ & +\sum _{\mathfrak{i}\in \mathtt{I}}{\sigma }_{\mathfrak{i}}^{\mathtt{K}}(-{\mathtt{K}}_{\mathfrak{i}})(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{i}})+\sum _{\mathfrak{i}\in \mathtt{I}}[{\tau }_{\mathfrak{i}}^{\mathtt{K}}{\mathtt{K}}_{\mathfrak{i}}(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{i}})+{\tau }_{\mathfrak{i}}^{\mathtt{L}}{\mathtt{L}}_{\mathfrak{i}}(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{i}})].\hfill \end{array}$$

(33)

Given the feasibility conditions of UNMPVC and WRD, we have ${\Psi }_j(\overline{\mathfrak{a}},{\overline{\mathfrak{u}}}_j)=0,$ $j\in {\mathfrak{T}}_{\Psi },$ ${\Theta }_{\mathfrak{s}}(\overline{\mathfrak{a}},{\overline{\mathfrak{u}}}_{\mathfrak{s}})=0,\mathfrak{s}\in \mathbb{S},{\mathtt{K}}_{\mathfrak{i}}(\overline{\mathfrak{a}},{\overline{\mathfrak{u}}}_{\mathfrak{i}})=0,\forall \mathfrak{i}\in {\mathfrak{T}}_0,{\mathtt{L}}_{\mathfrak{i}}(\overline{\mathfrak{a}},{\overline{\mathfrak{u}}}_{\mathfrak{i}})=0,\forall \mathfrak{i}\in {\mathfrak{T}}_{+0}.$ Therefore, we have

$$\begin{array}{cc}\hfill \mathcal{G}\left(\overline{\mathfrak{a}}\right)=\mathcal{G}\left(\overline{\mathfrak{a}}\right)& +\sum _{j\in {\mathfrak{T}}_{\Psi }}{\overline{\sigma }}_j^{\Psi }{\Psi }_j(\overline{\mathfrak{a}},{\overline{\mathfrak{u}}}_j)+\sum _{\mathfrak{s}\in \mathbb{S}}[{\overline{\sigma }}_{\mathfrak{s}}^{\Theta }{\Theta }_{\mathfrak{s}}(\overline{\mathfrak{a}},{\overline{\mathfrak{u}}}_{\mathfrak{s}})+{\overline{\tau }}_{\mathfrak{s}}^{\Theta }(-{\Theta }_{\mathfrak{s}})(\overline{\mathfrak{a}},{\overline{\mathfrak{u}}}_{\mathfrak{s}})]\hfill \\ & +\sum _{\mathfrak{i}\in \mathtt{I}}{\overline{\sigma }}_{\mathfrak{i}}^{\mathtt{K}}(-{\mathtt{K}}_{\mathfrak{i}})(\overline{\mathfrak{a}},{\overline{\mathfrak{u}}}_{\mathfrak{i}})+\sum _{\mathfrak{i}\in \mathtt{I}}[{\overline{\tau }}_{\mathfrak{i}}^{\mathtt{K}}{\mathtt{K}}_{\mathfrak{i}}(\overline{\mathfrak{a}},{\overline{\mathfrak{u}}}_{\mathfrak{i}})+{\overline{\tau }}_{\mathfrak{i}}^{\mathtt{L}}{\mathtt{L}}_{\mathfrak{i}}(\overline{\mathfrak{a}},{\overline{\mathfrak{u}}}_{\mathfrak{i}})].\hfill \end{array}$$

(34)

From Equations (33) and (34), we have

$$\begin{array}{cc}\hfill \mathcal{G}\left(\overline{\mathfrak{a}}\right)& +\sum _{j\in {\mathfrak{T}}_{\Psi }}{\overline{\sigma }}_j^{\Psi }{\Psi }_j(\overline{\mathfrak{a}},{\overline{\mathfrak{u}}}_j)+\sum _{\mathfrak{s}\in \mathbb{S}}[{\overline{\sigma }}_{\mathfrak{s}}^{\Theta }{\Theta }_{\mathfrak{s}}(\overline{\mathfrak{a}},{\overline{\mathfrak{u}}}_{\mathfrak{s}})+{\overline{\tau }}_{\mathfrak{s}}^{\Theta }(-{\Theta }_{\mathfrak{s}})(\overline{\mathfrak{a}},{\overline{\mathfrak{u}}}_{\mathfrak{s}})]\hfill \\ & +\sum _{\mathfrak{i}\in \mathtt{I}}{\overline{\sigma }}_{\mathfrak{i}}^{\mathtt{K}}(-{\mathtt{K}}_{\mathfrak{i}})(\overline{\mathfrak{a}},{\overline{\mathfrak{u}}}_{\mathfrak{i}})+\sum _{\mathfrak{i}\in \mathtt{I}}[{\overline{\tau }}_{\mathfrak{i}}^{\mathtt{K}}{\mathtt{K}}_{\mathfrak{i}}(\overline{\mathfrak{a}},{\overline{\mathfrak{u}}}_{\mathfrak{i}})+{\overline{\tau }}_{\mathfrak{i}}^{\mathtt{L}}{\mathtt{L}}_{\mathfrak{i}}(\overline{\mathfrak{a}},{\overline{\mathfrak{u}}}_{\mathfrak{i}})]\hfill \\ & \nprec \mathcal{G}\left(\mathfrak{b}\right)+\sum _{\mathfrak{i}\in {\mathfrak{T}}_{\Psi }}{\sigma }_j^{\Psi }{\Psi }_j(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{i}})+\sum _{\mathfrak{s}\in \mathbb{S}}[{\sigma }_{\mathfrak{s}}^{\Theta }{\Theta }_{\mathfrak{s}}(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{s}})+{\tau }_{\mathfrak{s}}^{\Theta }(-{\Theta }_{\mathfrak{s}})(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{s}})]\hfill \\ & +\sum _{\mathfrak{i}\in \mathtt{I}}{\sigma }_{\mathfrak{i}}^{\mathtt{K}}(-{\mathtt{K}}_{\mathfrak{i}})(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{s}})+\sum _{\mathfrak{i}\in \mathtt{I}}[{\tau }_{\mathfrak{i}}^{\mathtt{K}}{\mathtt{K}}_{\mathfrak{i}}(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{i}})+{\tau }_{\mathfrak{i}}^{\mathtt{L}}{\mathtt{L}}_{\mathfrak{i}}(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{i}})].\hfill \end{array}$$

Therefore, $(\overline{\mathfrak{a}},\overline{\sigma })$ is a robust local weak Pareto solution of WRD. Moreover, the corresponding objective function values are equal. □

Now, we establish the strict converse duality theorem relating WRD and UNMPVC.

Theorem 7. 

Let $(\widehat{\mathfrak{b}},\widehat{\sigma })$ be the global weak Pareto solution of WRD and $\overline{\mathfrak{a}}$ be a robust local weak Pareto solution of UNMPVC. Moreover, we suppose that the hypotheses of strong duality theorem hold at $\widehat{\mathfrak{b}}$, such that ${\mathcal{G}}_t,t\in \mathbb{T}$ are strictly ${\partial }^{*}$-convex at $\widehat{\mathfrak{b}}$ and ${\mathfrak{T}}_{0-}^{\mathtt{K}}\cup {\mathfrak{T}}_{0+}^{\mathtt{K}}\cup {\mathfrak{T}}_{+0}^{\mathtt{L}}=\varnothing$. Then $\overline{\mathfrak{a}}=\widehat{\mathfrak{b}}$.

Proof. 

Suppose that $\overline{\mathfrak{a}}\ne \widehat{\mathfrak{b}}$. Theorem 6 then stipulates that

$$\begin{array}{cc}\hfill \mathcal{G}\left(\overline{\mathfrak{a}}\right)=\mathcal{G}\left(\overline{\mathfrak{a}}\right)& +\sum _{j\in {\mathfrak{T}}_{\Psi }}{\overline{\sigma }}_j^{\Psi }{\Psi }_j(\overline{\mathfrak{a}},{\overline{\mathfrak{u}}}_j)e+\sum _{\mathfrak{s}\in \mathbb{S}}[{\overline{\sigma }}_{\mathfrak{s}}^{\Theta }{\Theta }_{\mathfrak{s}}(\overline{\mathfrak{a}},{\overline{\mathfrak{u}}}_{\mathfrak{s}})+{\overline{\tau }}_{\mathfrak{s}}^{\Theta }(-{\Theta }_{\mathfrak{s}})(\overline{\mathfrak{a}},{\overline{\mathfrak{u}}}_{\mathfrak{s}})]e\hfill \\ & +\sum _{\mathfrak{i}\in \mathtt{I}}{\overline{\sigma }}_{\mathfrak{i}}^{\mathtt{K}}(-{\mathtt{K}}_{\mathfrak{i}})(\overline{\mathfrak{a}},{\overline{\mathfrak{u}}}_{\mathfrak{i}})e+\sum _{\mathfrak{i}\in \mathtt{I}}[{\overline{\tau }}_{\mathfrak{i}}^{\mathtt{K}}{\mathtt{K}}_{\mathfrak{i}}(\overline{\mathfrak{a}},{\overline{\mathfrak{u}}}_{\mathfrak{i}})+{\overline{\tau }}_{\mathfrak{i}}^{\mathtt{L}}{\mathtt{L}}_{\mathfrak{i}}(\overline{\mathfrak{a}},{\overline{\mathfrak{u}}}_{\mathfrak{i}})]e.\hfill \\ \hfill & =\mathcal{G}\left(\widehat{\mathfrak{b}}\right)+\sum _{j\in {\mathfrak{T}}_{\Psi }}{\widehat{\sigma }}_j^{\Psi }{\Psi }_j(\widehat{\mathfrak{b}},{\mathfrak{u}}_j)e+\sum _{\mathfrak{s}\in \mathbb{S}}[{\widehat{\sigma }}_{\mathfrak{s}}^{\Theta }{\Theta }_{\mathfrak{s}}(\widehat{\mathfrak{b}},{\mathfrak{u}}_{\mathfrak{s}})+{\widehat{\tau }}_{\mathfrak{s}}^{\Theta }(-{\Theta }_{\mathfrak{s}})(\widehat{\mathfrak{b}},{\mathfrak{u}}_{\mathfrak{s}})]e\hfill \\ & +\sum _{\mathfrak{i}\in \mathtt{I}}{\widehat{\sigma }}_{\mathfrak{i}}^{\mathtt{K}}(-{\mathtt{K}}_{\mathfrak{i}})(\widehat{\mathfrak{b}},{\mathfrak{u}}_{\mathfrak{i}})e+\sum _{\mathfrak{i}\in \mathtt{I}}[{\widehat{\tau }}_{\mathfrak{i}}^{\mathtt{K}}{\mathtt{K}}_{\mathfrak{i}}(\widehat{\mathfrak{b}},{\mathfrak{u}}_{\mathfrak{i}})+{\widehat{\tau }}_{\mathfrak{i}}^{\mathtt{L}}{\mathtt{L}}_{\mathfrak{i}}(\widehat{\mathfrak{b}},{\mathfrak{u}}_{\mathfrak{i}})]e.\hfill \end{array}$$

(35)

Multiplying (35) by ${\widehat{\sigma }}_t^{\mathcal{G}}>0,t\in \mathbb{T}$, and since ${\left({\sigma }^{\mathcal{G}}\right)}^{T}e=1$, we get

$$\begin{array}{cc}\hfill \sum _{t\in \mathbb{T}}{\widehat{\sigma }}_t^{\mathcal{G}}{\mathcal{G}}_t\left(\mathfrak{a}\right)& =\sum _{t\in \mathbb{T}}{\widehat{\sigma }}_t^{\mathcal{G}}{\mathcal{G}}_t\left(\widehat{\mathfrak{b}}\right)+\sum _{j\in {\mathfrak{T}}_{\Psi }}{\widehat{\sigma }}_j^{\Psi }{\Psi }_j(\widehat{\mathfrak{b}},{\mathfrak{u}}_j)\hfill \\ & +\sum _{\mathfrak{s}\in \mathbb{S}}{\widehat{\sigma }}_{\mathfrak{s}}^{\Theta }{\Theta }_{\mathfrak{s}}(\widehat{\mathfrak{b}},{\mathfrak{u}}_{\mathfrak{s}})+\sum _{\mathfrak{s}\in \mathbb{S}}{\tau }_{\mathfrak{s}}^{\Theta }(-{\Theta }_{\mathfrak{s}})(\widehat{\mathfrak{b}},{\mathfrak{u}}_{\mathfrak{s}})\hfill \\ & +\sum _{\mathfrak{i}\in \mathtt{I}}{\widehat{\sigma }}_{\mathfrak{i}}^{\mathtt{K}}(-{\mathtt{K}}_{\mathfrak{i}})(\widehat{\mathfrak{b}},{\mathfrak{u}}_{\mathfrak{i}})\hfill \\ & +\sum _{\mathfrak{i}\in \mathtt{I}}[{\widehat{\sigma }}_{\mathfrak{i}}^{\mathtt{K}}{\mathtt{K}}_{\mathfrak{i}}(\widehat{\mathfrak{b}},{\mathfrak{u}}_{\mathfrak{i}})+{\widehat{\sigma }}_{\mathfrak{i}}^{\mathtt{L}}{\mathtt{L}}_{\mathfrak{i}}(\widehat{\mathfrak{b}},{\mathfrak{u}}_{\mathfrak{i}})].\hfill \end{array}$$

(36)

In view of the fact that ${\mathcal{G}}_t,t\in \mathbb{T}$ are strictly ${\partial }^{*}$-convex at $\widehat{\mathfrak{b}}$, we infer that

$${\mathcal{G}}_t\left(\overline{\mathfrak{a}}\right)-{\mathcal{G}}_t\left(\widehat{\mathfrak{b}}\right)>⟨{\xi }_t^1,\overline{\mathfrak{a}}-\widehat{\mathfrak{b}}⟩,\forall {\xi }_t^1\in {\partial }^{*}\mathcal{G}\left(\widehat{\mathfrak{b}}\right),\forall t\in \mathbb{T}.$$

(37)

In a similar manner, by employing the ${\partial }^{*}$-convexity assumptions of ${\Psi }_j(·,{\mathfrak{u}}_j),$ $j\in {\mathfrak{T}}_{\Psi },$ $\pm {\Theta }_{\mathfrak{s}}(·,{\mathfrak{u}}_{\mathfrak{s}}),$ $\mathfrak{s}\in \mathbb{S},$ $-{\mathtt{K}}_{\mathfrak{i}}(·,{\mathfrak{u}}_{\mathfrak{i}}),$ $\mathfrak{i}\in {\mathfrak{T}}_{0+}\cup {\mathfrak{T}}_{0-}\cup {\mathfrak{T}}_{00},$ ${\mathtt{K}}_{\mathfrak{i}}(·,{\mathfrak{u}}_{\mathfrak{i}}),\mathfrak{i}\in {\mathfrak{T}}_{0+},$ ${\mathtt{L}}_{\mathfrak{i}}(·,{\mathfrak{u}}_{\mathfrak{i}}),\mathfrak{i}\in {\mathfrak{T}}_{+0}$ at $\widehat{\mathfrak{b}}$, we obtain the subsequent inequalities:

$${\Psi }_j(\overline{\mathfrak{a}},{\mathfrak{u}}_j)-{\Psi }_j(\widehat{\mathfrak{b}},{\mathfrak{u}}_j)\ge ⟨{\xi }_j^2,\overline{\mathfrak{a}}-\widehat{\mathfrak{b}}⟩,\forall {\xi }_j^2\in {\partial }^{*}{\Psi }_j(\widehat{\mathfrak{b}},,{\mathfrak{u}}_j),\forall j\in {\mathfrak{T}}_{\Psi },$$

(38)

$${\Theta }_{\mathfrak{s}}(\overline{\mathfrak{a}},{\mathfrak{u}}_{\mathfrak{s}})-{\Theta }_{\mathfrak{s}}(\widehat{\mathfrak{b}},{\mathfrak{u}}_{\mathfrak{s}})\ge ⟨{\xi }_{\mathfrak{s}}^3,\overline{\mathfrak{a}}-\widehat{\mathfrak{b}}⟩,\forall {\xi }_{\mathfrak{s}}^3\in {\partial }^{*}{\Theta }_{\mathfrak{s}}(\widehat{\mathfrak{b}},{\mathfrak{u}}_{\mathfrak{s}}),\forall \mathfrak{s}\in \mathbb{S},$$

(39)

$$\begin{array}{cc}\hfill (-{\Theta }_{\mathfrak{s}})(\overline{\mathfrak{a}},{\mathfrak{u}}_{\mathfrak{s}})-(-{\Theta }_{\mathfrak{s}})(\widehat{\mathfrak{b}},{\mathfrak{u}}_{\mathfrak{s}})\ge ⟨{\xi }_{\mathfrak{s}}^4,\overline{\mathfrak{a}}-\widehat{\mathfrak{b}}⟩,\forall {\xi }_{\mathfrak{s}}^4\in & {\partial }^{*}(-{\Theta }_{\mathfrak{s}})(\widehat{\mathfrak{b}},{\mathfrak{u}}_{\mathfrak{s}}),\hfill \\ & \forall \mathfrak{s}\in \mathbb{S},\hfill \end{array}$$

(40)

$$\begin{array}{cc}\hfill (-{\mathtt{K}}_{\mathfrak{i}})(\overline{\mathfrak{a}},{\mathfrak{u}}_{\mathfrak{i}})-(-{\mathtt{K}}_{\mathfrak{i}})(\widehat{\mathfrak{b}},{\mathfrak{u}}_{\mathfrak{i}})\ge ⟨{\xi }_{\mathfrak{i}}^5,\overline{\mathfrak{a}}-\widehat{\mathfrak{b}}⟩,& \forall {\xi }_{\mathfrak{i}}^5\in {\partial }^{*}(-{\mathtt{K}}_{\mathfrak{i}})(\widehat{\mathfrak{b}},{\mathfrak{u}}_{\mathfrak{i}}),\hfill \\ & \forall \mathfrak{i}\in {\mathfrak{T}}_{0+}\cup {\mathfrak{T}}_{0-}\cup {\mathfrak{T}}_{00},\hfill \end{array}$$

(41)

$${\mathtt{K}}_{\mathfrak{i}}(\overline{\mathfrak{a}},{\mathfrak{u}}_{\mathfrak{i}})-{\mathtt{K}}_{\mathfrak{i}}(\widehat{\mathfrak{b}},{\mathfrak{u}}_{\mathfrak{i}})\ge ⟨{\xi }_{\mathfrak{i}}^6,\overline{\mathfrak{a}}-\widehat{\mathfrak{b}}⟩,\forall {\xi }_{\mathfrak{i}}^6\in {\partial }^{*}{\mathtt{L}}_{\mathfrak{i}}(\widehat{\mathfrak{b}},{\mathfrak{u}}_{\mathfrak{i}}),\forall \mathfrak{i}\in {\mathfrak{T}}_{0+}.$$

(42)

$${\mathtt{L}}_{\mathfrak{i}}(\overline{\mathfrak{a}},{\mathfrak{u}}_{\mathfrak{i}})-{\mathtt{L}}_{\mathfrak{i}}(\widehat{\mathfrak{b}},{\mathfrak{u}}_{\mathfrak{i}})\ge ⟨{\xi }_{\mathfrak{i}}^7,\overline{\mathfrak{a}}-\widehat{\mathfrak{b}}⟩,\forall {\xi }_{\mathfrak{i}}^7\in {\partial }^{*}{\mathtt{L}}_{\mathfrak{i}}(\widehat{\mathfrak{b}},{\mathfrak{u}}_{\mathfrak{i}}),\forall \mathfrak{i}\in {\mathfrak{T}}_{+0}.$$

(43)

If ${\mathfrak{T}}_{0-}^{\mathtt{K}}\cup {\mathfrak{T}}_{0+}^{\mathtt{K}}\cup {\mathfrak{T}}_{+0}^{\mathtt{L}}=\varnothing$, then multiplying (37)–(43) by ${\widehat{\sigma }}_t^{\mathcal{G}}>0,t\in \mathbb{T}$, ${\widehat{\sigma }}_j^{\Psi }\ge 0,j\in {\mathfrak{T}}_{\Psi }$, ${\widehat{\sigma }}_{\mathfrak{s}}^{\Theta }\ge 0,\mathfrak{s}\in \mathbb{S}$, ${\widehat{\tau }}_{\mathfrak{s}}^{\Theta }\ge 0,\mathfrak{s}\in \mathbb{S}$, ${\widehat{\sigma }}_{\mathfrak{i}}^{\mathtt{K}}>0,\mathfrak{i}\in {\mathfrak{T}}_{0+}\cup {\mathfrak{T}}_{0-}\cup {\mathfrak{T}}_{00}$, ${\widehat{\tau }}_{\mathfrak{i}}^{\mathtt{K}}>0,\mathfrak{i}\in {\mathfrak{T}}_{0+}$, ${\widehat{\tau }}_{\mathfrak{i}}^{\mathtt{L}}>0,$ $\mathfrak{i}\in {\mathfrak{T}}_{+0}$, respectively, and then by summing these inequalities, we obtain the following:

$$\begin{array}{cc}\hfill \sum _{t\in \mathbb{T}}{\widehat{\sigma }}_t^{\mathcal{G}}{\mathcal{G}}_t\left(\overline{\mathfrak{a}}\right)& -\sum _{t\in \mathbb{T}}{\widehat{\sigma }}_t^{\mathcal{G}}{\mathcal{G}}_t\left(\widehat{\mathfrak{b}}\right)+\sum _{j\in {\mathfrak{T}}_{\Psi }}{\widehat{\sigma }}_j^{\Psi }{\Psi }_j(\overline{\mathfrak{a}},{\mathfrak{u}}_j)-\sum _{j\in {\mathfrak{T}}_{\Psi }}{\widehat{\sigma }}_j^{\Psi }{\Psi }_j(\widehat{\mathfrak{b}},{\mathfrak{u}}_j)\hfill \\ & +\sum _{\mathfrak{s}\in \mathbb{S}}{\widehat{\sigma }}_{\mathfrak{s}}^{\Theta }{\Theta }_{\mathfrak{s}}(\overline{\mathfrak{a}},{\mathfrak{u}}_j)-\sum _{\mathfrak{s}\in \mathbb{S}}{\widehat{\sigma }}_{\mathfrak{s}}^{\Theta }{\Theta }_{\mathfrak{s}}(\widehat{\mathfrak{b}},{\mathfrak{u}}_{\mathfrak{s}})+\sum _{\mathfrak{s}\in \mathbb{S}}{\widehat{\tau }}_{\mathfrak{s}}^{\Theta }(-{\Theta }_{\mathfrak{s}})(\overline{\mathfrak{a}},{\mathfrak{u}}_{\mathfrak{s}})\hfill \\ & -\sum _{\mathfrak{s}\in \mathbb{S}}{\widehat{\tau }}_{\mathfrak{s}}^{\Theta }(-{\Theta }_{\mathfrak{s}})(\widehat{\mathfrak{b}},{\widehat{w}}_{\mathfrak{s}})+\sum _{\mathfrak{i}\in \mathtt{I}}{\widehat{\sigma }}_{\mathfrak{i}}^{\mathtt{K}}(-{\mathtt{K}}_{\mathfrak{i}})(\overline{\mathfrak{a}},{\mathfrak{u}}_{\mathfrak{i}})-\sum _{\mathfrak{i}\in \mathtt{I}}{\widehat{\sigma }}_{\mathfrak{i}}^{\mathtt{K}}(-{\mathtt{K}}_{\mathfrak{i}})(\widehat{\mathfrak{b}},{\mathfrak{u}}_{\mathfrak{i}})\hfill \\ \hfill & +\sum _{\mathfrak{i}\in \mathtt{I}}{\widehat{\tau }}_{\mathfrak{i}}^{\mathtt{K}}{\mathtt{K}}_{\mathfrak{i}}(\overline{\mathfrak{a}},{\mathfrak{u}}_{\mathfrak{i}})-\sum _{\mathfrak{i}\in \mathtt{I}}{\widehat{\tau }}_{\mathfrak{i}}^{\mathtt{K}}{\mathtt{K}}_{\mathfrak{i}}(\widehat{\mathfrak{b}},{\mathfrak{u}}_{\mathfrak{i}})+\sum _{\mathfrak{i}\in \mathtt{I}}{\widehat{\tau }}_{\mathfrak{i}}^{\mathtt{L}}{\mathtt{L}}_{\mathfrak{i}}(\overline{\mathfrak{a}},{\mathfrak{u}}_{\mathfrak{i}})\hfill \\ & -\sum _{\mathfrak{i}\in \mathtt{I}}{\widehat{\tau }}_{\mathfrak{i}}^{\mathtt{L}}{\mathtt{L}}_{\mathfrak{i}}(\widehat{\mathfrak{b}},{\mathfrak{u}}_{\mathfrak{i}})>⟨\sum _{t\in \mathbb{T}}{\widehat{\sigma }}_t^{\mathcal{G}}{\xi }_t^1+\sum _{j\in {\mathfrak{T}}_{\Psi }}{\widehat{\sigma }}_j^{\Psi }{\xi }_j^2+\sum _{\mathfrak{s}\in \mathbb{S}}[{\widehat{\sigma }}_{\mathfrak{s}}^{\Theta }{\xi }_{\mathfrak{s}}^3+{\widehat{\tau }}_{\mathfrak{s}}^{\Theta }{\xi }_{\mathfrak{s}}^4]\hfill \\ & +\sum _{\mathfrak{i}\in \mathtt{I}}{\widehat{\sigma }}_{\mathfrak{i}}^{\mathtt{K}}{\xi }_{\mathfrak{i}}^5+\sum _{\mathfrak{i}\in \mathtt{I}}[{\widehat{\tau }}_{\mathfrak{i}}^{\mathtt{K}}{\xi }_{\mathfrak{i}}^6+{\widehat{\sigma }}_{\mathfrak{i}}^{\mathtt{L}}{\xi }_{\mathfrak{i}}^7],\overline{\mathfrak{a}}-\widehat{\mathfrak{b}}⟩.\hfill \end{array}$$

Given the feasibility conditions of $\widehat{\mathfrak{b}}\in {\mathbb{F}}_{\mathcal{G}}^{WRD}$, there exist ${\widehat{\xi }}_t^1\in co{\partial }^{*}{\mathcal{G}}_t\left(\widehat{\mathfrak{b}}\right),$ $t\in \mathbb{T}$, ${\widehat{\xi }}_j^2\in co{\partial }^{*}{\Psi }_j(\widehat{\mathfrak{b}},{\mathfrak{u}}_j),$ $j\in {\mathfrak{T}}_{\Psi }$, ${\widehat{\xi }}_{\mathfrak{s}}^3\in co{\partial }^{*}{\Theta }_{\mathfrak{s}}(\widehat{\mathfrak{b}},{\mathfrak{u}}_{\mathfrak{s}}),$ $\mathfrak{s}\in \mathbb{S}$, ${\widehat{\xi }}_{\mathfrak{s}}^4\in co{\partial }^{*}(-{\Theta }_{\mathfrak{s}})(\widehat{\mathfrak{b}},{\mathfrak{u}}_{\mathfrak{s}}),$ $\mathfrak{s}\in \mathbb{S}$, ${\widehat{\xi }}_{\mathfrak{i}}^5\in co{\partial }^{*}(-{\mathtt{K}}_{\mathfrak{i}})(\widehat{\mathfrak{b}},{\mathfrak{u}}_{\mathfrak{i}}),$ $\mathfrak{i}\in \mathtt{I}$, ${\widehat{\xi }}_{\mathfrak{i}}^6\in co{\partial }^{*}{\mathtt{K}}_{\mathfrak{i}}(\widehat{\mathfrak{b}},{\mathfrak{u}}_{\mathfrak{i}}),$ $\mathfrak{i}\in \mathtt{I}$ and ${\widehat{\xi }}_{\mathfrak{i}}^7\in co{\partial }^{*}{\mathtt{L}}_{\mathfrak{i}}(\widehat{\mathfrak{b}},{\mathfrak{u}}_{\mathfrak{i}}),$ $\mathfrak{i}\in \mathtt{I}$, such that

$$\sum _{t\in \mathbb{T}}{\widehat{\sigma }}_t^{\mathcal{G}}{\widehat{\xi }}_t^1+\sum _{j\in {\mathfrak{T}}_{\Psi }}{\widehat{\sigma }}_j^{\Psi }{\widehat{\xi }}_j^2+\sum _{\mathfrak{s}\in \mathbb{S}}[{\widehat{\sigma }}_{\mathfrak{s}}^{\Theta }{\widehat{\xi }}_{\mathfrak{s}}^3+{\widehat{\tau }}_{\mathfrak{s}}^{\Theta }{\widehat{\xi }}_{\mathfrak{s}}^4]+\sum _{\mathfrak{i}\in \mathtt{I}}{\widehat{\sigma }}_{\mathfrak{i}}^{\mathtt{K}}{\widehat{\xi }}_{\mathfrak{i}}^5+\sum _{\mathfrak{i}\in \mathtt{I}}[{\widehat{\tau }}_{\mathfrak{i}}^{\mathtt{K}}{\widehat{\xi }}_{\mathfrak{i}}^6+{\widehat{\tau }}_{\mathfrak{i}}^{\mathtt{L}}{\widehat{\xi }}_{\mathfrak{i}}^6]=0.$$

Therefore,

$$\begin{array}{cc}\hfill \sum _{t\in \mathbb{T}}{\widehat{\sigma }}_t^{\mathcal{G}}{\mathcal{G}}_t\left(\overline{\mathfrak{a}}\right)& -\sum _{t\in \mathbb{T}}{\widehat{\sigma }}_t^{\mathcal{G}}{\mathcal{G}}_t\left(\widehat{\mathfrak{b}}\right)+\sum _{j\in {\mathfrak{T}}_{\Psi }}{\widehat{\sigma }}_j^{\Psi }{\Psi }_j(\overline{\mathfrak{a}},{\mathfrak{u}}_j)-\sum _{j\in {\mathfrak{T}}_{\Psi }}{\widehat{\sigma }}_j^{\Psi }{\Psi }_j(\widehat{\mathfrak{b}},{\mathfrak{u}}_j)\hfill \\ & +\sum _{\mathfrak{s}\in \mathbb{S}}{\widehat{\sigma }}_{\mathfrak{s}}^{\Theta }{\Theta }_{\mathfrak{s}}(\overline{\mathfrak{a}},{\mathfrak{u}}_{\mathfrak{s}})-\sum _{\mathfrak{s}\in \mathbb{S}}{\widehat{\sigma }}_{\mathfrak{s}}^{\Theta }{\Theta }_{\mathfrak{s}}(\widehat{\mathfrak{b}},{\mathfrak{u}}_{\mathfrak{s}})+\sum _{\mathfrak{s}\in \mathbb{S}}{\widehat{\tau }}_{\mathfrak{s}}^{\Theta }(-{\Theta }_{\mathfrak{s}})(\overline{\mathfrak{a}},{\mathfrak{u}}_{\mathfrak{s}})\hfill \\ & -\sum _{\mathfrak{s}\in \mathbb{S}}{\widehat{\tau }}_{\mathfrak{s}}^{\Theta }(-{\Theta }_{\mathfrak{s}})(\widehat{\mathfrak{b}},{\mathfrak{u}}_{\mathfrak{s}})+\sum _{\mathfrak{i}\in \mathtt{I}}{\widehat{\sigma }}_{\mathfrak{i}}^{\mathtt{K}}(-{\mathtt{K}}_{\mathfrak{i}})(\overline{\mathfrak{a}},{\mathfrak{u}}_{\mathfrak{i}})-\sum _{\mathfrak{i}\in \mathtt{I}}{\widehat{\sigma }}_{\mathfrak{i}}^{\mathtt{K}}(-{\mathtt{K}}_{\mathfrak{i}})(\widehat{\mathfrak{b}},{\mathfrak{u}}_{\mathfrak{i}})\hfill \\ & +\sum _{\mathfrak{i}\in \mathtt{I}}{\widehat{\tau }}_{\mathfrak{i}}^{\mathtt{K}}{\mathtt{K}}_{\mathfrak{i}}(\overline{\mathfrak{a}},{\mathfrak{u}}_{\mathfrak{i}})-\sum _{\mathfrak{i}\in \mathtt{I}}{\widehat{\tau }}_{\mathfrak{i}}^{\mathtt{K}}{\mathtt{K}}_{\mathfrak{i}}(\widehat{\mathfrak{b}},{\mathfrak{u}}_{\mathfrak{i}})+\sum _{\mathfrak{i}\in \mathtt{I}}{\widehat{\tau }}_{\mathfrak{i}}^{\mathtt{L}}{\mathtt{L}}_{\mathfrak{i}}(\overline{\mathfrak{a}},{\mathfrak{u}}_{\mathfrak{i}})\hfill \\ & -\sum _{\mathfrak{i}\in \mathtt{I}}{\widehat{\tau }}_{\mathfrak{i}}^{\mathtt{L}}{\mathtt{L}}_{\mathfrak{i}}(\widehat{\mathfrak{b}},{\mathfrak{u}}_{\mathfrak{i}})\ge 0.\hfill \end{array}$$

Since $\mathfrak{a}\in {\mathbb{F}}_{\mathcal{G}}$, we have ${\Psi }_j(\mathfrak{a},{\mathfrak{u}}_j)\le 0,j\in {\mathfrak{T}}_{\Psi },{\Theta }_{\mathfrak{s}}(\mathfrak{a},{\mathfrak{u}}_{\mathfrak{s}})=0,\mathfrak{s}\in \mathbb{S},-{\mathtt{K}}_{\mathfrak{i}}(\mathfrak{a},{\mathfrak{u}}_{\mathfrak{i}})<0,$ $\mathfrak{i}\in {\mathfrak{T}}_{+},$ ${\mathtt{K}}_{\mathfrak{i}}(\mathfrak{a},{\mathfrak{u}}_{\mathfrak{i}})=0,\mathfrak{i}\in {\mathfrak{T}}_0,{\mathtt{L}}_{\mathfrak{i}}(\mathfrak{a},{\mathfrak{u}}_{\mathfrak{i}})>0,\mathfrak{i}\in {\mathfrak{T}}_{0+},{\mathtt{L}}_{\mathfrak{i}}(\mathfrak{a},{\mathfrak{u}}_{\mathfrak{i}})=0,\mathfrak{i}\in {\mathfrak{T}}_{00}\cup {\mathfrak{T}}_{+0},$ ${\mathtt{L}}_{\mathfrak{i}}(\mathfrak{a},{\mathfrak{u}}_{\mathfrak{i}})=0,$ $\mathfrak{i}\in {\mathfrak{T}}_{0-}\cup {\mathfrak{T}}_{+-}.$ Therefore,

$$\begin{array}{cc}\hfill \sum _{t\in \mathbb{T}}{\widehat{\sigma }}_t^{\mathcal{G}}{\mathcal{G}}_t\left(\mathfrak{a}\right)& >\sum _{t\in \mathbb{T}}{\widehat{\sigma }}_t^{\mathcal{G}}{\mathcal{G}}_t\left(\mathfrak{b}\right)+\sum _{j\in {\mathfrak{T}}_{\Psi }}{\widehat{\sigma }}_j^{\Psi }{\Psi }_j(\mathfrak{b},{\mathfrak{u}}_j)\hfill \\ & +\sum _{\mathfrak{s}\in \mathbb{S}}[{\widehat{\sigma }}_{\mathfrak{s}}^{\Theta }{\Theta }_{\mathfrak{s}}(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{s}})+{\tau }_{\mathfrak{s}}^{\Theta }(-{\Theta }_{\mathfrak{s}})(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{s}})]\hfill \\ & +\sum _{\mathfrak{i}\in \mathtt{I}}{\widehat{\sigma }}_{\mathfrak{i}}^{\mathtt{K}}(-{\mathtt{K}}_{\mathfrak{i}})(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{i}})\hfill \\ & +\sum _{\mathfrak{i}\in \mathtt{I}}[{\widehat{\sigma }}_{\mathfrak{i}}^{\mathtt{K}}{\mathtt{K}}_{\mathfrak{i}}(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{i}})+{\widehat{\sigma }}_{\mathfrak{i}}^{\mathtt{L}}{\mathtt{L}}_{\mathfrak{i}}(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{i}})].\hfill \end{array}$$

This contradicts (36). Thus, $\overline{\mathfrak{a}}=\widehat{\mathfrak{b}}$. This concludes the proof. □

Remark 3. 

1 

It is worthwhile to note that the weak, strong, and strict converse duality theorems presented in this article for the Wolfe-type robust dual model generalize Theorem 3, Theorem 4, and Theorem 7, respectively, derived by Mishra et al. [45] for MPVC to a more general programming problem, UNMPVC.

2 

If all the functions are continuously differentiable and if ${\Omega }_j,j\in \mathbb{J},$ ${\Omega }_{\mathfrak{s}},\mathfrak{s}\in \mathbb{S}$, and ${\Omega }_{\mathfrak{i}},\mathfrak{i}\in \mathtt{I}$ are singleton sets, then the Theorem 5, Theorem 6, and Theorem 7 reduce to Theorem 3, Theorem 4 and Theorem 7, respectively, derived by Mishra et al. [45].

5. Mond–Weir-Type Robust Dual Model

This section deals with formulating a Mond–Weir-type robust dual model (MWRD) associated with UNMPVC. We then deduce various duality results, including weak, strong, and strict converse duality, revealing the intricate connections between UNMPVC and MWRD.

We present the MWRD corresponding to the problem, UNMPVC, as follows:

$$\begin{array}{cc}\hfill & \underset{(\mathfrak{b},\sigma )}{\mathrm{Maximize}}\mathcal{G}\left(\mathfrak{b}\right)=({\mathcal{G}}_1\left(\mathfrak{b}\right),\dots ,{\mathcal{G}}_r\left(\mathfrak{b}\right)),\hfill \\ \hfill & \mathrm{subject}\mathrm{to}:(\mathfrak{b},\sigma )\in {\mathbb{F}}_{\mathcal{G}}^{MW},\hfill \end{array}$$

where ${\mathbb{F}}_{\mathcal{G}}^{MW}$ represents the set of feasible elements of MWRD, which is defined as follows:

$$\begin{array}{cc}\hfill {\mathbb{F}}_{\mathcal{G}}^{MW}:=& \{\sigma =({\sigma }^{\mathcal{G}},{\sigma }^{\Psi },{\sigma }^{\Theta },{\tau }^{\Theta },{\sigma }^{\mathtt{K}},{\tau }^{\mathtt{K}},{\tau }^{\mathtt{L}})\in {\mathbb{R}}^r\times {\mathbb{R}}^p\times {\mathbb{R}}^{2\ell }\times {\mathbb{R}}^{3m},\hfill \\ \hfill & \mathfrak{b}\in {\mathbb{R}}^n:0\in \sum _{t\in \mathbb{T}}{\sigma }_t^{\mathcal{G}}co{\partial }^{*}{\mathcal{G}}_t\left(\mathfrak{b}\right)+\sum _{j\in {\mathfrak{T}}_{\Psi }}{\sigma }_j^{\Psi }co{\partial }^{*}{\Psi }_j(\mathfrak{b},{\mathfrak{u}}_j)\hfill \\ \hfill & +\sum _{\mathfrak{s}\in \mathbb{S}}[{\sigma }_{\mathfrak{s}}^{\Theta }co{\partial }^{*}{\Theta }_{\mathfrak{s}}(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{s}})+{\tau }_{\mathfrak{s}}^{\Theta }co{\partial }^{*}(-{\Theta }_{\mathfrak{s}})(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{s}})]\hfill \\ \hfill & +\sum _{\mathfrak{i}\in \mathtt{I}}{\sigma }_{\mathfrak{i}}^{\mathtt{K}}co{\partial }^{*}(-{\mathtt{K}}_{\mathfrak{i}})(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{i}})\hfill \\ \hfill & +\sum _{\mathfrak{i}\in \mathtt{I}}[{\tau }_{\mathfrak{i}}^{\mathtt{K}}co{\partial }^{*}{\mathtt{K}}_{\mathfrak{i}}(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{i}})+{\tau }_{\mathfrak{i}}^{\mathtt{L}}co{\partial }^{*}{\mathtt{L}}_{\mathfrak{i}}(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{i}})],\hfill \\ \hfill & {\Psi }_j(\mathfrak{b},{\mathfrak{u}}_j)\ge 0,j\in {\mathfrak{T}}_{\Psi },{\Theta }_{\mathfrak{s}}(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{s}})=0,\mathfrak{s}\in \mathbb{S},{\mathtt{K}}_{\mathfrak{i}}(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{i}})\le 0,\mathfrak{i}\in {\mathfrak{T}}_{+0},\hfill \\ & -{\mathtt{K}}_{\mathfrak{i}}(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{i}})\ge 0,\mathfrak{i}\in {\mathfrak{T}}_{0+}\cup {\mathfrak{T}}_{0-}\cup {\mathfrak{T}}_{00},{\mathtt{L}}_{\mathfrak{i}}(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{i}})\ge 0,\mathfrak{i}\in {\mathfrak{T}}_{+0},\hfill \\ \hfill & {\sigma }_t^{\mathcal{G}}>0,t\in \mathbb{T},{\sigma }_j^{\Psi }\ge 0,j\in {\mathfrak{T}}_{\Psi },{\sigma }_{\mathfrak{s}}^{\Theta }\ge 0,{\tau }_{\mathfrak{s}}^{\Theta }\ge 0,\mathfrak{s}\in \mathbb{S},\hfill \\ \hfill & {\sigma }_{\mathfrak{i}}^{\mathtt{K}}\ge 0,{\tau }_{\mathfrak{i}}^{\mathtt{K}}\ge 0,{\tau }_{\mathfrak{i}}^{\mathtt{L}}\ge 0,\mathfrak{i}\in \mathtt{I},\hfill \\ & {\sigma }_{{\mathfrak{T}}_{+0}\cup {\mathfrak{T}}_{+-}}^{\mathtt{K}}={\tau }_{{\mathfrak{T}}_{+0}\cup {\mathfrak{T}}_{+-}}^{\mathtt{K}}={\tau }_{{\mathfrak{T}}_{0+}\cup {\mathfrak{T}}_{+-}\cup {\mathfrak{T}}_{0-}}^{\mathtt{L}}=0,\hfill \\ \hfill & {\sigma }_i^{\mathtt{K}}-{\tau }_i^{\mathtt{K}}\ge 0,\mathfrak{i}\in {\mathfrak{T}}_{0-},{\tau }_{\mathfrak{i}}^{\mathtt{L}}=0,({\sigma }_{\mathfrak{i}}^{\mathtt{K}}-{\tau }_{\mathfrak{i}}^{\mathtt{K}})\ge 0,\forall \mathfrak{i}\in {\mathfrak{T}}_{00}\}.\hfill \end{array}$$

The subsequent theorem establishes a weak duality result connecting UNMPVC and MWRD.

Theorem 8 

(Weak Duality). Let $\mathfrak{a}\in {\mathbb{F}}_{\mathcal{G}}$ and $(\mathfrak{b},\sigma )\in {\mathbb{F}}_{\mathcal{G}}^{MW}$. We assume that ${\mathcal{G}}_t,t\in \mathbb{T},{\Psi }_j,$ $j\in {\mathfrak{T}}_{\Psi },\pm {\Theta }_{\mathfrak{s}}\mathfrak{s}\in \mathbb{S},-{\mathtt{K}}_{\mathfrak{i}},\mathfrak{i}\in {\mathfrak{T}}_{0+}\cup {\mathfrak{T}}_{0-}\cup {\mathfrak{T}}_{00},{\mathtt{K}}_{\mathfrak{i}},\mathfrak{i}\in {\mathfrak{T}}_{0+},{\mathtt{L}}_{\mathfrak{i}},\mathfrak{i}\in {\mathfrak{T}}_{+0}$ are ${\partial }^{*}$-convex at $\mathfrak{b}$ and admit bounded USRC. Assume also that ${\mathfrak{T}}_{0-}^{\mathtt{K}}\cup {\mathfrak{T}}_{0+}^{\mathtt{K}}\cup {\mathfrak{T}}_{+0}^{\mathtt{L}}=\varnothing$. Then,

$$\mathcal{G}\left(\mathfrak{a}\right)\nprec \mathcal{G}\left(\mathfrak{b}\right).$$

(44)

Proof. 

On the other hand, suppose that

$$\begin{array}{c}\hfill \mathcal{G}\left(\mathfrak{a}\right)\prec \mathcal{G}\left(\mathfrak{b}\right).\end{array}$$

(45)

Multiplying (45) by ${\sigma }_t^{\mathcal{G}}>0,t\in \mathbb{T}$, we obtain

$$\sum _{t\in \mathbb{T}}{\sigma }_t^{\mathcal{G}}{\mathcal{G}}_t\left(\mathfrak{a}\right)<\sum _{t\in \mathbb{T}}{\sigma }_t^{\mathcal{G}}{\mathcal{G}}_t\left(\mathfrak{b}\right).$$

(46)

Since ${\mathcal{G}}_t,t\in \mathbb{T}$ are ${\partial }^{*}$-convex at $\mathfrak{b}$, we have

$${\mathcal{G}}_t\left(\mathfrak{a}\right)-{\mathcal{G}}_t\left(\mathfrak{b}\right)\ge ⟨{\xi }_t^1,\mathfrak{a}-\mathfrak{b}⟩,\forall {\xi }_t^1\in {\partial }^{*}{\mathcal{G}}_t\left(\mathfrak{b}\right),\forall t\in \mathbb{T}.$$

(47)

Similarly, according to the ${\partial }^{*}$-convexity of ${\Psi }_j(·,{\mathfrak{u}}_j)$ $j\in {\mathfrak{T}}_{\Psi },$ $\pm {\Theta }_{\mathfrak{s}}(·,{\mathfrak{u}}_{\mathfrak{s}}),$ $\mathfrak{s}\in \mathbb{S},$ $-{\mathtt{K}}_{\mathfrak{i}}(·,{\mathfrak{u}}_{\mathfrak{i}}),$ $\mathfrak{i}\in {\mathfrak{T}}_{0+}\cup {\mathfrak{T}}_{0-}\cup {\mathfrak{T}}_{00}$, ${\mathtt{K}}_{\mathfrak{i}}(·,{\mathfrak{u}}_{\mathfrak{i}}),$ $\mathfrak{i}\in {\mathfrak{T}}_{0+}$, and ${\mathtt{L}}_{\mathfrak{i}}(·,{\mathfrak{u}}_{\mathfrak{i}}),$ $\mathfrak{i}\in {\mathfrak{T}}_{+0}$ at $\mathfrak{b}$, we have

$$\begin{array}{c}\hfill {\Psi }_j(\mathfrak{a},{\mathfrak{u}}_j)-{\Psi }_j(\mathfrak{b},{\mathfrak{u}}_j)\ge ⟨{\xi }_j^2,\mathfrak{a}-\mathfrak{b}⟩,\forall {\xi }_j^2\in {\partial }^{*}{\Psi }_j(\mathfrak{b},{\mathfrak{u}}_j),\forall j\in {\mathfrak{T}}_{\Psi },\end{array}$$

(48)

$${\Theta }_{\mathfrak{s}}(\mathfrak{a},{\mathfrak{u}}_{\mathfrak{s}})-{\Theta }_{\mathfrak{s}}(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{s}})\ge ⟨{\xi }_{\mathfrak{s}}^3,\mathfrak{a}-\mathfrak{b}⟩,\forall {\xi }_{\mathfrak{s}}^3\in {\partial }^{*}{\Theta }_{\mathfrak{s}}(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{s}}),\forall \mathfrak{s}\in \mathbb{S},$$

(49)

$$\begin{array}{cc}\hfill (-{\Theta }_{\mathfrak{s}})(\mathfrak{a},{\mathfrak{u}}_{\mathfrak{s}})-(-{\Theta }_{\mathfrak{s}})(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{s}})\ge ⟨{\xi }_{\mathfrak{s}}^4,\mathfrak{a}-\mathfrak{b}⟩,\forall {\xi }_{\mathfrak{s}}^4\in {\partial }^{*}(-{\Theta }_{\mathfrak{s}})& (\mathfrak{b},{\mathfrak{u}}_{\mathfrak{s}}),\hfill \\ & \forall \mathfrak{s}\in \mathbb{S},\hfill \end{array}$$

(50)

$$\begin{array}{cc}\hfill (-{\mathtt{K}}_{\mathfrak{i}})(\mathfrak{a},{\mathfrak{u}}_{\mathfrak{i}})-(-{\mathtt{K}}_{\mathfrak{i}})(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{i}})\ge ⟨{\xi }_{\mathfrak{i}}^5,\mathfrak{a}-\mathfrak{b}⟩,& \forall {\xi }_{\mathfrak{i}}^5\in {\partial }^{*}(-{\mathtt{K}}_{\mathfrak{i}})(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{i}}),\hfill \\ & \forall \mathfrak{i}\in {\mathfrak{T}}_{0+}\cup {\mathfrak{T}}_{0-}\cup {\mathfrak{T}}_{00},\hfill \end{array}$$

(51)

$${\mathtt{K}}_{\mathfrak{i}}(\mathfrak{a},{\mathfrak{u}}_{\mathfrak{i}})-{\mathtt{K}}_{\mathfrak{i}}(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{i}})\ge ⟨{\xi }_{\mathfrak{i}}^6,\mathfrak{a}-\mathfrak{b}⟩,\forall {\xi }_{\mathfrak{i}}^6\in {\partial }^{*}{\mathtt{K}}_{\mathfrak{i}}(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{i}}),\forall \mathfrak{i}\in {\mathfrak{T}}_{0+},$$

(52)

$${\mathtt{L}}_{\mathfrak{i}}(\mathfrak{a},{\mathfrak{u}}_{\mathfrak{i}})-{\mathtt{L}}_{\mathfrak{i}}(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{i}})\ge ⟨{\xi }_{\mathfrak{i}}^7,\mathfrak{a}-\mathfrak{b}⟩,\forall {\xi }_{\mathfrak{i}}^7\in {\partial }^{*}{\mathtt{K}}_{\mathfrak{i}}(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{i}}),\forall \mathfrak{i}\in {\mathfrak{T}}_{+0},$$

(53)

If ${\mathfrak{T}}_{0-}^{\mathtt{K}}\cup {\mathfrak{T}}_{0+}^{\mathtt{K}}\cup {\mathfrak{T}}_{+0}^{\mathtt{L}}=\varnothing$, then multiplying (47)–(53) by ${\sigma }_t^{\mathcal{G}}>0,t\in \mathbb{T}$, ${\sigma }_j^{\Psi }\ge 0,j\in {\mathfrak{T}}_{\Psi }$, ${\sigma }_{\mathfrak{s}}^{\Theta }\ge 0,\mathfrak{s}\in \mathbb{S}$, ${\tau }_{\mathfrak{s}}^{\Theta }\ge 0,\mathfrak{s}\in \mathbb{S}$, ${\sigma }_{\mathfrak{i}}^{\mathtt{K}}\ge 0,\mathfrak{i}\in {\mathfrak{T}}_{0+}\cup {\mathfrak{T}}_{0-}\cup {\mathfrak{T}}_{00}$, ${\tau }_{\mathfrak{i}}^{\mathtt{K}}\ge 0,\mathfrak{i}\in {\mathfrak{T}}_{0+},{\tau }_{\mathfrak{i}}^{\mathtt{L}}\ge 0,$ $\mathfrak{i}\in {\mathfrak{T}}_{+0}$, respectively, and then by summing these inequalities, we obtain the following:

$$\begin{array}{cc}\hfill \sum _{t\in \mathbb{T}}{\sigma }_t^{\mathcal{G}}{\mathcal{G}}_t\left(\mathfrak{a}\right)& -\sum _{t\in \mathbb{T}}{\sigma }_t^{\mathcal{G}}{\mathcal{G}}_t\left(\mathfrak{b}\right)+\sum _{j\in {\mathfrak{T}}_{\Psi }}{\sigma }_j^{\Psi }{\Psi }_j(\mathfrak{a},{\mathfrak{u}}_j)-\sum _{j\in {\mathfrak{T}}_{\Psi }}{\sigma }_j^{\Psi }{\Psi }_j(\mathfrak{b},{\mathfrak{u}}_j)\hfill \\ & +\sum _{\mathfrak{s}\in \mathbb{S}}{\sigma }_{\mathfrak{s}}^{\Theta }{\Theta }_{\mathfrak{s}}(\mathfrak{a},{\mathfrak{u}}_{\mathfrak{s}})-\sum _{\mathfrak{s}\in \mathbb{S}}{\sigma }_{\mathfrak{s}}^{\Theta }{\Theta }_{\mathfrak{s}}(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{s}})+\sum _{\mathfrak{s}\in \mathbb{S}}{\tau }_{\mathfrak{s}}^{\Theta }(-{\Theta }_{\mathfrak{s}})(\mathfrak{a},{\mathfrak{u}}_{\mathfrak{s}})\hfill \\ & -\sum _{\mathfrak{s}\in \mathbb{S}}{\tau }_{\mathfrak{s}}^{\Theta }(-{\Theta }_{\mathfrak{s}})(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{s}})+\sum _{\mathfrak{i}\in \mathtt{I}}{\sigma }_{\mathfrak{i}}^{\mathtt{K}}(-{\mathtt{K}}_{\mathfrak{i}})(\mathfrak{a},{\mathfrak{u}}_{\mathfrak{i}})-\sum _{\mathfrak{i}\in \mathtt{I}}{\sigma }_{\mathfrak{i}}^{\mathtt{K}}(-{\mathtt{K}}_{\mathfrak{i}})(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{i}})\hfill \\ & +\sum _{\mathfrak{i}\in \mathtt{I}}{\tau }_{\mathfrak{i}}^{\mathtt{K}}{\mathtt{K}}_{\mathfrak{i}}(\mathfrak{a},{\mathfrak{u}}_{\mathfrak{i}})-\sum _{\mathfrak{i}\in \mathtt{I}}{\tau }_{\mathfrak{i}}^{\mathtt{K}}{\mathtt{K}}_{\mathfrak{i}}(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{i}})+\sum _{\mathfrak{i}\in \mathtt{I}}{\tau }_{\mathfrak{i}}^{\mathtt{L}}{\mathtt{L}}_{\mathfrak{i}}(\mathfrak{a},{\mathfrak{u}}_{\mathfrak{i}})\hfill \\ & -\sum _{\mathfrak{i}\in \mathtt{I}}{\tau }_{\mathfrak{i}}^{\mathtt{L}}{\mathtt{L}}_{\mathfrak{i}}(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{i}})\ge ⟨\sum _{t\in \mathbb{T}}{\sigma }_t^{\mathcal{G}}{\xi }_t^1+\sum _{j\in {\mathfrak{T}}_{\Psi }}{\sigma }_j^{\Psi }{\xi }_j^2+\sum _{\mathfrak{s}\in \mathbb{S}}[{\sigma }_{\mathfrak{s}}^{\Theta }{\xi }_{\mathfrak{s}}^3+{\tau }_{\mathfrak{s}}^{\Theta }{\xi }_{\mathfrak{s}}^4]\hfill \\ & +\sum _{\mathfrak{i}\in \mathtt{I}}{\sigma }_{\mathfrak{i}}^{\mathtt{K}}{\xi }_{\mathfrak{i}}^5+\sum _{\mathfrak{i}\in \mathtt{I}}[{\tau }_{\mathfrak{i}}^{\mathtt{K}}{\xi }_{\mathfrak{i}}^6+{\tau }_{\mathfrak{i}}^{\mathtt{L}}{\xi }_{\mathfrak{i}}^7],\mathfrak{a}-\mathfrak{b}⟩.\hfill \end{array}$$

Given the feasibility conditions of $\mathfrak{b}\in {\mathbb{F}}_{\mathcal{G}}^{MW}$, there exist ${\overline{\xi }}_t^1\in co{\partial }^{*}{\mathcal{G}}_t\left(\mathfrak{b}\right),$ $t\in \mathbb{T}$, ${\overline{\xi }}_j^2\in co{\partial }^{*}{\Psi }_j(\mathfrak{b},{\mathfrak{u}}_j),$ $j\in {\mathfrak{T}}_{\Psi }$, ${\overline{\xi }}_{\mathfrak{s}}^3\in co{\partial }^{*}{\Theta }_{\mathfrak{s}}(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{s}}),$ $\mathfrak{s}\in \mathbb{S}$, ${\overline{\xi }}_{\mathfrak{s}}^4\in co{\partial }^{*}(-{\Theta }_{\mathfrak{s}})(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{s}}),$ $\mathfrak{s}\in \mathbb{S}$, ${\overline{\xi }}_{\mathfrak{i}}^5\in co{\partial }^{*}(-{\mathtt{K}}_{\mathfrak{i}})(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{i}}),$ $\mathfrak{i}\in \mathtt{I}$, ${\overline{\xi }}_{\mathfrak{i}}^6\in co{\partial }^{*}{\mathtt{K}}_{\mathfrak{i}}(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{i}}),$ $\mathfrak{i}\in \mathtt{I}$, and ${\overline{\xi }}_{\mathfrak{i}}^7\in co{\partial }^{*}{\mathtt{L}}_{\mathfrak{i}}(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{i}}),$ $\mathfrak{i}\in \mathtt{I}$, such that

$$\sum _{t\in \mathbb{T}}{\sigma }_t^{\mathcal{G}}{\overline{\xi }}_t^1+\sum _{j\in {\mathfrak{T}}_{\Psi }}{\sigma }_j^{\Psi }{\overline{\xi }}_j^2+\sum _{\mathfrak{s}\in \mathbb{S}}[{\sigma }_{\mathfrak{s}}^{\Theta }{\overline{\xi }}_{\mathfrak{s}}^3+{\tau }_{\mathfrak{s}}^{\Theta }{\overline{\xi }}_{\mathfrak{s}}^4]+\sum _{\mathfrak{i}\in \mathtt{I}}{\sigma }_{\mathfrak{i}}^{\mathtt{K}}{\overline{\xi }}_{\mathfrak{i}}^5+\sum _{\mathfrak{i}\in \mathtt{I}}[{\tau }_{\mathfrak{i}}^{\mathtt{K}}{\overline{\xi }}_{\mathfrak{i}}^5+{\tau }_{\mathfrak{i}}^{\mathtt{L}}{\overline{\xi }}_{\mathfrak{i}}^6]=0.$$

Therefore,

$$\begin{array}{cc}\hfill \sum _{t\in \mathbb{T}}{\sigma }_t^{\mathcal{G}}{\mathcal{G}}_t\left(\mathfrak{a}\right)& -\sum _{t\in \mathbb{T}}{\sigma }_t^{\mathcal{G}}{\mathcal{G}}_t\left(\mathfrak{b}\right)+\sum _{j\in {\mathfrak{T}}_{\Psi }}{\sigma }_j^{\Psi }{\Psi }_j(\mathfrak{a},{\mathfrak{u}}_j)-\sum _{j\in {\mathfrak{T}}_{\Psi }}{\sigma }_j^{\Psi }{\Psi }_j(\mathfrak{b},{\mathfrak{u}}_j)\hfill \\ & +\sum _{\mathfrak{s}\in \mathbb{S}}{\sigma }_{\mathfrak{s}}^{\Theta }{\Theta }_{\mathfrak{s}}(\mathfrak{a},{\mathfrak{u}}_{\mathfrak{s}})-\sum _{\mathfrak{s}\in \mathbb{S}}{\sigma }_{\mathfrak{s}}^{\Theta }{\Theta }_{\mathfrak{s}}(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{s}})+\sum _{\mathfrak{s}\in \mathbb{S}}{\tau }_{\mathfrak{s}}^{\Theta }(-{\Theta }_{\mathfrak{s}})(\mathfrak{a},{\mathfrak{u}}_{\mathfrak{s}})\hfill \\ & -\sum _{\mathfrak{s}\in \mathbb{S}}{\tau }_{\mathfrak{s}}^{\Theta }(-{\Theta }_{\mathfrak{s}})(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{s}})+\sum _{\mathfrak{i}\in \mathtt{I}}{\sigma }_{\mathfrak{i}}^{\mathtt{K}}(-{\mathtt{K}}_{\mathfrak{i}})(\mathfrak{a},{\mathfrak{u}}_{\mathfrak{i}})-\sum _{\mathfrak{i}\in \mathtt{I}}{\sigma }_{\mathfrak{i}}^{\mathtt{K}}(-{\mathtt{K}}_{\mathfrak{i}})(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{i}})\hfill \\ & +\sum _{\mathfrak{i}\in \mathtt{I}}{\tau }_{\mathfrak{i}}^{\mathtt{K}}{\mathtt{K}}_{\mathfrak{i}}(\mathfrak{a},{\mathfrak{u}}_{\mathfrak{i}})-\sum _{\mathfrak{i}\in \mathtt{I}}{\tau }_{\mathfrak{i}}^{\mathtt{K}}{\mathtt{K}}_{\mathfrak{i}}(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{i}})+\sum _{\mathfrak{i}\in \mathtt{I}}{\tau }_{\mathfrak{i}}^{\mathtt{L}}{\mathtt{L}}_{\mathfrak{i}}(\mathfrak{a},{\mathfrak{u}}_{\mathfrak{i}})\hfill \\ & -\sum _{\mathfrak{i}\in \mathtt{I}}{\tau }_{\mathfrak{i}}^{\mathtt{L}}{\mathtt{L}}_{\mathfrak{i}}(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{i}})\ge 0.\hfill \end{array}$$

Since $\mathfrak{a}\in {\mathbb{F}}_{\mathcal{G}}$, we have ${\Psi }_j(\mathfrak{a},{\mathfrak{u}}_j)\le 0,$ $j\in {\mathfrak{T}}_{\Psi },$ ${\Theta }_{\mathfrak{s}}(\mathfrak{a},{\mathfrak{u}}_{\mathfrak{s}})=0,$ $\mathfrak{s}\in \mathbb{S},$ $-{\mathtt{K}}_{\mathfrak{i}}(\mathfrak{a},{\mathfrak{u}}_{\mathfrak{i}})<0,$ $\mathfrak{i}\in {\mathfrak{T}}_{+},$ ${\mathtt{K}}_{\mathfrak{i}}(\mathfrak{a},{\mathfrak{u}}_{\mathfrak{i}})=0,$ $\mathfrak{i}\in {\mathfrak{T}}_0,$ ${\mathtt{L}}_{\mathfrak{i}}(\mathfrak{a},{\mathfrak{u}}_{\mathfrak{i}})>0,$ $\mathfrak{i}\in {\mathfrak{T}}_{0+},$ ${\mathtt{L}}_{\mathfrak{i}}(\mathfrak{a},{\mathfrak{u}}_{\mathfrak{i}})=0,$ $\mathfrak{i}\in {\mathfrak{T}}_{00}\cup {\mathfrak{T}}_{+0},$ ${\mathtt{L}}_{\mathfrak{i}}(\mathfrak{a},{\mathfrak{u}}_{\mathfrak{i}})=0,$ $\mathfrak{i}\in {\mathfrak{T}}_{0-}\cup {\mathfrak{T}}_{+-}.$ Since $y\in {\mathbb{F}}_{\mathcal{G}}^{MW},$ we have ${\Psi }_j(\mathfrak{b},{\mathfrak{u}}_j)\ge 0,$ $j\in {\mathfrak{T}}_{\Psi },$ ${\Theta }_{\mathfrak{s}}(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{s}})=0,$ $\mathfrak{s}\in \mathbb{S},$ $-{\mathtt{K}}_{\mathfrak{i}}(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{i}})>0,$ $\mathfrak{i}\in {\mathfrak{T}}_{+},$ ${\mathtt{K}}_{\mathfrak{i}}(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{i}})=0,$ $\mathfrak{i}\in {\mathfrak{T}}_0,$ ${\mathtt{L}}_{\mathfrak{i}}(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{i}})<0,$ $\mathfrak{i}\in {\mathfrak{T}}_{0+},$ ${\mathtt{L}}_{\mathfrak{i}}(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{i}})=0,$ $\mathfrak{i}\in {\mathfrak{T}}_{00}\cup {\mathfrak{T}}_{+0},$ ${\mathtt{L}}_{\mathfrak{i}}(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{i}})>0,$ $\mathfrak{i}\in {\mathfrak{T}}_{0-}\cup {\mathfrak{T}}_{+-}.$ Hence,

$$\sum _{t\in \mathbb{T}}{\sigma }_t^{\mathcal{G}}{\mathcal{G}}_t\left(\mathfrak{a}\right)\ge \sum _{t\in \mathbb{T}}{\sigma }_t^{\mathcal{G}}{\mathcal{G}}_t\left(\mathfrak{b}\right).$$

Hence, from (46), we arrive at a contradiction. This concludes the proof of the weak duality theorem. □

In the subsequent theorem, a strong duality result is established, which relates UNMPVC to the associated MWRD dual model.

Theorem 9. 

Let $\overline{\mathfrak{a}}$ be a local robust Pareto solution UNMPVC. Moreover, we assume that ${\mathcal{G}}_t,$ $t\in \mathbb{T},$ ${\Psi }_j,j\in {\mathfrak{T}}_{\Psi },\pm {\Theta }_{\mathfrak{s}},\mathfrak{s}\in \mathbb{S},-{\mathtt{K}}_{\mathfrak{i}},\mathfrak{i}\in {\mathfrak{T}}_{0+}\cup {\mathfrak{T}}_{0-}\cup {\mathfrak{T}}_{00},$ ${\mathtt{K}}_{\mathfrak{i}},\mathfrak{i}\in {\mathfrak{T}}_{0+}$, and ${\mathtt{L}}_{\mathfrak{i}},\mathfrak{i}\in {\mathfrak{T}}_{+0}$ admit bounded USRC at $\overline{\mathfrak{a}}$, such that each function is a ${\partial }^{*}$-convex function at $\overline{\mathfrak{a}}$. Furthermore, we suppose that at $\overline{\mathfrak{a}}$, GS-ACQ holds. Then, there exists $\overline{\sigma }=({\overline{\sigma }}^{\mathcal{G}},{\overline{\sigma }}^{\Psi },{\overline{\sigma }}^{\Theta },{\overline{\tau }}^{\Theta },{\overline{\sigma }}^{\mathtt{K}},{\overline{\tau }}^{\mathtt{K}},{\overline{\tau }}^{\mathtt{L}})\in {\mathbb{R}}^r\times {\mathbb{R}}^p\times {\mathbb{R}}^{2\ell }\times {\mathbb{R}}^{3m}$, such that $(\overline{\mathfrak{a}},\overline{\sigma })$ is a robust local weak Pareto solution of MWRD. Moreover, the corresponding objective function values are equal.

Proof. 

Given that GS-ACQ holds as a robust local weak Pareto solution $\overline{\mathfrak{a}}$ of UNMPVC, then, in light of Theorem 3, there exists $\overline{\sigma }=({\overline{\sigma }}^{\mathcal{G}},{\overline{\sigma }}^{\Psi },{\overline{\sigma }}^{\Theta },{\overline{\tau }}^{\Theta },{\overline{\sigma }}^{\mathtt{K}},{\overline{\tau }}^{\mathtt{K}},{\overline{\tau }}^{\mathtt{L}})\in {\mathbb{R}}^r\times {\mathbb{R}}^p\times {\mathbb{R}}^{2\ell }\times {\mathbb{R}}^{3m}$, such that $\overline{\mathfrak{a}}$ is a RS-stationary point of UNMPVC. Thus, there exist ${\overline{\xi }}_t^1\in co{\partial }^{*}{\mathcal{G}}_t\left(\overline{\mathfrak{a}}\right),$ $t\in \mathbb{T},$ ${\overline{\xi }}_j^2\in co{\partial }^{*}{\Psi }_{\mathfrak{i}}(\overline{\mathfrak{a}},$ ${\overline{\mathfrak{u}}}_j),$ $j\in {\mathfrak{T}}_{\Psi },$ ${\overline{\mathfrak{u}}}_j\in {\Omega }_j\left(\overline{\mathfrak{a}}\right)$, ${\overline{\xi }}_{\mathfrak{s}}^3\in co{\partial }^{*}{\Theta }_{\mathfrak{s}}(\overline{\mathfrak{a}},{\overline{\mathfrak{u}}}_{\mathfrak{s}}),$ ${\overline{\xi }}_{\mathfrak{s}}^4\in co{\partial }^{*}(-{\Theta }_{\mathfrak{s}})(\overline{\mathfrak{a}},{\overline{\mathfrak{u}}}_{\mathfrak{s}}),$ $\mathfrak{s}\in \mathbb{S},{\overline{\mathfrak{u}}}_{\mathfrak{s}}\in {\Omega }_{\mathfrak{s}}\left(\overline{\mathfrak{a}}\right)$, ${\overline{\xi }}_{\mathfrak{i}}^5\in co{\partial }^{*}(-{\mathtt{K}}_{\mathfrak{i}})(\overline{\mathfrak{a}},{\overline{\mathfrak{u}}}_{\mathfrak{i}}),$ $\mathfrak{i}\in \mathtt{I}$, ${\overline{\xi }}_{\mathfrak{i}}^6\in co{\partial }^{*}{\mathtt{K}}_{\mathfrak{i}}(\overline{\mathfrak{a}},{\overline{\mathfrak{u}}}_{\mathfrak{i}}),$ $\mathfrak{i}\in \mathtt{I},$ and ${\overline{\xi }}_{\mathfrak{i}}^7\in co{\partial }^{*}{\mathtt{L}}_{\mathfrak{i}}(\overline{\mathfrak{a}},{\overline{\mathfrak{u}}}_{\mathfrak{i}}),$ $\mathfrak{i}\in \mathtt{I},{\overline{\mathfrak{u}}}_{\mathfrak{i}}\in {\Omega }_{\mathfrak{i}}\left(\overline{\mathfrak{a}}\right)$, such that

$$\sum _{t\in \mathbb{T}}{\overline{\sigma }}_t^{\mathcal{G}}{\overline{\xi }}_t^1+\sum _{j\in {\mathfrak{T}}_{\Psi }}{\overline{\sigma }}_j^{\Psi }{\overline{\xi }}_j^2+\sum _{\mathfrak{s}\in \mathbb{S}}[{\overline{\sigma }}_{\mathfrak{s}}^{\Theta }{\overline{\xi }}_{\mathfrak{s}}^3+{\overline{\tau }}_{\mathfrak{s}}^{\Theta }{\overline{\xi }}_{\mathfrak{s}}^4]+\sum _{\mathfrak{i}\in \mathtt{I}}{\overline{\sigma }}_{\mathfrak{i}}^{\mathtt{K}}{\overline{\xi }}_{\mathfrak{i}}^5+\sum _{\mathfrak{i}\in \mathtt{I}}[{\overline{\tau }}_{\mathfrak{i}}^{\mathtt{K}}{\overline{\xi }}_{\mathfrak{i}}^6+{\overline{\tau }}_{\mathfrak{i}}^{\mathtt{L}}{\overline{\xi }}_{\mathfrak{i}}^7]=0,$$

$$\begin{array}{cc}& {\overline{\sigma }}_t^{\mathcal{G}}>0,t\in \mathbb{T},{\overline{\sigma }}_j^{\Psi }\ge 0,j\in {\mathfrak{T}}_{\Psi },{\overline{\sigma }}_{\mathfrak{s}}^{\Theta }\ge 0,{\overline{\tau }}_{\mathfrak{s}}^{\Theta }\ge 0,\mathfrak{s}\in \mathbb{S},{\overline{\sigma }}_{\mathfrak{i}}^{\mathtt{K}}\ge 0,{\overline{\tau }}_{\mathfrak{i}}^{\mathtt{K}}\ge 0,{\overline{\tau }}_{\mathfrak{i}}^{\mathtt{L}}\ge 0,\mathfrak{i}\in \mathtt{I}.\hfill \end{array}$$

Therefore, $(\overline{\mathfrak{a}},\overline{\sigma })\in {\mathbb{F}}_{\mathcal{G}}^{MW}$. Theorem 8 then implies that for any $(\mathfrak{b},\sigma )\in {\mathbb{F}}_{\mathcal{G}}^{MW}$, we have

$$\begin{array}{c}\hfill \mathcal{G}\left(\overline{\mathfrak{a}}\right)\nprec \mathcal{G}\left(\mathfrak{b}\right).\end{array}$$

Therefore, $(\overline{\mathfrak{a}},\overline{\sigma })$ is a robust local weak Pareto solution of MWRD. Moreover, the corresponding objective function values are equal. □

Now, we establish the strict converse duality theorem relating WRD and UNMPVC.

Theorem 10. 

Let $(\widehat{\mathfrak{b}},\widehat{\sigma })$ be the global weak Pareto solution of WRD and $\overline{\mathfrak{a}}$ be a robust local weak Pareto solution of UNMPVC. Moreover, we suppose that the hypotheses of the strong duality theorem hold at $\widehat{\mathfrak{b}}$, such that ${\mathcal{G}}_t,t\in \mathbb{T}$ are strictly ${\partial }^{*}$-convex at $\widehat{\mathfrak{b}}$, then $\overline{\mathfrak{a}}=\widehat{\mathfrak{b}}$.

Proof. 

Suppose that $\overline{\mathfrak{a}}\ne \widehat{\mathfrak{b}}$. Theorem 9 then stipulates that

$$\overline{\sigma }=({\overline{\sigma }}^{\mathcal{G}},{\overline{\sigma }}^{\Psi },{\overline{\sigma }}^{\Theta },{\overline{\tau }}^{\Theta },{\overline{\sigma }}^{\mathtt{K}},{\overline{\tau }}^{\mathtt{K}},{\overline{\tau }}^{\mathtt{L}})\in {\mathbb{R}}^r\times {\mathbb{R}}^p\times {\mathbb{R}}^{2\ell }\times {\mathbb{R}}^{3m},$$

such that $(\overline{\mathfrak{a}},\overline{\sigma })\in {\mathbb{F}}_{\mathcal{G}}^{MW}$ and

$$\mathcal{G}\left(\overline{\mathfrak{a}}\right)=\mathcal{G}\left(\widehat{\mathfrak{b}}\right).$$

(54)

Multiplying (54) by ${\widehat{\sigma }}_t^{\mathcal{G}},t\in \mathbb{T}$, we obtain

$$\sum _{t\in \mathbb{T}}{\widehat{\sigma }}_t^{\mathcal{G}}{\mathcal{G}}_t\left(\mathfrak{a}\right)=\sum _{t\in \mathbb{T}}{\widehat{\sigma }}_t^{\mathcal{G}}{\mathcal{G}}_t\left(\mathfrak{b}\right).$$

(55)

In view of the fact that ${\mathcal{G}}_t,t\in \mathbb{T}$ are strictly ${\partial }^{*}$-convex at $\widehat{\mathfrak{b}}$, we infer that

$${\mathcal{G}}_t\left(\overline{\mathfrak{a}}\right)-{\mathcal{G}}_t\left(\widehat{\mathfrak{b}}\right)>⟨{\xi }_t^1,\overline{\mathfrak{a}}-\widehat{\mathfrak{b}}⟩,\forall {\xi }_t^1\in {\partial }^{*}\mathcal{G}\left(\widehat{\mathfrak{b}}\right),t\in \mathbb{T}.$$

(56)

Similarly, according to the ${\partial }^{*}$-convexity of ${\Psi }_j,$ $j\in {\mathfrak{T}}_{\Psi },$ $\pm {\Theta }_{\mathfrak{s}},$ $\mathfrak{s}\in \mathbb{S},$ $(-{\mathtt{K}}_{\mathfrak{i}}),$ $\mathfrak{i}\in {\mathfrak{T}}_{0+}\cup {\mathfrak{T}}_{0-}\cup {\mathfrak{T}}_{00},$ ${\mathtt{K}}_{\mathfrak{i}},$ $\mathfrak{i}\in {\mathfrak{T}}_{0+},$ ${\mathtt{L}}_{\mathfrak{i}},$ $\mathfrak{i}\in {\mathfrak{T}}_{+0}$ at $\widehat{\mathfrak{b}}$, we have

$${\Psi }_j(\overline{\mathfrak{a}},{\mathfrak{u}}_j)-{\Psi }_j(\widehat{\mathfrak{b}},{\mathfrak{u}}_j)\ge ⟨{\xi }_j^2,\overline{\mathfrak{a}}-\widehat{\mathfrak{b}}⟩,\forall {\xi }_j^2\in {\partial }^{*}{\Psi }_j(\widehat{\mathfrak{b}},,{\mathfrak{u}}_j),\forall j\in {\mathfrak{T}}_{\Psi },$$

(57)

$${\Theta }_{\mathfrak{s}}(\overline{\mathfrak{a}},{\mathfrak{u}}_{\mathfrak{s}})-{\Theta }_{\mathfrak{s}}(\widehat{\mathfrak{b}},{\mathfrak{u}}_{\mathfrak{s}})\ge ⟨{\xi }_{\mathfrak{s}}^3,\overline{\mathfrak{a}}-\widehat{\mathfrak{b}}⟩,\forall {\xi }_{\mathfrak{s}}^3\in {\partial }^{*}{\Theta }_{\mathfrak{s}}(\widehat{\mathfrak{b}},{\mathfrak{u}}_{\mathfrak{s}}),\forall \mathfrak{s}\in \mathbb{S},$$

(58)

$$\begin{array}{cc}\hfill (-{\Theta }_{\mathfrak{s}})(\overline{\mathfrak{a}},{\mathfrak{u}}_{\mathfrak{s}})-(-{\Theta }_{\mathfrak{s}})(\widehat{\mathfrak{b}},{\mathfrak{u}}_{\mathfrak{s}})\ge ⟨{\xi }_{\mathfrak{s}}^4,\overline{\mathfrak{a}}-\widehat{\mathfrak{b}}⟩,\forall {\xi }_{\mathfrak{s}}^4\in {\partial }^{*}& (-{\Theta }_{\mathfrak{s}})(\widehat{\mathfrak{b}},{\mathfrak{u}}_{\mathfrak{s}}),\hfill \\ & \forall \mathfrak{s}\in \mathbb{S},\hfill \end{array}$$

(59)

$$\begin{array}{cc}\hfill (-{\mathtt{K}}_{\mathfrak{i}})(\overline{\mathfrak{a}},{\mathfrak{u}}_{\mathfrak{i}})-(-{\mathtt{K}}_{\mathfrak{i}},{\mathfrak{u}}_{\mathfrak{i}})(\widehat{\mathfrak{b}},{\mathfrak{u}}_{\mathfrak{i}})\ge ⟨{\xi }_{\mathfrak{i}}^5,\overline{\mathfrak{a}}-\widehat{\mathfrak{b}}⟩,& \forall {\xi }_{\mathfrak{i}}^5\in {\partial }^{*}(-{\mathtt{K}}_{\mathfrak{i}})(\widehat{\mathfrak{b}},{\mathfrak{u}}_{\mathfrak{i}}),\hfill \\ & \forall \mathfrak{i}\in {\mathfrak{T}}_{0+}\cup {\mathfrak{T}}_{0-}\cup {\mathfrak{T}}_{00},\hfill \end{array}$$

(60)

$${\mathtt{K}}_{\mathfrak{i}}(\overline{\mathfrak{a}},{\mathfrak{u}}_{\mathfrak{i}})-{\mathtt{K}}_{\mathfrak{i}}(\widehat{\mathfrak{b}},{\mathfrak{u}}_{\mathfrak{i}})\ge ⟨{\xi }_{\mathfrak{i}}^6,\overline{\mathfrak{a}}-\widehat{\mathfrak{b}}⟩,\forall {\xi }_{\mathfrak{i}}^6\in {\partial }^{*}{\mathtt{L}}_{\mathfrak{i}}(\widehat{\mathfrak{b}},{\mathfrak{u}}_{\mathfrak{i}}),\forall \mathfrak{i}\in {\mathfrak{T}}_{0+}.$$

(61)

$${\mathtt{L}}_{\mathfrak{i}}(\overline{\mathfrak{a}},{\mathfrak{u}}_{\mathfrak{i}})-{\mathtt{L}}_{\mathfrak{i}}(\widehat{\mathfrak{b}},{\mathfrak{u}}_{\mathfrak{i}})\ge ⟨{\xi }_{\mathfrak{i}}^7,\overline{\mathfrak{a}}-\widehat{\mathfrak{b}}⟩,\forall {\xi }_{\mathfrak{i}}^7\in {\partial }^{*}{\mathtt{L}}_{\mathfrak{i}}(\widehat{\mathfrak{b}},{\mathfrak{u}}_{\mathfrak{i}}),\forall \mathfrak{i}\in {\mathfrak{T}}_{+0}.$$

(62)

If ${\mathfrak{T}}_{0-}^{\mathtt{K}}\cup {\mathfrak{T}}_{0+}^{\mathtt{K}}\cup {\mathfrak{T}}_{+0}^{\mathtt{L}}=\varnothing$, then multiplying (56)–(62) by ${\widehat{\sigma }}_t^{\mathcal{G}}>0,t\in \mathbb{T}$, ${\widehat{\sigma }}_j^{\Psi }\ge 0,j\in {\mathfrak{T}}_{\Psi }$, ${\widehat{\sigma }}_{\mathfrak{s}}^{\Theta }\ge 0,\mathfrak{s}\in \mathbb{S}$, ${\widehat{\tau }}_{\mathfrak{s}}^{\Theta }\ge 0,\mathfrak{s}\in \mathbb{S}$, ${\widehat{\sigma }}_{\mathfrak{i}}^{\mathtt{K}}>0,\mathfrak{i}\in {\mathfrak{T}}_{0+}\cup {\mathfrak{T}}_{0-}\cup {\mathfrak{T}}_{00}$, ${\widehat{\tau }}_{\mathfrak{i}}^{\mathtt{K}}>0,\mathfrak{i}\in {\mathfrak{T}}_{0+}$, ${\widehat{\tau }}_{\mathfrak{i}}^{\mathtt{L}}>0,$ $\mathfrak{i}\in {\mathfrak{T}}_{+0}$, respectively, and then by summing these inequalities, we obtain the following:

$$\begin{array}{cc}\hfill \sum _{t\in \mathbb{T}}{\widehat{\sigma }}_t^{\mathcal{G}}{\mathcal{G}}_t\left(\overline{\mathfrak{a}}\right)& -\sum _{t\in \mathbb{T}}{\widehat{\sigma }}_t^{\mathcal{G}}{\mathcal{G}}_t\left(\widehat{\mathfrak{b}}\right)+\sum _{j\in {\mathfrak{T}}_{\Psi }}{\widehat{\sigma }}_j^{\Psi }{\Psi }_j(\overline{\mathfrak{a}},{\mathfrak{u}}_j)-\sum _{j\in {\mathfrak{T}}_{\Psi }}{\widehat{\sigma }}_j^{\Psi }{\Psi }_j(\widehat{\mathfrak{b}},{\mathfrak{u}}_j)\hfill \\ & +\sum _{\mathfrak{s}\in \mathbb{S}}{\widehat{\sigma }}_{\mathfrak{s}}^{\Theta }{\Theta }_{\mathfrak{s}}(\overline{\mathfrak{a}},{\mathfrak{u}}_j)-\sum _{\mathfrak{s}\in \mathbb{S}}{\widehat{\sigma }}_{\mathfrak{s}}^{\Theta }{\Theta }_{\mathfrak{s}}(\widehat{\mathfrak{b}},{\mathfrak{u}}_{\mathfrak{s}})+\sum _{\mathfrak{s}\in \mathbb{S}}{\widehat{\tau }}_{\mathfrak{s}}^{\Theta }(-{\Theta }_{\mathfrak{s}})(\overline{\mathfrak{a}},{\mathfrak{u}}_{\mathfrak{s}})\hfill \\ & -\sum _{\mathfrak{s}\in \mathbb{S}}{\widehat{\tau }}_{\mathfrak{s}}^{\Theta }(-{\Theta }_{\mathfrak{s}})(\widehat{\mathfrak{b}},{\widehat{w}}_{\mathfrak{s}})+\sum _{\mathfrak{i}\in \mathtt{I}}{\widehat{\sigma }}_{\mathfrak{i}}^{\mathtt{K}}(-{\mathtt{K}}_{\mathfrak{i}})(\overline{\mathfrak{a}},{\mathfrak{u}}_{\mathfrak{i}})-\sum _{\mathfrak{i}\in \mathtt{I}}{\widehat{\sigma }}_{\mathfrak{i}}^{\mathtt{K}}(-{\mathtt{K}}_{\mathfrak{i}})(\widehat{\mathfrak{b}},{\mathfrak{u}}_{\mathfrak{i}})\hfill \\ \hfill & +\sum _{\mathfrak{i}\in \mathtt{I}}{\widehat{\tau }}_{\mathfrak{i}}^{\mathtt{K}}{\mathtt{K}}_{\mathfrak{i}}(\overline{\mathfrak{a}},{\mathfrak{u}}_{\mathfrak{i}})-\sum _{\mathfrak{i}\in \mathtt{I}}{\widehat{\tau }}_{\mathfrak{i}}^{\mathtt{K}}{\mathtt{K}}_{\mathfrak{i}}(\widehat{\mathfrak{b}},{\mathfrak{u}}_{\mathfrak{i}})+\sum _{\mathfrak{i}\in \mathtt{I}}{\widehat{\tau }}_{\mathfrak{i}}^{\mathtt{L}}{\mathtt{L}}_{\mathfrak{i}}(\overline{\mathfrak{a}},{\mathfrak{u}}_{\mathfrak{i}})\hfill \\ & -\sum _{\mathfrak{i}\in \mathtt{I}}{\widehat{\tau }}_{\mathfrak{i}}^{\mathtt{L}}{\mathtt{L}}_{\mathfrak{i}}(\widehat{\mathfrak{b}},{\mathfrak{u}}_{\mathfrak{i}})>⟨\sum _{t\in \mathbb{T}}{\widehat{\sigma }}_t^{\mathcal{G}}{\xi }_t^1+\sum _{j\in {\mathfrak{T}}_{\Psi }}{\widehat{\sigma }}_j^{\Psi }{\xi }_j^2+\sum _{\mathfrak{s}\in \mathbb{S}}[{\widehat{\sigma }}_{\mathfrak{s}}^{\Theta }{\xi }_{\mathfrak{s}}^3+{\widehat{\tau }}_{\mathfrak{s}}^{\Theta }{\xi }_{\mathfrak{s}}^4]\hfill \\ & +\sum _{\mathfrak{i}\in \mathtt{I}}{\widehat{\sigma }}_{\mathfrak{i}}^{\mathtt{K}}{\xi }_{\mathfrak{i}}^5+\sum _{\mathfrak{i}\in \mathtt{I}}[{\widehat{\tau }}_{\mathfrak{i}}^{\mathtt{K}}{\xi }_{\mathfrak{i}}^6+{\widehat{\sigma }}_{\mathfrak{i}}^{\mathtt{L}}{\xi }_{\mathfrak{i}}^7],\overline{\mathfrak{a}}-\widehat{\mathfrak{b}}⟩.\hfill \end{array}$$

Given the feasibility conditions of $\widehat{\mathfrak{b}}\in {\mathbb{F}}_{\mathcal{G}}^{WRD}$, there exist ${\widehat{\xi }}_t^1\in co{\partial }^{*}{\mathcal{G}}_t\left(\widehat{\mathfrak{b}}\right),$ $t\in \mathbb{T}$, ${\widehat{\xi }}_j^2\in co{\partial }^{*}{\Psi }_j(\widehat{\mathfrak{b}},{\mathfrak{u}}_j),$ $j\in {\mathfrak{T}}_{\Psi }$, ${\widehat{\xi }}_{\mathfrak{s}}^3\in co{\partial }^{*}{\Theta }_{\mathfrak{s}}(\widehat{\mathfrak{b}},{\mathfrak{u}}_{\mathfrak{s}}),$ $\mathfrak{s}\in \mathbb{S}$, ${\widehat{\xi }}_{\mathfrak{s}}^4\in co{\partial }^{*}(-{\Theta }_{\mathfrak{s}})(\widehat{\mathfrak{b}},{\mathfrak{u}}_{\mathfrak{s}}),$ $\mathfrak{s}\in \mathbb{S}$, ${\widehat{\xi }}_{\mathfrak{i}}^5\in co{\partial }^{*}(-{\mathtt{K}}_{\mathfrak{i}})(\widehat{\mathfrak{b}},{\mathfrak{u}}_{\mathfrak{i}}),$ $\mathfrak{i}\in \mathtt{I}$, ${\widehat{\xi }}_{\mathfrak{i}}^6\in co{\partial }^{*}{\mathtt{K}}_{\mathfrak{i}}(\widehat{\mathfrak{b}},{\mathfrak{u}}_{\mathfrak{i}}),\mathfrak{i}\in \mathtt{I}$, and ${\widehat{\xi }}_{\mathfrak{i}}^7\in co{\partial }^{*}{\mathtt{L}}_{\mathfrak{i}}(\widehat{\mathfrak{b}},{\mathfrak{u}}_{\mathfrak{i}}),\mathfrak{i}\in \mathtt{I}$, such that

$$\sum _{t\in \mathbb{T}}{\widehat{\sigma }}_t^{\mathcal{G}}{\widehat{\xi }}_t^1+\sum _{j\in {\mathfrak{T}}_{\Psi }}{\widehat{\sigma }}_j^{\Psi }{\widehat{\xi }}_j^2+\sum _{\mathfrak{s}\in \mathbb{S}}[{\widehat{\sigma }}_{\mathfrak{s}}^{\Theta }{\widehat{\xi }}_{\mathfrak{s}}^3+{\widehat{\tau }}_{\mathfrak{s}}^{\Theta }{\widehat{\xi }}_{\mathfrak{s}}^4]+\sum _{\mathfrak{i}\in \mathtt{I}}{\widehat{\sigma }}_{\mathfrak{i}}^{\mathtt{K}}{\widehat{\xi }}_{\mathfrak{i}}^5+\sum _{\mathfrak{i}\in \mathtt{I}}[{\widehat{\tau }}_{\mathfrak{i}}^{\mathtt{K}}{\widehat{\xi }}_{\mathfrak{i}}^6+{\widehat{\tau }}_{\mathfrak{i}}^{\mathtt{L}}{\widehat{\xi }}_{\mathfrak{i}}^6]=0.$$

Therefore,

$$\begin{array}{cc}\hfill \sum _{t\in \mathbb{T}}{\widehat{\sigma }}_t^{\mathcal{G}}{\mathcal{G}}_t\left(\overline{\mathfrak{a}}\right)& -\sum _{t\in \mathbb{T}}{\widehat{\sigma }}_t^{\mathcal{G}}{\mathcal{G}}_t\left(\widehat{\mathfrak{b}}\right)+\sum _{j\in {\mathfrak{T}}_{\Psi }}{\widehat{\sigma }}_j^{\Psi }{\Psi }_j(\overline{\mathfrak{a}},{\mathfrak{u}}_j)-\sum _{j\in {\mathfrak{T}}_{\Psi }}{\widehat{\sigma }}_j^{\Psi }{\Psi }_j(\widehat{\mathfrak{b}},{\mathfrak{u}}_j)\hfill \\ & +\sum _{\mathfrak{s}\in \mathbb{S}}{\widehat{\sigma }}_{\mathfrak{s}}^{\Theta }{\Theta }_{\mathfrak{s}}(\overline{\mathfrak{a}},{\mathfrak{u}}_{\mathfrak{s}})-\sum _{\mathfrak{s}\in \mathbb{S}}{\widehat{\sigma }}_{\mathfrak{s}}^{\Theta }{\Theta }_{\mathfrak{s}}(\widehat{\mathfrak{b}},{\mathfrak{u}}_{\mathfrak{s}})+\sum _{\mathfrak{s}\in \mathbb{S}}{\widehat{\tau }}_{\mathfrak{s}}^{\Theta }(-{\Theta }_{\mathfrak{s}})(\overline{\mathfrak{a}},{\mathfrak{u}}_{\mathfrak{s}})\hfill \\ & -\sum _{\mathfrak{s}\in \mathbb{S}}{\widehat{\tau }}_{\mathfrak{s}}^{\Theta }(-{\Theta }_{\mathfrak{s}})(\widehat{\mathfrak{b}},{\mathfrak{u}}_{\mathfrak{s}})+\sum _{\mathfrak{i}\in \mathtt{I}}{\widehat{\sigma }}_{\mathfrak{i}}^{\mathtt{K}}(-{\mathtt{K}}_{\mathfrak{i}})(\overline{\mathfrak{a}},{\mathfrak{u}}_{\mathfrak{i}})-\sum _{\mathfrak{i}\in \mathtt{I}}{\widehat{\sigma }}_{\mathfrak{i}}^{\mathtt{K}}(-{\mathtt{K}}_{\mathfrak{i}})(\widehat{\mathfrak{b}},{\mathfrak{u}}_{\mathfrak{i}})\hfill \\ & +\sum _{\mathfrak{i}\in \mathtt{I}}{\widehat{\tau }}_{\mathfrak{i}}^{\mathtt{K}}{\mathtt{K}}_{\mathfrak{i}}(\overline{\mathfrak{a}},{\mathfrak{u}}_{\mathfrak{i}})-\sum _{\mathfrak{i}\in \mathtt{I}}{\widehat{\tau }}_{\mathfrak{i}}^{\mathtt{K}}{\mathtt{K}}_{\mathfrak{i}}(\widehat{\mathfrak{b}},{\mathfrak{u}}_{\mathfrak{i}})+\sum _{\mathfrak{i}\in \mathtt{I}}{\widehat{\tau }}_{\mathfrak{i}}^{\mathtt{L}}{\mathtt{L}}_{\mathfrak{i}}(\overline{\mathfrak{a}},{\mathfrak{u}}_{\mathfrak{i}})\hfill \\ & -\sum _{\mathfrak{i}\in \mathtt{I}}{\widehat{\tau }}_{\mathfrak{i}}^{\mathtt{L}}{\mathtt{L}}_{\mathfrak{i}}(\widehat{\mathfrak{b}},{\mathfrak{u}}_{\mathfrak{i}})>0.\hfill \end{array}$$

Since $\overline{\mathfrak{a}}\in {\mathbb{F}}_{\mathcal{G}}$ and $\widehat{\mathfrak{b}}\in {\mathbb{F}}_{\mathcal{G}}^{MW}$, we have

$$\begin{array}{cc}\hfill \sum _{t\in \mathbb{T}}{\widehat{\sigma }}_t^{\mathcal{G}}{\mathcal{G}}_t\left(\overline{\mathfrak{a}}\right)>& \sum _{t\in \mathbb{T}}{\widehat{\sigma }}_t^{\mathcal{G}}{\mathcal{G}}_t\left(\widehat{\mathfrak{b}}\right),\hfill \end{array}$$

which contradicts (55). Therefore, $\overline{\mathfrak{a}}=\widehat{\mathfrak{b}}$. □

Remark 4. 

Theorem 8, Theorem 9, and Theorem 10 presented in this article for the Mond–Weir-type robust dual model generalize Theorem 8, Theorem 9 and Theorem 12, respectively, deduced by Mishra et al. [45] for MPVC to a broader class of programming problems involving data uncertainty.

To illustrate the duality results deduced in the manuscript for WRD- and MWRD-type dual problems related to UNMPVC, we present the following non-trivial example in the Euclidean space setting.

Example 4. 

Let us examine the subsequent robust multiobjective programming problem with vanishing constraints under data uncertainty given by

$$\begin{array}{cc}\hfill \left(P4\right)\mathit{Minimize}& \mathcal{G}\left(\mathfrak{a}\right)=({\mathcal{G}}_1\left(\mathfrak{a}\right),{\mathcal{G}}_2\left(\mathfrak{a}\right)):=(e^{-{\mathfrak{a}}_1},e^{-2{\mathfrak{a}}_2}),\hfill \\ \hfill \mathit{subject}\mathit{to}& {\Psi }_1(\mathfrak{a},{\mathfrak{u}}_1):=-\left|{\mathfrak{a}}_1\right|{\mathfrak{u}}_1^2\le 0,\forall {\mathfrak{u}}_1\in {\Omega }_1:=[1,+\infty ),\hfill \\ & {\Psi }_2(\mathfrak{a},{\mathfrak{u}}_2):={\mathfrak{a}}_{2}cos{\mathfrak{u}}_2\le 0,\forall {\mathfrak{u}}_2\in {\Omega }_2:=\left[0,{\displaystyle \frac{\pi }{2}}\right],\hfill \\ & \mathtt{K}(\mathfrak{a},{\mathfrak{u}}_3)=-{\mathfrak{a}}_1{\mathfrak{u}}_3\ge 0,\forall {\mathfrak{u}}_3\in {\Omega }_3:=\left[1,+\infty \right),\hfill \\ & \mathtt{L}(\mathfrak{a},{\mathfrak{u}}_3)\mathtt{K}(\mathfrak{a},{\mathfrak{u}}_3)=-{\mathfrak{a}}_1{\mathfrak{a}}_2{\mathfrak{u}}_3\le 0,\forall {\mathfrak{u}}_3\in {\Omega }_3:=\left[1,+\infty \right),\hfill \end{array}$$

where ${\mathcal{G}}_t:{\mathbb{R}}^2\to \overline{\mathbb{R}},t\in \{1,2\}$, $\Psi :{\mathbb{R}}^2\times {\Omega }_1\to \overline{\mathbb{R}}$, $\mathtt{K}(\mathfrak{a},{\mathfrak{u}}_2):{\mathbb{R}}^2\times {\Omega }_2\to \overline{\mathbb{R}}$, $\mathtt{L}(\mathfrak{a},{\mathfrak{u}}_2):{\mathbb{R}}^2\times {\Omega }_2\to \overline{\mathbb{R}}$ and $z=({\mathfrak{a}}_1,{\mathfrak{a}}_2)\in {\mathbb{R}}^2.$ The set of feasible elements of (P4) is given by

$$\begin{array}{cc}\hfill {\mathbb{F}}_{\mathcal{G}}=\{({\mathfrak{a}}_1,{\mathfrak{a}}_2)\in {\mathbb{R}}^2:& -|{\mathfrak{a}}_1|{\mathfrak{u}}_1^2\le 0,{\mathfrak{a}}_{2}cos{\mathfrak{u}}_2\le 0,-{\mathfrak{a}}_{1}sin{\mathfrak{u}}_3\ge 0,\hfill \\ & -{\mathfrak{a}}_1{\mathfrak{a}}_2{\mathfrak{u}}_3\ge 0,{\mathfrak{u}}_k\in {\Omega }_k,k\in \{1,2,3\}\}.\hfill \end{array}$$

It is evident that

$$\begin{array}{cc}& {\Psi }_1^{*}\left(\mathfrak{a}\right)=\underset{{\mathfrak{u}}_1\in {\Omega }_1}{max}-|{\mathfrak{a}}_1|{\mathfrak{u}}_1^2=-\left|{\mathfrak{a}}_1\right|,\hfill \\ & {\Psi }_2^{*}\left(\mathfrak{a}\right)=\underset{{\mathfrak{u}}_2\in {\Omega }_2}{max}{\mathfrak{a}}_{2}cos{\mathfrak{u}}_2={\mathfrak{a}}_2,\hfill \\ & {\mathtt{K}}^{*}\left(\mathfrak{a}\right)=\underset{{\mathfrak{u}}_3\in {\Omega }_3}{max}-{\mathfrak{a}}_1{\mathfrak{u}}_3=-{\mathfrak{a}}_1,\hfill \\ & {\mathtt{L}}^{*}\left(\mathfrak{a}\right)=\underset{{\mathfrak{u}}_2\in {\Omega }_2}{max}{\mathfrak{a}}_2={\mathfrak{a}}_2.\hfill \end{array}$$

It can be verified that given any feasible solution to the primal problem, inequality (23) is satisfied. Hence, Theorem 5 is satisfied for the considered problem (P4). Furthermore, GS-ACQ is satisfied at $\overline{\mathfrak{a}}=(0,0)$ and there exists $\overline{\sigma }=({\overline{\sigma }}^{\mathcal{G}},{\overline{\sigma }}^{\Psi },{\overline{\sigma }}^{\mathtt{K}},{\overline{\tau }}^{\mathtt{K}},{\overline{\tau }}^{\mathtt{L}})\in {\mathbb{R}}^r\times {\mathbb{R}}^p\times {\mathbb{R}}^{3m}$, such that $(\overline{\mathfrak{a}},\overline{\sigma })$ is a feasible solution of the dual problem. Hence, Theorem 6 holds for the considered problem (P4). Moreover, objective functions are strictly ${\partial }^{*}$-convex at $\overline{\mathfrak{a}}$, and according to Theorem 4, $\overline{\mathfrak{a}}$ is a robust local weak Pareto solution of the primal problem (P4). It can be verified from the weak duality theorem that $\overline{\mathfrak{a}}=(0,0)$ is the global robust weak Pareto solution of the Wolfe-type dual problem (P4). Hence, Theorem 7 is verified for the problem (P4).

Now, corresponding to (P4), we formulate the MWRD as follows:

$$\begin{array}{cc}\hfill & \underset{(\mathfrak{b},\sigma )}{\mathit{Maximize}}\mathcal{G}\left(\mathfrak{b}\right)=(e^{-y_1},e^{-2y_2}),\hfill \\ \hfill & \mathit{subject}\mathit{to}{\Psi }_1(\mathfrak{b},{\mathfrak{u}}_1):=-\left|y_1\right|{\mathfrak{u}}_1^2\ge 0,\forall {\mathfrak{u}}_1\in {\Omega }_1:=[1,+\infty ),\hfill \\ \hfill & {\Psi }_2(\mathfrak{b},{\mathfrak{u}}_2):=y_{2}cos{\mathfrak{u}}_2\ge 0,\forall {\mathfrak{u}}_2\in {\Omega }_2:=\left[0,{\displaystyle \frac{\pi }{2}}\right],\hfill \\ \hfill & \mathtt{K}(\mathfrak{b},{\mathfrak{u}}_3)=-y_1{\mathfrak{u}}_3\le 0,\forall {\mathfrak{u}}_3\in {\Omega }_3:=\left[1,+\infty \right),\hfill \\ \hfill & \mathtt{L}(\mathfrak{b},{\mathfrak{u}}_3)\mathtt{K}(\mathfrak{b},{\mathfrak{u}}_3)=-y_1{y}_2{\mathfrak{u}}_3\ge 0,\forall {\mathfrak{u}}_3\in {\Omega }_3:=\left[1,+\infty \right),\hfill \end{array}$$

Similar to the Wolfe-type robust dual model, it can be verified that given any feasible solution to the considered problem (P4), inequality (44) is satisfied. Hence, Theorem 8 holds for the considered problem (P4). Moreover, at $\overline{\mathfrak{a}}=(0,0)$, GS-ACQ is satisfied and there exists $\overline{\sigma }=({\overline{\sigma }}^{\mathcal{G}},{\overline{\sigma }}^{\Psi },{\overline{\sigma }}^{\mathtt{K}},{\overline{\tau }}^{\mathtt{K}},{\overline{\tau }}^{\mathtt{L}})\in {\mathbb{R}}^r\times {\mathbb{R}}^p\times {\mathbb{R}}^{3m}$, such that $(\overline{\mathfrak{a}},\overline{\sigma })$ is a feasible solution of the dual problem. Hence, Theorem 9 holds for the considered problem (P4). Moreover, objective functions are strictly ${\partial }^{*}$-convex at $\overline{\mathfrak{a}}$, and from Theorem 4, $\overline{\mathfrak{a}}$ is a robust local weak Pareto solution of the primal problem (P4). It can be verified from Theorem 8 that $\overline{\mathfrak{a}}=(0,0)$ is the global robust weak Pareto solution of the Mond–Weir-type dual problem of (P4). Hence, Theorem 10 is verified for the problem (P4).

6. Application and Significance

In this section, we explore the application of UNMPVC in truss topology optimization with uncertainty due to variations in temperature. We examine a truss of size $2\times 1$ consisting of six nodes and ten potential bars, with the two leftmost nodes fixed. The cross-sectional area of each of the 10 bars ($i=1,\dots ,10$) is denoted by ${\mathfrak{a}}_{\mathfrak{i}}$. Moreover, the length of each bar is denoted by $L_{\mathfrak{i}}$. Let $\mathfrak{a}\in {\mathbb{R}}^{10}$, and $y_1,y_2,y_3\in {\mathbb{R}}^8$ are three auxiliary vector variables as defined in [59]. The global stiffness matrix of the truss, $\mathcal{K}\left(\mathfrak{a}\right)$, is given by

$$\mathcal{K}\left(\mathfrak{a}\right)=\sum _{i=1}^{10}{\displaystyle \frac{E{\mathfrak{a}}_{\mathfrak{i}}}{L_{\mathfrak{i}}}}{\gamma }_{\mathfrak{i}}{\gamma }_{\mathfrak{i}}^T\in {\mathbb{R}}^{8\times 8},$$

with some vectors ${{\rm Y}}_{\mathfrak{i}}\in {\mathbb{R}}^8$ and the Young’s modulus E.

It is worth noting that $\mathcal{K}\left(\mathfrak{a}\right)$ depends on the Young’s modulus, which is influenced by factors such as temperature, pressure, and material composition. Suppose that there is uncertainty due to variation in temperature, i.e., $t\in \Omega :=[T^L,T^U]$, where $T^L$ and $T^U$ represent the minimum and maximum temperature of the environment, respectively. Therefore, we suppose that $\mathcal{K}(\mathfrak{a},t)$, $t\in \Omega$ represents the uncertain global stiffness matrix of the truss.

Building on the above discussions, we now consider the following uncertain truss optimization problem:

$$\begin{array}{cc}\hfill \left(\mathrm{P}5\right)\mathrm{Minimize}& \sum _{\mathfrak{i}=1}^{10}{L}_{\mathfrak{i}}{\mathfrak{a}}_{\mathfrak{i}},\hfill \\ \hfill \mathrm{subject}\mathrm{to}& \mathcal{K}(\mathfrak{a},t)y_j-f_j=0,\forall j\in \{1,2,3\},\hfill \\ & f_j^T{y}_j-c\le 0,\forall j\in \{1,2,3\},\hfill \\ & -{\mathfrak{a}}_{\mathfrak{i}}\le 0,\forall \mathfrak{i}\in \{1,\dots ,10\},\hfill \\ & {\mathfrak{a}}_{\mathfrak{i}}-A_{max}\le 0,\forall i=1,\dots ,10,\hfill \\ & ({\sigma }_{\mathfrak{i}}{(a,y_j)}^2-{\sigma }_{max}^2){\mathfrak{a}}_{\mathfrak{i}}\le 0,\forall \mathfrak{i}\in \{1,\dots ,10\},\forall j\in \{1,2,3\},\hfill \\ & (A_{min}-{\mathfrak{a}}_{\mathfrak{i}}){\mathfrak{a}}_{\mathfrak{i}}\le 0\forall \mathfrak{i}\in \{1,\dots ,10\},\hfill \end{array}$$

where the inequality $f_j^T{y}_j\le c$ bounds the work caused by force $f_j$, $c>0$ is a constant, and $t\in \Omega$ is an uncertain parameter. Moreover, $A_{\max }$ and $A_{\min }$ represent the maximum and minimum cross-sectional area, respectively. Furthermore, for each $\mathfrak{i}\in \{1,\dots ,10\}$, the function ${\sigma }_{\mathfrak{i}}(a,y_j)$ denotes the stress on each bar.

Then, the robust counterpart of the considered problem (P5) is given by

$$\begin{array}{cc}\hfill \left(\mathrm{RC}1\right)\mathrm{Minimize}& \sum _{i=1}^{10}{L}_{\mathfrak{i}}{\mathfrak{a}}_{\mathfrak{i}},\hfill \\ \hfill \mathrm{subject}\mathrm{to}& \mathcal{K}(\mathfrak{a},t)y_j-f_j=0,\forall t\in \Omega ,\forall j\in \{1,2,3\},\hfill \\ & f_j^T{y}_j-c\le 0,\forall j\in \{1,2,3\},\hfill \\ & 0\le {\mathfrak{a}}_{\mathfrak{i}}\le A_{max},\forall \mathfrak{i}\in \{1,\dots ,10\},\hfill \\ & ({\sigma }_{\mathfrak{i}}{(a,y_j)}^2-{\sigma }_{max}^2){\mathfrak{a}}_{\mathfrak{i}}\le 0,\forall \mathfrak{i}\in \{1,\dots ,10\},\forall j\in \{1,2,3\},\hfill \\ & (A_{min}-{\mathfrak{a}}_{\mathfrak{i}}){\mathfrak{a}}_{\mathfrak{i}}\le 0\forall \mathfrak{i}\in \{1,\dots ,10\}.\hfill \end{array}$$

The problem (P5) can be effectively solved by employing the results established in this article.

Remark 5. 

If Ω is a singleton set, then the problem (P5) reduces to the well-known deterministic truss optimization problem considered by Hoheisel et al. [59].

To illustrate the importance of the results deduced in the present article, we consider the subsequent example:

Example 5. 

We consider the following problem with vanishing constraints under data uncertainty:

$$\begin{array}{cc}\hfill \left(P6\right)\mathit{Minimize}& \mathcal{G}\left(\mathfrak{a}\right)=({\mathcal{G}}_1\left(\mathfrak{a}\right),{\mathcal{G}}_2\left(\mathfrak{a}\right)):=(e^{-{\mathfrak{a}}_1},e^{-2{\mathfrak{a}}_2}),\hfill \\ \hfill \mathit{subject}\mathit{to}& \Psi (\mathfrak{a},{\mathfrak{u}}_1):=-{\mathfrak{a}}_1{\mathfrak{u}}_1^2\le 0,\forall {\mathfrak{u}}_1\in {\Omega }_1:=[1,+\infty ),\hfill \\ & \mathtt{K}(\mathfrak{a},{\mathfrak{u}}_2):={\mathfrak{a}}_{2}sin{\mathfrak{u}}_2\ge 0,\forall {\mathfrak{u}}_2\in {\Omega }_2:=\left[0,{\displaystyle \frac{\pi }{2}}\right],\hfill \\ & \mathtt{L}(\mathfrak{a},{\mathfrak{u}}_3)\mathtt{K}(\mathfrak{a},{\mathfrak{u}}_3):=-{\mathfrak{a}}_1{\mathfrak{a}}_{2}sin{\mathfrak{u}}_2\le 0,\forall {\mathfrak{u}}_2\in {\Omega }_2:=\left[0,{\displaystyle \frac{\pi }{2}}\right],\hfill \end{array}$$

where ${\mathcal{G}}_t:{\mathbb{R}}^2\to \overline{\mathbb{R}},t\in \{1,2\}$, $\Psi :{\mathbb{R}}^2\times {\Omega }_1\to \overline{\mathbb{R}}$, $\mathtt{K}(\mathfrak{a},{\mathfrak{u}}_2):{\mathbb{R}}^2\times {\Omega }_2\to \overline{\mathbb{R}}$, $\mathtt{L}(\mathfrak{a},{\mathfrak{u}}_2):{\mathbb{R}}^2\times {\Omega }_2\to \overline{\mathbb{R}}$ and $z=({\mathfrak{a}}_1,{\mathfrak{a}}_2)\in {\mathbb{R}}^2.$ It is evident that

$$\begin{array}{cccc}& {\Psi }^{*}\left(\mathfrak{a}\right)=\underset{{\mathfrak{u}}_1\in {\Omega }_1}{max}-{\mathfrak{a}}_1{\mathfrak{u}}_1^2=-{\mathfrak{a}}_1,\hfill & & {\mathtt{K}}^{*}\left(\mathfrak{a}\right)=\underset{{\mathfrak{u}}_2\in {\Omega }_2}{max}{\mathfrak{a}}_{2}sin{\mathfrak{u}}_2={\mathfrak{a}}_2,\hfill \\ & {\mathtt{L}}^{*}\left(\mathfrak{a}\right){\mathtt{K}}^{*}\left(\mathfrak{a}\right)=-{\mathfrak{a}}_1{\mathfrak{a}}_2,\hfill \end{array}$$

and $\overline{\mathfrak{a}}=(0,0)$ is a robust local weak Pareto solution in (P6). Furthermore, in view of Definition 3, the upper semi-regular convexificators at $\overline{\mathfrak{a}}$ of each function constituting (P6) are given by

$$\begin{array}{cccc}& {\partial }^{*}{\mathcal{G}}_1\left(\overline{\mathfrak{a}}\right)=\left\{(1,0)\right\},\hfill & & {\partial }^{*}{\mathcal{G}}_2\left(\overline{\mathfrak{a}}\right)=\left\{(0,1)\right\},\hfill \\ & {\partial }^{*}{\Psi }^{*}\left(\overline{\mathfrak{a}}\right)=\left\{(-1,0)\right\},\hfill & & {\partial }^{*}{\mathtt{K}}^{*}\left(\overline{\mathfrak{a}}\right)=\left\{(0,1)\right\},\hfill \\ & {\partial }^{*}(-{\mathtt{K}}^{*})\left(\overline{\mathfrak{a}}\right)=\left\{(0,-1)\right\},\hfill & & {\partial }^{*}{\mathtt{L}}^{*}\left(\overline{\mathfrak{a}}\right)=\left\{(-1,0)\right\}.\hfill \end{array}$$

It can be verified that, at $\overline{\mathfrak{a}}=(0,0)$, GS-ACQ holds, and $D^t$ is a closed set. Moreover, $\overline{\mathfrak{a}}=(0,0)$ is a RS-stationary point of (P6). In addition to this, at $\overline{\mathfrak{a}}$, ${\mathcal{G}}_t,t\in \{1,2\}$ are ${\partial }^{*}$-pseudoconvex; Ψ, $\pm \mathtt{K}$, and $\mathtt{L}$ are ${\partial }^{*}$-quasiconvex. Hence, the hypothesis of Theorem 4 is fulfilled at $\overline{\mathfrak{a}}$.

We formulate WRD corresponding to the primal problem (P6) as follows:

$$\begin{array}{cc}\hfill \left(\mathit{WD}\right)\underset{(\mathfrak{b},\sigma )}{\mathit{Maximize}}\{\mathcal{G}\left(\mathfrak{b}\right)+{\sigma }^{\Psi }(-{\mathfrak{a}}_1{\mathfrak{u}}_1^2)e+& {\sigma }^{\mathtt{K}}(-{\mathfrak{a}}_{2}sin{\mathfrak{u}}_2)e\hfill \\ & +{\tau }^{\mathtt{K}}({\mathfrak{a}}_{2}sin{\mathfrak{u}}_2)e+{\tau }^{\mathtt{L}}(-{\mathfrak{a}}_1)e,e=(1,1)\in {\mathbb{R}}^2\},\hfill \end{array}$$

where $\mathcal{G}\left(\mathfrak{b}\right)=(e^{-{\mathfrak{a}}_1},e^{-2{\mathfrak{a}}_2})$. The set of feasible solutions is defined as

$$\begin{array}{cc}\hfill {\mathbb{F}}_{\mathcal{G}}^{WRD}:=\{\sigma =& ({\sigma }^{\mathcal{G}},{\sigma }^{\Psi },{\sigma }^{\mathtt{K}},{\tau }^{\mathtt{K}},{\tau }^{\mathtt{L}})\in {\mathbb{R}}^r\times {\mathbb{R}}^p\times {\mathbb{R}}^{3m},{\left({\sigma }^{\mathcal{G}}\right)}^{T}e=1,\hfill \\ & \mathfrak{b}\in {\mathbb{R}}^n:0\in \sum _{t\in \{1,2\}}{\sigma }_t^{\mathcal{G}}co{\partial }^{*}{\mathcal{G}}_t\left(\mathfrak{b}\right)+{\sigma }_j^{\Psi }co{\partial }^{*}{\Psi }_j(\mathfrak{b},{\mathfrak{u}}_j)\hfill \\ \hfill & +\sum _{\mathfrak{i}\in \mathtt{I}}{\sigma }_{\mathfrak{i}}^{\mathtt{K}}co{\partial }^{*}(-{\mathtt{K}}_{\mathfrak{i}})(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{i}})\hfill \\ \hfill & +\sum _{\mathfrak{i}\in \mathtt{I}}[{\tau }_{\mathfrak{i}}^{\mathtt{K}}co{\partial }^{*}{\mathtt{K}}_{\mathfrak{i}}(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{i}})+{\tau }_{\mathfrak{i}}^{\mathtt{L}}co{\partial }^{*}{\mathtt{L}}_{\mathfrak{i}}(\mathfrak{b},{\mathfrak{u}}_{\mathfrak{i}})],\hfill \\ \hfill & {\sigma }_t^{\mathcal{G}}>0,t\in \mathbb{T},{\sigma }_j^{\Psi }\ge 0,j\in {\mathfrak{T}}_{\Psi },\hfill \\ \hfill & {\sigma }_{\mathfrak{i}}^{\mathtt{K}}\ge 0,{\tau }_{\mathfrak{i}}^{\mathtt{K}}\ge 0,{\tau }_{\mathfrak{i}}^{\mathtt{L}}\ge 0,\mathfrak{i}\in \mathtt{I},\hfill \\ & {\sigma }_{{\mathfrak{T}}_{+0}\cup {\mathfrak{T}}_{+-}}^{\mathtt{K}}={\tau }_{{\mathfrak{T}}_{+0}\cup {\mathfrak{T}}_{+-}}^{\mathtt{K}}={\tau }_{{\mathfrak{T}}_{0+}\cup {\mathfrak{T}}_{+-}\cup {\mathfrak{T}}_{0-}}^{\mathtt{L}}=0,\hfill \\ \hfill & {\sigma }_i^{\mathtt{K}}-{\tau }_i^{\mathtt{K}}\ge 0,\mathfrak{i}\in {\mathfrak{T}}_{0-},{\tau }_{\mathfrak{i}}^{\mathtt{L}}=0,({\sigma }_{\mathfrak{i}}^{\mathtt{K}}-{\tau }_{\mathfrak{i}}^{\mathtt{K}})\ge 0,\forall \mathfrak{i}\in {\mathfrak{T}}_{00}\}.\hfill \end{array}$$

Since, at $\overline{\mathfrak{a}}=(0,0)$, GS-ACQ holds, and there exists $\overline{\sigma }=({\overline{\sigma }}^{\mathcal{G}},{\overline{\sigma }}^{\Psi },{\overline{\sigma }}^{\mathtt{K}},{\overline{\tau }}^{\mathtt{K}},{\overline{\tau }}^{\mathtt{L}})\in {\mathbb{R}}^r\times {\mathbb{R}}^p\times {\mathbb{R}}^{3m}$, such that $(\overline{\mathfrak{a}},\overline{\sigma })$ is a feasible solution of the dual problem. Hence, Theorem 6 holds for the considered problem $\left(P6\right)$. Moreover, objective functions are strictly ${\partial }^{*}$-convex at $\overline{\mathfrak{a}}$ and from Theorem 4, $\overline{\mathfrak{a}}$ is a robust local weak Pareto solution of the primal problem $\left(P6\right)$. It can be verified from the Theorem 5 that $\overline{\mathfrak{a}}=(0,0)$ is the global robust weak Pareto solution of the Wolfe-type dual problem of $\left(P6\right)$. Hence, Theorem 7 is verified for the problem $\left(P6\right)$.

Now, corresponding to the problem $\left(P6\right)$, the MWRD problem can be stated as

$$\begin{array}{cc}\hfill \underset{(\mathfrak{b},\sigma )}{\mathit{Maximize}}& \mathcal{G}\left(\mathfrak{b}\right)=(e^{y_1},e^{y_2}),\hfill \\ \hfill \mathit{subject}\mathit{to}& \Psi (\mathfrak{b},{\mathfrak{u}}_1):=-y_1{\mathfrak{u}}_1^2\le 0,\forall {\mathfrak{u}}_1\in {\Omega }_1:=[1,+\infty ),\hfill \\ \hfill & \mathtt{K}(\mathfrak{b},{\mathfrak{u}}_2):=y_{2}sin{\mathfrak{u}}_2\ge 0,\forall {\mathfrak{u}}_2\in {\Omega }_2:=\left[0,{\displaystyle \frac{\pi }{2}}\right],\hfill \\ \hfill & \mathtt{L}(\mathfrak{b},{\mathfrak{u}}_3)\mathtt{K}(\mathfrak{b},{\mathfrak{u}}_3):=-y_1{y}_{2}sin{\mathfrak{u}}_2\le 0,\forall {\mathfrak{u}}_2\in {\Omega }_2:=\left[0,{\displaystyle \frac{\pi }{2}}\right].\hfill \end{array}$$

Similar to the WRD, it can be verified that given any feasible solution to the primal problem, inequality (44) is satisfied. Hence, Theorem 8 holds for the considered problem $\left(P6\right)$. Moreover, at $\overline{\mathfrak{a}}=(0,0)$, GS-ACQ holds and there exists $\overline{\sigma }=({\overline{\sigma }}^{\mathcal{G}},{\overline{\sigma }}^{\Psi },{\overline{\sigma }}^{\mathtt{K}},{\overline{\tau }}^{\mathtt{K}},{\overline{\tau }}^{\mathtt{L}})\in {\mathbb{R}}^r\times {\mathbb{R}}^p\times {\mathbb{R}}^{3m}$, such that $(\overline{\mathfrak{a}},\overline{\sigma })$ is a feasible solution of the dual problem. Hence, Theorem 9 holds for the considered problem $\left(P6\right)$. Moreover, the objective functions are strictly ${\partial }^{*}$-convex at $\overline{\mathfrak{a}}$, and from Theorem 4, $\overline{\mathfrak{a}}$ is a robust local weak Pareto solution of the primal problem $\left(P6\right)$. It can be verified from Theorem 8 that $\overline{\mathfrak{a}}=(0,0)$ is the global robust weak Pareto solution of the MWRD problem of $\left(P6\right)$. Hence, Theorem 10 is verified for the problem $\left(P6\right)$.

7. Conclusions and Future Research Directions

In this article, a class of nonsmooth multiobjective programming problems with vanishing constraints under data uncertainty (UNMPVC) have been investigated. We introduced GS-ACQ for the considered problem UNMPVC. Moreover, we have introduced RW-stationary, RM-stationary and RS-stationary conditions for the problem UNMPVC. By employing GS-ACQ, we have established that the necessary local weak Pareto efficiency conditions for UNMPVC. Subsequently, under generalized convexity assumptions, we have derived sufficient optimality criteria. Furthermore, we have formulated WRD and MWRD dual models and derived duality results, namely weak, strong, and strict converse duality theorems, that relate the original problem UNMPVC to the corresponding dual models.

The Pareto efficiency conditions and duality theorems deduced in the present article extend various well-known results from the literature on nonsmooth MOP to a more general programming problem, UNMPVC, via convexificators. Specifically, the optimality conditions and duality results presented in this article generalize the corresponding optimality conditions established in [40,55] and duality results established in [45] to a broader category of optimization problems UNMPVC.

The findings presented in this article open up several directions for future research. In view of the work presented in this article, it would be interesting to explore various other CQs and deduce interrelationships among them for nonsmooth multiobjective programming problems with vanishing constraints under data uncertainty.