On Mond–Weir-Type Robust Duality for a Class of Uncertain Fractional Optimization Problems

Abstract

This article is focused on the investigation of Mond–Weir-type robust duality for a class of semi-infinite multi-objective fractional optimization with uncertainty in the constraint functions. We first establish a Mond–Weir-type robust dual problem for this fractional optimization problem. Then, by combining a new robust-type subdifferential constraint qualification condition and a generalized convex-inclusion assumption, we present robust $\epsilon$-quasi-weak and strong duality properties between this uncertain fractional optimization and its uncertain Mond–Weir-type robust dual problem. Moreover, we also investigate robust $\epsilon$-quasi converse-like duality properties between them.

1. Introduction

Let T be a nonempty infinite index set. Suppose that $f_i:{\mathbb{R}}^n\to \mathbb{R}$, $i=1,\dots ,p$, and $h_t:{\mathbb{R}}^n\to \mathbb{R}$, $t\in T$. Let us consider the semi-infinite optimization problem:

$$\begin{array}{c}\hfill \left(\mathrm{MP}\right)\left\{\begin{array}{c}{\mathrm{Min}}_{{\mathbb{R}}_{+}^p}\left(f_1\left(x\right),\dots ,f_p\left(x\right)\right)\hfill \\ s.t.h_t\left(x\right)\le 0,\forall t\in T,\hfill \\ x\in {\mathbb{R}}^n.\hfill \end{array}\right.\end{array}$$

The study of optimization problem $\left(\mathrm{MP}\right)$ is a very interesting topic and has been considered extensively by many scholars from different points of view, see [1,2,3,4,5,6,7,8,9,10,11,12,13]. However, most semi-infinite optimization models of real-world problems are contaminated by prediction errors or asymmetry knowledge. Thus, it is necessary to consider semi-infinite optimization problems under uncertain data. This optimization problem $\left(\mathrm{MP}\right)$ with uncertainty can be captured by

$$\begin{array}{c}\hfill \left(\mathrm{UMP}\right)\left\{\begin{array}{c}{\mathrm{Min}}_{{\mathbb{R}}_{+}^p}\left(f_1\left(x\right),\dots ,f_p\left(x\right)\right)\hfill \\ s.t.h_t(x,v_t)\le 0,\forall t\in T,\hfill \\ x\in {\mathbb{R}}^n.\hfill \end{array}\right.\end{array}$$

Here, $h_t:{\mathbb{R}}^n\times {\mathbb{R}}^q\to \mathbb{R}$, $t\in T$, are given functions, $v_t$, $t\in T$, are uncertain parameters which belongs to compact sets ${\mathcal{V}}_t\subseteq {\mathbb{R}}^q$.

As we know, robust optimization [14,15,16] is an useful approach to solve optimization problems with uncertainty. Following robust optimization methodology, we usually associate UMP with its robust counterpart

$$\begin{array}{c}\hfill \left(\mathrm{RMP}\right)\left\{\begin{array}{c}{\mathrm{Min}}_{{\mathbb{R}}_{+}^p}\left(f_1\left(x\right),\dots ,f_p\left(x\right)\right)\hfill \\ s.t.h_t(x,v_t)\le 0,\forall v_t\in {\mathcal{V}}_t,t\in T,\hfill \\ x\in {\mathbb{R}}^n.\hfill \end{array}\right.\end{array}$$

Recently, following robust optimization methodology, many interesting results devoted to $\left(\mathrm{UMP}\right)$ and its generalizations have been obtained from several different perspectives. By using scalarizing methods and robust optimization, Lee and Lee [17] establish necessary optimality theorems for robust weakly and properly efficient solutions of a multi-objective optimization problem with uncertainty. By virtue of a new concept of generalized convexity and robust type constraint qualification conditions, Chen et al. [18] give some optimality conditions and duality results for an uncertain nonconvex and nonsmooth multi-objective optimization problem. Guo and Yu [19] obtain optimality conditions for robust approximate quasi-weakly efficient solutions for uncertain multi-objective convex optimization problems. By combining robust optimization and scalarization technique, Sun et al. [20] give some new characterizations of Wolfe type robust approximate duality and saddle point theorems for a nonsmooth robust multi-objective optimization problem. Sun et al. [21] investigate optimality conditions for robust $ϵ$-quasi efficient solutions of a class of uncertain semi-infinite multi-objective optimization under some tools of non-smooth analysis and a new modified scalarization technique. In addition, nonsmooth robust $ϵ$-duality properties and $ϵ$-quasi saddle point theorems are also established. New results on optimality and duality results for uncertain multiobjective polynomial optimization problems are given in [22]. By using tangential subdifferential and robust optimization, Liu et al. [23] obtained some characterizations of robust optimal solution sets for nonconvex uncertain semi-infinite optimization problems.

On the other hand, the fractional multi-objective optimization problem is an important subclass of multi-objective optimization problems. In the last decades, a wide variety of interesting works devoted to fractional multi-objective optimization problems and its generalizations have been given, see, for example, [24,25,26,27,28,29,30,31,32,33]. We observe that there are some papers devoted to the study of uncertain fractional multi-objective optimization problems under a robust optimization approach. In [34], the authors study approximate optimality conditions and Wolfe-type robust approximate duality of robust approximate weakly efficient solutions for uncertain fractional multi-objective optimization problems. Li et al. [35] establish optimality theorems and robust duality properties for minimax convex–concave fractional optimization problems with uncertainty. Antczak [36] establish a new parametric approach for robust approximate quasi-efficient solutions of robust fractional multi-objective optimization problems. Feng and Sun [37] obtain some new results for robust weakly $ϵ$-efficient solutions for an uncertain fractional multi-objective semi-infinite optimization by employing conjugate analysis. Very recently, by employing robust limiting constraint qualification conditions and generalized convexity assumptions, Thuy and Su [38] consider optimality conditions and duality results for nonsmooth fractional multi-objective semi-infinite optimization problems with uncertain data.

In this paper, our main concern is to give new duality results of robust $ϵ$-quasi-efficient solutions for fractional multi-objective semi-infinite optimization problems ($\mathrm{UFP}$, for brevity) with uncertainty appearing in the constraint functions. We first introduce the robust counterpart model ($\mathrm{RFP}$, for brevity) for $\mathrm{UFP}$. Then, with the help of a robust-type subdifferential constraint qualification, we present a necessary approximate optimality condition for robust $ϵ$-quasi-efficient solutions for ($\mathrm{UFP}$). Subsequently, we introduce a Mond–Weir-type robust approximate dual problem of ($\mathrm{UFP}$) based on the obtained necessary optimality conditions. Then, we investigate robust weak, strong and converse-like duality results between them under a new assumption of generalized convex-inclusion for Lipschitz functions.

This paper is organized as follows. In Section 2, we first recall some basic concepts in nonsmooth analysis and present approximate optimality results for robust $ϵ$-quasi-efficient solutions of ($\mathrm{UFP}$). In Section 3, we introduce a Mond–Weir-type robust approximate dual problem for ($\mathrm{UFP}$), and establish the robust $ϵ$-quasi duality results between them. As a special case, we also deal with robust $ϵ$-quasi duality results of the uncertain multi-objective optimization problem ($\mathrm{UMP}$) and its robust approximate dual problem.

2. Mathematical Preliminaries

In this paper, let us recall some concepts and preliminary results [39,40]. Let ${\mathbb{R}}^p$ be the p-dimensional Euclidean space. We use the notation $\parallel ·\parallel$ for the Euclidean norm for ${\mathbb{R}}^p$. The nonnegative orthant of ${\mathbb{R}}^p$ is defined by ${\mathbb{R}}_{+}^p:=\{x=(x_1,\dots ,x_n)|x_k\ge 0,k=1,\dots ,n\}$. We always use the symbol $⟨·,·⟩$ for the inner product in ${\mathbb{R}}^p$. The closed unit ball of ${\mathbb{R}}^p$ is denoted by ${\mathbb{B}}^{*}$. For a nonempty infinite index set T, the linear space ${\mathbb{R}}^{\left(T\right)}$[41] is denoted by

$$\begin{array}{c}\hfill {\mathbb{R}}^{\left(T\right)}:=\{{\gamma }_T={\left({\gamma }_t\right)}_{t\in T}|{\gamma }_t=0\mathrm{for}\mathrm{all}t\in T\mathrm{except}\mathrm{for}\mathrm{finitely}\mathrm{many}{\gamma }_t\ne 0\}.\end{array}$$

Let ${\mathbb{R}}_{+}^{\left(T\right)}$ be the nonnegative cone of ${\mathbb{R}}^{\left(T\right)}$, i.e.,

$$\begin{array}{c}\hfill {\mathbb{R}}_{+}^{\left(T\right)}:=\{{\gamma }_T\in {\mathbb{R}}^{\left(T\right)}|{\gamma }_t\ge 0,\forall t\in T\}.\end{array}$$

Let $\varphi :{\mathbb{R}}^p\to \mathbb{R}$ be a locally Lipschitz function. The Clarke generalized directional derivative of $\varphi$ at $x\in {\mathbb{R}}^p$ in the direction $d\in {\mathbb{R}}^p$ is defined by

$${\varphi }^c(x;d):={\displaystyle \underset{y\to x,t\downarrow 0}{lim\; sup}}\frac{\varphi (y+td)-\varphi \left(y\right)}{t}.$$

The one-sided directional derivative of $\varphi$ at $x\in {\mathbb{R}}^p$ in direction $d\in {\mathbb{R}}^p$ is defined by

$${\varphi }^{\prime }(x;d):={\displaystyle \underset{t\downarrow 0}{lim}}\frac{\varphi (x+td)-\varphi \left(x\right)}{t}.$$

We say that $\varphi$ is quasidifferentiable at $x\in {\mathbb{R}}^p$ iff, for each $d\in {\mathbb{R}}^n$, ${\phi }^{\prime }(x;d)$ exists and ${\phi }^{\prime }(x;d)={\phi }^c(x;d)$. The Clarke subdifferential ${\partial }^c\varphi \left(x\right)$ of $\varphi$ at $x\in {\mathbb{R}}^p$ is defined by

$${\partial }^c\varphi \left(x\right):=\left\{{\xi }^{*}\in {\mathbb{R}}^p|{\varphi }^c(x;d)\ge ⟨{\xi }^{*},d⟩,\forall d\in {\mathbb{R}}^p\right\}.$$

Obviously,

$${\varphi }^c(x;d)={\displaystyle \underset{\xi \in {\partial }^c\varphi \left(x\right)}{sup}}⟨\xi ,d⟩,\forall d\in {\mathbb{R}}^n.$$

On the other hand, if $\varphi :{\mathbb{R}}^p\to \mathbb{R}$ is a convex function, ${\partial }^c\varphi \left(x\right)$ coincides with the convex subdifferential $\partial \varphi \left(x\right)$, that is

$$\begin{array}{c}\hfill \partial \varphi \left(x\right):=\{{\xi }^{*}\in {\mathbb{R}}^p|\varphi \left(y\right)-\varphi \left(x\right)\ge ⟨{\xi }^{*},y-x⟩,\forall y\in {\mathbb{R}}^p\}.\end{array}$$

Let $\Omega \subseteq {\mathbb{R}}^p$ be a nonempty subset. The Clarke normal cone to $\Omega$ at $x\in \Omega$ is defined by

$$N^c(\Omega ,x):=\{\xi \in {\mathbb{R}}^p|⟨{\xi }^{*},w⟩\le 0,\forall w\in T_{\Omega }\left(x\right)\}.$$

Here, $T_{\Omega }\left(x\right)$ is the Clarke tangent cone to $\Omega$ at $x\in \Omega$. Clearly, if $\Omega \subseteq {\mathbb{R}}^n$ is a nonempty closed convex set, $N^c(\Omega ,x)$ becomes the following normal cone:

$$N(\Omega ,x):=\{{\xi }^{*}\in {\mathbb{R}}^p|⟨{\xi }^{*},y-x⟩\le 0,\forall y\in \Omega \}.$$

In what follows, let $f_i,g_i:{\mathbb{R}}^n\to \mathbb{R}$, $i=1,\dots ,p$, and $h_t:{\mathbb{R}}^n\to \mathbb{R}$, $t\in T$. We consider the following fractional multi-objective optimization problem

$$\begin{array}{c}\hfill \left(\mathrm{FP}\right)\left\{\begin{array}{c}{\mathrm{Min}}_{{\mathbb{R}}_{+}^p}\left(\frac{f_1\left(x\right)}{g_1\left(x\right)},\dots ,\frac{f_p\left(x\right)}{g_p\left(x\right)}\right)\hfill \\ s.t.h_t\left(x\right)\le 0,\forall t\in T,\hfill \\ x\in {\mathbb{R}}^n.\hfill \end{array}\right.\end{array}$$

The fractional optimization problem $\left(\mathrm{FP}\right)$ under uncertain data in the constraint functions becomes

$$\begin{array}{c}\hfill \left(\mathrm{UFP}\right)\left\{\begin{array}{c}{\mathrm{Min}}_{{\mathbb{R}}_{+}^p}\left(\frac{f_1\left(x\right)}{g_1\left(x\right)},\dots ,\frac{f_p\left(x\right)}{g_p\left(x\right)}\right)\hfill \\ s.t.h_t(x,v_t)\le 0,\forall t\in T,\hfill \\ x\in {\mathbb{R}}^n.\hfill \end{array}\right.\end{array}$$

Here $h_t:{\mathbb{R}}^n\times {\mathbb{R}}^q\to \mathbb{R}$. $v_t\in {\mathcal{V}}_t\subseteq {\mathbb{R}}^q$, $t\in T$ are uncertain parameters.

For $\left(\mathrm{UFP}\right)$, we consider its robust counterpart, namely

$$\begin{array}{c}\hfill \left(\mathrm{RFP}\right)\left\{\begin{array}{c}{\mathrm{Min}}_{{\mathbb{R}}_{+}^p}\left(\frac{f_1\left(x\right)}{g_1\left(x\right)},\dots ,\frac{f_p\left(x\right)}{g_p\left(x\right)}\right)\hfill \\ s.t.h_t(x,v_t)\le 0,\forall v_t\in {\mathcal{V}}_t,t\in T,\hfill \\ x\in {\mathbb{R}}^n.\hfill \end{array}\right.\end{array}$$

In this paper, without special statements, let $f_i$, $i=1,\dots ,p$, be locally Lipschitz functions with $f_i\left(x\right)\ge 0$, $\forall x\in {\mathbb{R}}^n$, and $g_i$, $i=1,\dots ,p$, be locally Lipschitz functions with $g_i\left(x\right)>0$, $\forall x\in {\mathbb{R}}^n$.

Now, we give the following important notations, which will be used later in this paper.

Definition 1. 

For $\left(\mathrm{UFP}\right)$. We say that $\mathcal{F}$ is the robust feasible set of $\left(\mathrm{UFP}\right)$ iff

$$\mathcal{F}:=\left\{x\in {\mathbb{R}}^n|h_t(x,v_t)\le 0,\forall v_t\in {\mathcal{V}}_t,t\in T\right\}.$$

Now, we consider the concept of robust $ϵ$-quasi efficient solution for $\left(\mathrm{UFP}\right)$. We refer the readers to [19,21,37] for other kinds of robust approximate efficient solutions.

Definition 2. 

Let $ϵ\in {\mathbb{R}}_{+}^p\setminus \left\{0\right\}$. $\overline{x}\in \mathcal{F}$ is a robust ϵ-quasi efficient solution of $\left(\mathrm{UFP}\right)$ if there is not $x\in \mathcal{F}$, such that

$$\begin{array}{c}\hfill \frac{f_i\left(x\right)}{g_i\left(x\right)}\le \frac{f_i\left(\overline{x}\right)}{g_i\left(\overline{x}\right)}-{ϵ}_i\parallel x-\overline{x}\parallel ,foralli=1,\dots ,p,\end{array}$$

and

$$\begin{array}{c}\hfill \frac{f_j\left(x\right)}{g_j\left(x\right)}<\frac{f_j\left(\overline{x}\right)}{g_j\left(\overline{x}\right)}-{ϵ}_j\parallel x-\overline{x}\parallel ,forsomej\in \{1,\dots ,p\}.\end{array}$$

Remark 1. 

Note that $g_i\equiv 1$, the concept of robust ε-quasi efficient solution of $\left(\mathrm{UFP}\right)$ deduces to the robust ε-quasi efficient solution of $\left(\mathrm{UMP}\right)$, i.e., there is not $x\in \mathcal{F}$, such that

$$\begin{array}{c}\hfill f_i\left(x\right)\le f_i\left(\overline{x}\right)-{ϵ}_i\parallel x-\overline{x}\parallel ,foralli=1,\dots ,p,\end{array}$$

and

$$\begin{array}{c}\hfill f_j\left(x\right)<f_j\left(\overline{x}\right)-{ϵ}_j\parallel x-\overline{x}\parallel ,forsomej\in \{1,\dots ,p\}.\end{array}$$

For more details, see [20,21,42].

Definition 3 

([43] (Definition 3.2)). Consider $\left(\mathrm{UFP}\right)$. We say that the robust-type subdifferential constraint qualification condition $\mathrm{RSCQ}$ holds at $\overline{x}\in \mathcal{F}$, iff

$$N^c(\mathcal{F},\overline{x})\subseteq {\displaystyle \bigcup _{\genfrac{}{}{0pt}{}{{\lambda }_T\in T\left(\overline{x}\right),}{v_T\in {\mathcal{V}}_T}}}\left[{\displaystyle \sum _{t\in T}}{\lambda }_t{\partial }_x^c{h}_t(\overline{x},v_t)\right],$$

where $T\left(\overline{x}\right)=\left\{{\lambda }_T\in {\mathbb{R}}_{+}^{\left(T\right)}|{\lambda }_t{h}_t(\overline{x},v_t)=0,\forall v_t\in {\mathcal{V}}_t,t\in T\right\}$.

Next, we recall the following necessary optimality conditions for robust $ϵ$-quasi-efficient solutions for $\left(\mathrm{UFP}\right)$ under the $\mathrm{RSCQ}$. For convenience, let $ϵ:=({ϵ}_1,\dots ,{ϵ}_p)\in {\mathbb{R}}_{+}^p\setminus \left\{0\right\}$.

Proposition 1 

([44] (Theorem 1)). Let $ϵ\in {\mathbb{R}}_{+}^p\setminus \left\{0\right\}$. Assume that $\left(\mathrm{RSCQ}\right)$ holds at $\overline{x}\in \mathcal{F}$. If $\overline{x}$ is a robust ϵ-quasi-efficient solution of $\left(\mathrm{UFP}\right)$, then there exist ${\overline{\eta }}_t\ge 0$, and ${\overline{v}}_t\in V_t$, $t\in T$, such that

$$0\in {\displaystyle \sum _{i=1}^p}{\partial }^c{f}_i\left(\overline{x}\right)+{\displaystyle \sum _{i=1}^p}{\varphi }_i\left(\overline{x}\right){\partial }^c(-g_i)\left(\overline{x}\right)+{\displaystyle \sum _{t\in T}}{\overline{\eta }}_t{\partial }^c{h}_t(·,{\overline{v}}_t)\left(\overline{x}\right)+2{\displaystyle \sum _{i=1}^p}{\epsilon }_i{g}_i\left(\overline{x}\right){\mathbb{B}}^{*},$$

(1)

and

$${\overline{\eta }}_t{h}_t(\overline{x},{\overline{v}}_t)=0,\forall t\in T.$$

(2)

Here, ${\varphi }_i(·)=\frac{f_i\left(\overline{x}\right)}{g_i\left(\overline{x}\right)}-{ϵ}_i\parallel ·-\overline{x}\parallel$, $i=1,\dots ,p$.

Remark 2. 

Proposition 1 extends [45] (Theorem 3.1) from the case of scalar optimization to the multi-objective setting.

In the case that $g_i\equiv 1$, the following result can be easily obtained by Proposition 1.

Proposition 2. 

Let $ϵ\in {\mathbb{R}}_{+}^p\setminus \left\{0\right\}$. Assume that $\left(\mathrm{RSCQ}\right)$ holds at $\overline{x}\in \mathcal{F}$. If $\overline{x}$ is a robust ϵ-quasi-efficient solution of $\left(\mathrm{UMP}\right)$, then there exist ${\overline{\eta }}_t\ge 0$, and ${\overline{v}}_t\in V_t$, $t\in T$, such that

$$0\in {\displaystyle \sum _{i=1}^p}{\partial }^c{f}_i\left(\overline{x}\right)+{\displaystyle \sum _{t\in T}}{\overline{\eta }}_t{\partial }^c{h}_t(·,{\overline{v}}_t)\left(\overline{x}\right)+2{\displaystyle \sum _{i=1}^p}{\epsilon }_i{\mathbb{B}}^{*},$$

(3)

and

$${\overline{\eta }}_t{h}_t(\overline{x},{\overline{v}}_t)=0,\forall t\in T.$$

(4)

3. Main Results

In this section, based on the optimality conditions obtained in Proposition 1, we establish a robust Mond-Weir-type approximate dual problem for ($\mathrm{UMFP}$), and then investigate robust duality properties between them. Here, we only consider their robust $ϵ$-quasi-efficient solutions. For the sake of convenience in the sequel, we set $f:=(f_1,\dots ,f_p)$, $g:=(g_1,\dots ,g_p)$, $h_T:={\left(h_t\right)}_{t\in T}$, ${\eta }_T:={\left({\eta }_t\right)}_t\in {\mathbb{R}}_{+}^{\left(T\right)}$, ${\mathcal{V}}_T:={\prod }_{t\in T}{\mathcal{V}}_t$, and $v_T:={\left(v_t\right)}_{t\in T}\in {\mathcal{V}}_T.$

Let $y\in {\mathbb{R}}^n$ and $ϵ\in {\mathbb{R}}_{+}^p\setminus \left\{0\right\}$. For given $v_t\in {\mathcal{V}}_t$, $t\in T$, the Mond-Weir-type uncertain approximate dual problem $\left(\mathrm{UFD}\right)$ of ($\mathrm{UFP}$) is

$$\begin{array}{c}\hfill \left(\mathrm{UFD}\right)\left\{\begin{array}{c}{\mathrm{Max}}_{{\mathbb{R}}_{+}^p}\left(\frac{f_1\left(y\right)}{g_1\left(y\right)},\dots ,\frac{f_p\left(y\right)}{g_p\left(y\right)}\right)\hfill \\ s.t.0\in {\displaystyle \sum _{i=1}^p}{\partial }^c{f}_i\left(y\right)+{\displaystyle \sum _{i=1}^p}\frac{f_i\left(y\right)}{g_i\left(y\right)}{\partial }^c(-g_i)\left(y\right)+{\displaystyle \sum _{t\in T}}{\eta }_t{\partial }^c{h}_t(·,v_t)\left(y\right)+2{\displaystyle \sum _{i=1}^p}{ϵ}_i{g}_i\left(y\right){\mathbb{B}}^{*},\hfill \\ {\eta }_t{h}_t(y,v_t)\ge 0,t\in T,\hfill \\ y\in {\mathbb{R}}^n,{ϵ}_i\ge 0,i=1,\dots ,p,{\eta }_t\ge 0,t\in T.\hfill \end{array}\right.\end{array}$$

The optimistic counterpart of $\left(\mathrm{UFD}\right)$ is defined by

$$\begin{array}{c}\hfill \left(\mathrm{OFD}\right)\left\{\begin{array}{c}{\mathrm{Max}}_{{\mathbb{R}}_{+}^p}\left(\frac{f_1\left(y\right)}{g_1\left(y\right)},\dots ,\frac{f_p\left(y\right)}{g_p\left(y\right)}\right)\hfill \\ s.t.0\in {\displaystyle \sum _{i=1}^p}{\partial }^c{f}_i\left(y\right)+{\displaystyle \sum _{i=1}^p}\frac{f_i\left(y\right)}{g_i\left(y\right)}{\partial }^c(-g_i)\left(y\right)+{\displaystyle \sum _{t\in T}}{\eta }_t{\partial }^c{h}_t(·,v_t)\left(y\right)+2{\displaystyle \sum _{i=1}^p}{ϵ}_i{g}_i\left(y\right){\mathbb{B}}^{*},\hfill \\ {\eta }_t{h}_t(y,v_t)\ge 0,t\in T,\hfill \\ y\in {\mathbb{R}}^n,{ϵ}_i\ge 0,i=1,\dots ,p,{\eta }_t\ge 0,v_t\in {\mathcal{V}}_t,t\in T.\hfill \end{array}\right.\end{array}$$

Here, the maximization is also over all the parameters $v_t\in {\mathcal{V}}_t$, $t\in T$. The feasible set of $\left(\mathrm{OFD}\right)$ is defined as

$$\begin{array}{ccc}& & \widehat{\mathcal{F}}:=\left\{(y,{\eta }_T,v_T)\in {\mathbb{R}}^n\times {\mathbb{R}}_{+}^{\left(T\right)}\times {\mathcal{V}}_T|0\in {\displaystyle \sum _{i=1}^p}{\partial }^c{f}_i\left(y\right)+{\displaystyle \sum _{i=1}^p}\frac{f_i\left(y\right)}{g_i\left(y\right)}{\partial }^c(-g_i)\left(y\right)+{\displaystyle \sum _{t\in T}}{\eta }_t{\partial }^c{h}_t(·,v_t)\left(y\right)\right.\hfill \\ & & \left.+2{\displaystyle \sum _{i=1}^p}{ϵ}_i{g}_i\left(y\right){\mathbb{B}}^{*},{\eta }_t{h}_t(y,v_t)\ge 0,t\in T\right\}.\hfill \end{array}$$

Remark 3. 

(i)

Obviously, if $g_i\left(x\right)\equiv 1$, $i=1,\dots ,p$, $\left(\mathrm{UFD}\right)$ becomes the following conventional Mond-Weir-type uncertain approximate dual problem of ($\mathrm{UMP}$)

$$\begin{array}{c}\hfill \left(\mathrm{UMD}\right)\left\{\begin{array}{c}{\mathrm{Max}}_{{\mathbb{R}}_{+}^p}\left(f_1\left(y\right),\dots ,f_p\left(y\right)\right)\hfill \\ s.t.0\in {\displaystyle \sum _{i=1}^p}{\partial }^c{f}_i\left(y\right)+{\displaystyle \sum _{t\in T}}{\eta }_t{\partial }^c{h}_t(·,v_t)\left(y\right)+2{\displaystyle \sum _{i=1}^p}{ϵ}_i{\mathbb{B}}^{*},\hfill \\ {\eta }_t{h}_t(y,v_t)\ge 0,t\in T,\hfill \\ y\in {\mathbb{R}}^n,{ϵ}_i\ge 0,i=1,\dots ,p,{\eta }_t\ge 0,t\in T.\hfill \end{array}\right.\end{array}$$

and $\left(\mathrm{OFD}\right)$ becomes the following Mond-Weir-type optimistic dual problem of $\left(\mathrm{UMP}\right)$

$$\begin{array}{c}\hfill \left(\mathrm{OMD}\right)\left\{\begin{array}{c}{\mathrm{Max}}_{{\mathbb{R}}_{+}^p}\left(f_1\left(y\right),\dots ,f_p\left(y\right)\right)\hfill \\ s.t.0\in {\displaystyle \sum _{i=1}^p}{\partial }^c{f}_i\left(y\right)+{\displaystyle \sum _{t\in T}}{\eta }_t{\partial }^c{h}_t(·,v_t)\left(y\right)+2{\displaystyle \sum _{i=1}^p}{ϵ}_i{\mathbb{B}}^{*},\hfill \\ {\eta }_t{h}_t(y,v_t)\ge 0,t\in T,\hfill \\ y\in {\mathbb{R}}^n,{ϵ}_i\ge 0,i=1,\dots ,p,{\eta }_t\ge 0,v_t\in {\mathcal{V}}_t,t\in T.\hfill \end{array}\right.\end{array}$$

Here, we denote the feasible set of $\left(\mathrm{OMD}\right)$ by

$$\begin{array}{ccc}& & \overline{\mathcal{F}}:=\left\{(y,{\eta }_T,v_T)\in {\mathbb{R}}^n\times {\mathbb{R}}_{+}^{\left(T\right)}\times {\mathcal{V}}_T|0\in {\displaystyle \sum _{i=1}^p}{\partial }^c{f}_i\left(y\right)+{\displaystyle \sum _{t\in T}}{\eta }_t{\partial }^c{h}_t(·,v_t)\left(y\right)\right.\hfill \\ & & \left.+2{\displaystyle \sum _{i=1}^p}{ϵ}_i{g}_i\left(y\right){\mathbb{B}}^{*},{\eta }_t{h}_t(y,v_t)\ge 0,t\in T\right\}.\hfill \end{array}$$

(ii)

In the case that $ϵ=0$ and there is no uncertainty in the constraint functions. Then, $\left(\mathrm{UFP}\right)$ becomes $\left(\mathrm{FP}\right)$, and $\left(\mathrm{OMD}\right)$ collapses to

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\mathrm{Max}}_{{\mathbb{R}}_{+}^p}\left(\frac{f_1\left(y\right)}{g_1\left(y\right)},\dots ,\frac{f_p\left(y\right)}{g_p\left(y\right)}\right)\hfill \\ s.t.0\in {\displaystyle \sum _{i=1}^p}\partial f_i\left(y\right)+{\displaystyle \sum _{i=1}^p}\frac{f_i\left(y\right)}{g_i\left(y\right)}{\partial }^c(-g_i)\left(y\right)+{\displaystyle \sum _{t\in T}}{\eta }_t{\partial }^c{h}_t\left(y\right),\hfill \\ {\eta }_t{h}_t\left(y\right)\ge 0,t\in T,\hfill \\ y\in {\mathbb{R}}^n,{\eta }_t\ge 0,t\in T.\hfill \end{array}\right.\end{array}$$

Now, similar to Definition 2, we introduce robust $ϵ$-quasi efficient solutions for $\left(\mathrm{UFD}\right)$.

Definition 4. 

Let $\epsilon \in {\mathbb{R}}_{+}^p\setminus \left\{0\right\}.$ $(\overline{y},{\overline{\eta }}_T,{\overline{v}}_T)\in \widehat{\mathcal{F}}$ is said to be a robust ε-quasi efficient solution of $\left(\mathrm{UFD}\right)$, iff it is an ε-quasi efficient solution of $\left(\mathrm{OFD}\right)$, i.e., there is no $(y,{\eta }_T,v_T)\in \widehat{\mathcal{F}}$, such that

$$\begin{array}{c}\hfill \frac{f_i\left(y\right)}{g_i\left(y\right)}\ge \frac{f_i\left(\overline{y}\right)}{g_i\left(\overline{y}\right)}+{ϵ}_i\parallel y-\overline{y}\parallel ,foralli=1,\dots ,p,\end{array}$$

and

$$\begin{array}{c}\hfill \frac{f_j\left(y\right)}{g_j\left(y\right)}>\frac{f_j\left(\overline{y}\right)}{g_j\left(\overline{y}\right)}+{ϵ}_j\parallel y-\overline{y}\parallel ,forsomej\in \{1,\dots ,p\}.\end{array}$$

Remark 4. 

In particular, if $g_i\equiv 1$, the concept of robust ε-quasi efficient solution of $\left(\mathrm{UFD}\right)$ deduces to the robust ε-quasi efficient solution of $\left(\mathrm{UMD}\right)$, i.e., there is no $(y,{\eta }_T,v_T)\in \overline{\mathcal{F}}$, such that

$$\begin{array}{c}\hfill f_i\left(y\right)\ge f_i\left(\overline{y}\right)+{ϵ}_i\parallel y-\overline{y}\parallel ,foralli=1,\dots ,p,\end{array}$$

and

$$\begin{array}{c}\hfill f_j\left(y\right)>f_j\left(\overline{y}\right)+{ϵ}_j\parallel y-\overline{y}\parallel ,forsomej\in \{1,\dots ,p\}.\end{array}$$

In order to give robust duality relations for $\left(\mathrm{UFP}\right)$ and $\left(\mathrm{UFD}\right)$, we introduce the new definition of generalized convex-inclusion for Lipschitz functions, which is inspired by [32] (Definition 3.4) and [21] (Definition 3.3).

Definition 5. 

Let $\Omega \subseteq {\mathbb{R}}^n$. $(f,-g,h_T)$ is said to generalized convex-inclusion on Ω at $x\in \Omega$, iff for any $y\in \Omega$, ${\xi }_i^{*}\in {\partial }^c{f}_i\left(x\right)$, ${\xi }_i^{**}\in {\partial }^c(-g_i)\left(x\right)$, $i=1,\dots ,p$, and ${\gamma }_t^{*}\in {\partial }_x^c{h}_t(x,v_t)$, $v_t\in {\mathcal{V}}_t$, $t\in T$, there exists $\omega \in {\mathbb{R}}^n$, such that

$$f_i\left(y\right)-f_i\left(x\right)>⟨{\xi }_i^{*},\omega ⟩,i=1,\dots ,p,$$

$$-g_i\left(y\right)+g_i\left(x\right)\ge ⟨{\xi }_i^{**},\omega ⟩,i=1,\dots ,p,$$

$$h_t(y,v_t)-h_t(x,v_t)\ge ⟨{\gamma }_t^{*},\omega ⟩,t\in T,$$

$$⟨b^{*},\omega ⟩\le \parallel y-x\parallel ,\forall b^{*}\in {\mathbb{B}}^{*},$$

and

$$0\in {\partial }^c{g}_i\left(y\right),i=1,\dots ,p.$$

Remark 5. 

(i)

In the special case that $g_i\equiv 1$, the concept of generalized convex-inclusion reduces to the concept of generalized convexity, i.e., $(f,h_T)$ is generalized convex on Ω at $x\in \Omega$, iff for any $y\in \Omega$, ${\xi }_i^{*}\in {\partial }^c{f}_i\left(x\right)$, $i=1,\dots ,p$, and ${\gamma }_t^{*}\in {\partial }_x^c{g}_t(x,v_t)$, $v_t\in {\mathcal{V}}_t$, $t\in T$, there exists $\omega \in {\mathbb{R}}^n$, such that

$$f_i\left(y\right)-f_i\left(x\right)>⟨{\xi }_i^{*},\omega ⟩,i=1,\dots ,p,$$

$$h_t(y,v_t)-h_t(x,v_t)\ge ⟨{\gamma }_t^{*},\omega ⟩,t\in T,$$

and

$$⟨b^{*},\omega ⟩\le \parallel y-x\parallel ,\forall b^{*}\in {\mathbb{B}}^{*}.$$

(ii)

If $g_i\equiv 1$ and there is uncertain data on $f_i$, $i=1,\dots ,p,$ Definition 5 reduces to [21] (Definition 3.3).

(iii)

If $g_i\equiv 1$ and there is no uncertain data on $h_t$, $t\in T,$ Definition 5 reduces to the concept of generalized convexity-inclusion introduced in [32] (Definition 3.4), i.e., for any $y\in \Omega$, ${\xi }_i^{*}\in {\partial }^c{f}_i\left(x\right)$, ${\xi }_i^{**}\in {\partial }^c(-g_i)\left(x\right)$, $i=1,\dots ,p$, and ${\gamma }_t^{*}\in {\partial }^c{h}_t\left(x\right)$, $t\in T$, there exists $\omega \in {\mathbb{R}}^n$, such that

$$f_i\left(y\right)-f_i\left(x\right)>⟨{\xi }_i^{*},\omega ⟩,i=1,\dots ,p,$$

$$-g_i\left(y\right)+g_i\left(x\right)\ge ⟨{\xi }_i^{**},\omega ⟩,i=1,\dots ,p,$$

$$h_t\left(y\right)-h_t\left(x\right)\ge ⟨{\gamma }_t^{*},\omega ⟩,t\in T,$$

$$⟨b^{*},\omega ⟩\le \parallel y-x\parallel ,\forall b^{*}\in {\mathbb{B}}^{*},$$

and

$$0\in {\partial }^c{g}_i\left(y\right),i=1,\dots ,p.$$

Note that this concept has been used to establish sufficient optimality conditions for weakly ϵ-quasi-efficient solution for fractional optimization problem. For more details, please see [32] (Theorem 3.5).

Now, we show robust approximate duality properties for $\left(\mathrm{UFP}\right)$ and $\left(\mathrm{UFD}\right)$ by showing approximate duality properties between the robust counterpart $\left(\mathrm{RMP}\right)$ and the optimistic counterpart $\left(\mathrm{OFD}\right)$. In what follows, we set

$${\omega }_1⪯{\omega }_2\iff {\omega }_2-{\omega }_1\in {\mathbb{R}}_{+}^p\setminus \left\{0\right\},\forall {\omega }_1,{\omega }_2\in {\mathbb{R}}^p,$$

$${\omega }_1\cancel{⪯}{\omega }_2\iff {\omega }_2-{\omega }_1\notin {\mathbb{R}}_{+}^p\setminus \left\{0\right\},\forall {\omega }_1,{\omega }_2\in {\mathbb{R}}^p.$$

The following result gives robust $ϵ$-quasi-weak duality between $\left(\mathrm{UFP}\right)$ and $\left(\mathrm{UFD}\right)$.

Theorem 1. 

Let $ϵ\in {\mathbb{R}}_{+}^p\setminus \left\{0\right\}$. Suppose that $x\in \mathcal{F}$ and $(y,{\eta }_T,v_T)\in \widehat{\mathcal{F}}$. If $(f,-g,h_T)$ is generalized convex-inclusion on ${\mathbb{R}}^n$ at $y\in {\mathbb{R}}^n$, then,

$$\left(\frac{f_1\left(x\right)}{g_1\left(x\right)},\dots ,\frac{f_p\left(x\right)}{g_p\left(x\right)}\right)\cancel{⪯}\left(\frac{f_1\left(y\right)}{g_1\left(y\right)}-2{ϵ}_1\parallel x-y\parallel ,\dots ,\frac{f_p\left(y\right)}{g_p\left(y\right)}-2{ϵ}_p\parallel x-y\parallel \right).$$

Proof. 

Suppose to the contrary that

$$\left(\frac{f_1\left(x\right)}{g_1\left(x\right)},\dots ,\frac{f_p\left(x\right)}{g_p\left(x\right)}\right)⪯\left(\frac{f_1\left(y\right)}{g_1\left(y\right)}-2{ϵ}_1\parallel y-x\parallel ,\dots ,\frac{f_p\left(y\right)}{g_p\left(y\right)}-2{ϵ}_p\parallel y-x\parallel \right).$$

Then,

$$\begin{array}{c}\hfill \frac{f_i\left(x\right)}{g_i\left(x\right)}\le \frac{f_i\left(y\right)}{g_i\left(y\right)}-2{ϵ}_i\parallel y-x\parallel ,\mathrm{for}\mathrm{all}i=1,\dots ,p,\end{array}$$

(5)

and

$$\begin{array}{c}\hfill \frac{f_i\left(x\right)}{g_i\left(x\right)}<\frac{f_j\left(y\right)}{g_j\left(y\right)}-2{ϵ}_j\parallel y-x\parallel ,\mathrm{for}\mathrm{some}j\in \{1,\dots ,p\}.\end{array}$$

(6)

On the other hand, note that $(y,{\eta }_T,v_T)\in \widehat{\mathcal{F}}$. Then, $y\in {\mathbb{R}}^n$, ${\eta }_t\ge 0$, $v_t\in {\mathcal{V}}_t$, $t\in T$, and

$$\begin{array}{c}\hfill 0\in {\displaystyle \sum _{i=1}^p}{\partial }^c{f}_i\left(y\right)+{\displaystyle \sum _{i=1}^p}\frac{f_i\left(y\right)}{g_i\left(y\right)}{\partial }^c(-g_i)\left(y\right)+{\displaystyle \sum _{t\in T}}{\eta }_t{\partial }^c{h}_t(·,v_t)\left(y\right)+2{\displaystyle \sum _{i=1}^p}{ϵ}_i{g}_i\left(y\right){\mathbb{B}}^{*},\end{array}$$

(7)

and

$$\begin{array}{c}\hfill {\eta }_t{h}_t(y,v_t)\ge 0,t\in T.\end{array}$$

(8)

By (5), there exist ${\xi }_i^{*}\in {\partial }^c{f}_i\left(y\right)$, ${\xi }_i^{**}\in {\partial }^c(-g_i)\left(y\right)$, $i=1,\dots ,p,$${\zeta }_t^{*}\in {\partial }^c{h}_t(·,v_t)\left(y\right)$, $t\in T$, and $b^{*}\in {\mathbb{B}}^{*}$, such that

$$\begin{array}{c}\hfill {\displaystyle \sum _{i=1}^p}{\xi }_i^{*}+{\displaystyle \sum _{i=1}^p}\frac{f_i\left(y\right)}{g_i\left(y\right)}{\xi }_i^{**}+{\displaystyle \sum _{t\in T}}{\eta }_t{\zeta }_t^{*}+2{\displaystyle \sum _{i=1}^p}{ϵ}_i{g}_i\left(y\right)b^{*}=0.\end{array}$$

(9)

Since $(f,-g,h_T)$ is generalized convex-inclusion on ${\mathbb{R}}^n$ at $y\in {\mathbb{R}}^n$, we have for such ${\xi }_i^{*}\in {\partial }^c{f}_i\left(y\right)$, ${\xi }_i^{**}\in {\partial }^c(-g_i)\left(y\right)$, $i=1,\dots ,p,$ and ${\zeta }_t^{*}\in {\partial }^c{h}_t(·,v_t)\left(y\right)$, $t\in T$, there exists $\vartheta \in {\mathbb{R}}^n$, such that

$$f_i\left(x\right)-f_i\left(y\right)>⟨{\xi }_i^{*},\vartheta ⟩,i=1,\dots ,p,$$

$$-g_i\left(x\right)+g_i\left(y\right)\ge ⟨{\xi }_i^{**},\vartheta ⟩,i=1,\dots ,p,$$

$$h_t(x,v_t)-h_t(y,v_t)\ge ⟨{\zeta }_t^{*},\vartheta ⟩,t\in T,$$

$$⟨b^{*},\vartheta ⟩\le \parallel x-y\parallel ,\forall b^{*}\in {\mathbb{B}}^{*},$$

and

$$0\in {\partial }^c{g}_i\left(y\right),i=1,\dots ,p.$$

Together with (7)–(9), these follow that

$$\begin{array}{cc}\hfill & {\displaystyle \sum _{i=1}^p}\left(f_i\left(x\right)-\frac{f_i\left(y\right)}{g_i\left(y\right)}{g}_i\left(x\right)+2{ϵ}_i{g}_i\left(y\right)\parallel y-x\parallel \right)\hfill \\ \hfill & >{\displaystyle \sum _{i=1}^p}\left(f_i\left(y\right)+⟨{\xi }_i^{*},\vartheta ⟩-\frac{f_i\left(y\right)}{g_i\left(y\right)}{g}_i\left(y\right)+\frac{f_i\left(y\right)}{g_i\left(y\right)}⟨{\xi }_i^{**},\vartheta ⟩+2{ϵ}_i{g}_i\left(y\right)⟨b^{*},\vartheta ⟩\right)\hfill \\ \hfill & =⟨{\displaystyle \sum _{i=1}^p}{\xi }_i^{*}+{\displaystyle \sum _{i=1}^p}\frac{f_i\left(y\right)}{g_i\left(y\right)}{\xi }_i^{**}+2{\displaystyle \sum _{i=1}^p}{ϵ}_i{g}_i\left(y\right)b^{*},\vartheta ⟩\hfill \\ \hfill & =-⟨{\displaystyle \sum _{i=1}^p}{\eta }_t{\zeta }_t^{*},\vartheta ⟩\hfill \\ \hfill & \ge -{\displaystyle \sum _{t\in T}}{\eta }_t{h}_t(x,v_t)+{\displaystyle \sum _{t\in T}}{\eta }_t{h}_t(y,v_t).\hfill \end{array}$$

Together with ${\eta }_t{h}_t(x,v_t)\le 0$, $\forall x\in F$, and ${\eta }_t{h}_t(y,v_t\ge 0$, we have

$${\displaystyle \sum _{i=1}^p}\left(f_i\left(x\right)-\frac{f_i\left(y\right)}{g_i\left(y\right)}{g}_i\left(x\right)+2{ϵ}_i{g}_i\left(y\right)\parallel y-x\parallel \right)>0.$$

Then, there exists $i_0\in \{1,\dots ,p\}$, such that

$$f_{i_0}\left(x\right)-\frac{f_{i_0}\left(y\right)}{g_{i_0}\left(y\right)}{g}_{i_0}\left(x\right)+2{ϵ}_{i_0}{g}_{i_0}\left(y\right)\parallel y-x\parallel >0,$$

which follows that

$$\begin{array}{c}\hfill \frac{f_{i_0}\left(x\right)}{g_{i_0}\left(x\right)}-\frac{f_{i_0}\left(y\right)}{g_{i_0}\left(y\right)}+2{ϵ}_{i_0}\frac{g_{i_0}\left(y\right)}{g_{i_0}\left(x\right)}\parallel y-x\parallel >0.\end{array}$$

(10)

Moreover, it follows from $0\in {\partial }^c{g}_i\left(y\right),i=1,\dots ,p,$ that

$$\begin{array}{c}\hfill g_{i_0}\left(x\right)\ge g_{i_0}\left(y\right).\end{array}$$

(11)

Together with (10) and (11), we have

$$\begin{array}{c}\hfill \frac{f_{i_0}\left(x\right)}{g_{i_0}\left(x\right)}-\frac{f_{i_0}\left(y\right)}{g_{i_0}\left(y\right)}+2{ϵ}_{i_0}\parallel y-x\parallel >0.\end{array}$$

which is a contradiction to (5) and (6). Thus, the conclusion holds. □

Now, we give the following example to justify the importance of the assumption of generalized convex-inclusion in Theorem 1.

Example 1. 

Let ${\mathcal{V}}_t:=[1-t,1+t]$, $t\in T:=\left[0,\frac{1}{2}\right]$. Let $f_1,f_2,g_1,g_2:\mathbb{R}\to \mathbb{R}$ and $g_t:\mathbb{R}\times \mathbb{R}\to \mathbb{R},t\in T$, be defined by

$$f_1\left(x\right)=f_2\left(x\right):=\frac{1}{2}\left|x\right|+\frac{1}{6}{x}^3,g_1\left(x\right)=g_2\left(x\right):=\left|x\right|+1,$$

and

$$h_t(x,v_t):=t{x}^2-tx-2v_t,$$

where $x\in \mathbb{R}$ and $v_t\in {\mathcal{V}}_t$, $t\in T$. Then, $\left(\mathrm{UFP}\right)$ becomes

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\mathrm{Min}}_{{\mathbb{R}}_{+}^2}\left(\frac{\frac{1}{2}\left|x\right|+\frac{1}{6}{x}^3}{\left|x\right|+1},\frac{\frac{1}{2}\left|x\right|+\frac{1}{6}{x}^3}{\left|x\right|+1}\right)\hfill \\ s.t.t{x}^2-tx-2v_t\le 0,\forall t\in \left[0,\frac{1}{2}\right],\hfill \\ x\in \mathbb{R},\hfill \end{array}\right.\end{array}$$

and $\left(\mathrm{RFP}\right)$ becomes

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\mathrm{Min}}_{{\mathbb{R}}_{+}^2}\left(\frac{\frac{1}{2}\left|x\right|+\frac{1}{6}{x}^3}{\left|x\right|+1},\frac{\frac{1}{2}\left|x\right|+\frac{1}{6}{x}^3}{\left|x\right|+1}\right)\hfill \\ s.t.t{x}^2-tx-2v_t\le 0,\forall v_t\in [1-t,1+t],t\in \left[0,\frac{1}{2}\right],\hfill \\ x\in \mathbb{R}.\hfill \end{array}\right.\end{array}$$

Obviously, $\mathcal{F}=[-1,2]$. Let us consider $\overline{x}:=-1\in \mathcal{F}$. Then,

$$\left(\frac{f_1\left(\overline{x}\right)}{g_1\left(\overline{x}\right)},\frac{f_2\left(\overline{x}\right)}{g_2\left(\overline{x}\right)}\right)=\left(\frac{1}{6},\frac{1}{6}\right).$$

Now, consider the dual problem $\left(\mathrm{UFD}\right)$. In this setting, $\left(\mathrm{OFD}\right)$ becomes

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\mathrm{Max}}_{{\mathbb{R}}_{+}^2}\left(\frac{f_1\left(y\right)}{g_1\left(y\right)},\frac{f_2\left(y\right)}{g_2\left(y\right)}\right)\hfill \\ s.t.0\in {\partial }^c{f}_1\left(y\right)+{\partial }^c{f}_2\left(y\right)+\frac{f_1\left(y\right)}{g_1\left(y\right)}{\partial }^c(-g_1)\left(y\right)+\frac{f_2\left(y\right)}{g_2\left(y\right)}{\partial }^c(-g_2)\left(y\right)\hfill \\ +{\displaystyle \sum _{t\in T}}{\eta }_t{\partial }^c{h}_t(·,v_t)\left(y\right)+2{ϵ}_1{g}_1\left(y\right){\mathbb{B}}^{*}+2{ϵ}_2{g}_2\left(y\right){\mathbb{B}}^{*},\hfill \\ {\eta }_t{h}_t(y,v_t)\ge 0,t\in \left[0,\frac{1}{2}\right],\hfill \\ y\in \mathbb{R},{ϵ}_1\ge 0,{ϵ}_2\ge 0,{\eta }_t\ge 0,v_t\in [1-t,1+t],t\in \left[0,\frac{1}{2}\right].\hfill \end{array}\right.\end{array}$$

Clearly, for any $y\in \mathbb{R}$ and $v_T\in {\mathcal{V}}_T$, we have

$${\partial }^c{f}_1\left(y\right)={\partial }^c{f}_2\left(y\right)=\left[\frac{1}{2}{y}^2-\frac{1}{2},\frac{1}{2}{y}^2+\frac{1}{2}\right],$$

$${\partial }^c(-g_1)\left(y\right)={\partial }^c(-g_2)\left(y\right)=\left[-1,1\right],$$

and

$${\partial }^c{h}_t(·,v_t)\left(y\right)=\left\{2ty-t\right\},\forall t\in T.$$

By selecting $\overline{y}:=1$, ${\overline{\eta }}_t:=0$, and ${\overline{v}}_t:=-t$, we have

$$\begin{array}{cc}\hfill & {\partial }^c{f}_1\left(\overline{y}\right)+{\partial }^c{f}_2\left(\overline{y}\right)+\frac{f_1\left(\overline{y}\right)}{g_1\left(\overline{y}\right)}{\partial }^c(-g_1)\left(\overline{y}\right)+\frac{f_2\left(\overline{y}\right)}{g_2\left(\overline{y}\right)}{\partial }^c(-g_2)\left(\overline{y}\right)\hfill \\ \hfill & +{\displaystyle \sum _{t\in T}}{\overline{\eta }}_t{\partial }^c{h}_t(·,{\overline{v}}_t)\left(\overline{y}\right)+2{ϵ}_1{g}_1\left(\overline{y}\right){\mathbb{B}}^{*}+2{ϵ}_2{g}_2\left(\overline{y}\right){\mathbb{B}}^{*}\hfill \\ \hfill & =\left[-4{ϵ}_1-4{ϵ}_2-\frac{1}{3},4{ϵ}_1+4{ϵ}_2+\frac{7}{3}\right],\hfill \end{array}$$

and

$${\overline{\eta }}_t{h}_t(\overline{y},{\overline{v}}_t)\ge 0,t\in \left[0,\frac{1}{2}\right].$$

These mean that $(\overline{y},{\overline{\eta }}_T,{\overline{v}}_T)\in \widehat{\mathcal{F}}$.

Now, take an arbitrarily $ϵ=({ϵ}_1,{ϵ}_2)\in {\mathbb{R}}_{+}^2\setminus \left\{0\right\}$ such that ${\epsilon }_i<\frac{1}{12},i=1,2$. Clearly,

$$\begin{array}{cc}\hfill \left(\frac{f_1\left(\overline{y}\right)}{g_1\left(\overline{y}\right)}-2{ϵ}_1\parallel \overline{x}-\overline{y}\parallel ,\frac{f_2\left(\overline{y}\right)}{g_2\left(\overline{y}\right)}-2{ϵ}_2\parallel \overline{x}-\overline{y}\parallel \right)& =\left(\frac{1}{3}-2{\epsilon }_1,\frac{1}{3}-2{\epsilon }_2\right)\hfill \\ \hfill & \succ \left(\frac{1}{6},\frac{1}{6}\right)=\left(\frac{f_1\left(\overline{x}\right)}{g_1\left(\overline{x}\right)},\frac{f_2\left(\overline{x}\right)}{g_2\left(\overline{x}\right)}\right).\hfill \end{array}$$

Thus, Theorem 1 is not applicable since $(f,-g,h_T)$ is not generalized convex-inclusion at $\overline{y}$. To do this, by choosing ${\overline{\xi }}_i:=0\in {\partial }^c{f}_i\left(\overline{y}\right)$, $i=1,2$, we have

$$f_i\left(\overline{x}\right)-f_i\left(\overline{y}\right)=-\frac{2}{3}<0=⟨{\overline{\xi }}_k,\omega ⟩,\forall \omega \in \mathbb{R}.$$

Similarly, we obtain the following robust weak duality between $\left(\mathrm{UMP}\right)$ and $\left(\mathrm{UMD}\right)$.

Corollary 1. 

Let $ϵ\in {\mathbb{R}}_{+}^p\setminus \left\{0\right\}$. Suppose that $x\in \mathcal{F}$ and $(y,{\eta }_T,v_T)\in \overline{\mathcal{F}}$. If $(f,h_T)$ is generalized convex on ${\mathbb{R}}^n$ at $y\in {\mathbb{R}}^n$, then,

$$\left(f_1\left(x\right),\dots ,f_p\left(x\right)\right)\cancel{⪯}\left(f_1\left(y\right)-2{ϵ}_1\parallel x-y\parallel ,\dots ,f_p\left(y\right)-2{ϵ}_p\parallel x-y\parallel \right).$$

Remark 6. 

Clearly, by virtue of Example 1, we can also illustrate that the assumption of generalized convexity imposed in Corollary 1 is indispensable.

Now, we give robust strong duality results between $\left(\mathrm{UFP}\right)$ and $\left(\mathrm{UFD}\right)$.

Theorem 2. 

Let $ϵ\in {\mathbb{R}}_{+}^p\setminus \left\{0\right\}$. Assume that $\left(\mathrm{RSCQ}\right)$ holds at $\overline{x}\in \mathcal{F}$. Suppose that $(f,-g,h_T)$ is generalized convex-inclusion on ${\mathbb{R}}^n$ at $y\in {\mathbb{R}}^n$. If $\overline{x}$ is a robust ϵ-quasi-efficient solution of $\left(\mathrm{UFP}\right)$, then there exist ${\overline{\eta }}_T\in {\mathbb{R}}_{+}^{\left(T\right)}$ and ${\overline{v}}_T\in {\mathcal{V}}_T$, such that $(\overline{x},{\overline{\eta }}_T,{\overline{v}}_T)\in \widehat{\mathcal{F}}$ is a robust $2ϵ$-quasi-efficient solution of $\left(\mathrm{UFD}\right)$.

Proof. 

Assume that $\overline{x}\in \mathcal{F}$ is a robust $ϵ$-quasi-efficient solution of $\left(\mathrm{UFP}\right)$. By Theorem 1, there exist ${\overline{\eta }}_t\ge 0$, and ${\overline{v}}_t\in {\mathcal{V}}_t$, $t\in T$, such that

$$0\in {\displaystyle \sum _{i=1}^p}\partial f_i\left(\overline{x}\right)-{\displaystyle \sum _{i=1}^p}{\varphi }_i\left(\overline{x}\right)\partial g_i\left(\overline{x}\right)+{\displaystyle \sum _{t\in T}}{\overline{\eta }}_t\partial h_t(·,{\overline{v}}_t)\left(\overline{x}\right)+2{\displaystyle \sum _{i=1}^p}{\epsilon }_i{g}_i\left(\overline{x}\right){\mathbb{B}}^{*},$$

(12)

and

$${\overline{\eta }}_t{h}_t(\overline{x},{\overline{v}}_t)=0,\forall t\in T.$$

(13)

From (12), (13) and ${\varphi }_i\left(\overline{x}\right)=\frac{f_i\left(\overline{x}\right)}{g_i\left(\overline{x}\right)}$, we have

$$(\overline{x},{\overline{\eta }}_T,{\overline{v}}_T)\in \widehat{\mathcal{F}}.$$

By Theorem 1, for all $(y,{\eta }_T,v_T)\in \widehat{\mathcal{F}}$, we have

$$\left(\frac{f_1\left(\overline{x}\right)}{g_1\left(\overline{x}\right)},\dots ,\frac{f_p\left(\overline{x}\right)}{g_p\left(\overline{x}\right)}\right)\cancel{⪯}\left(\frac{f_1\left(y\right)}{g_1\left(y\right)}-2{ϵ}_1\parallel \overline{x}-y\parallel ,\dots ,\frac{f_p\left(y\right)}{g_p\left(y\right)}-2{ϵ}_p\parallel \overline{x}-y\parallel \right).$$

Thus, $(\overline{x},{\overline{\eta }}_T,{\overline{v}}_T)$ is a robust $2ϵ$-quasi-efficient solutions of $\left(\mathrm{UFD}\right)$. Thus, the conclusion holds. □

Remark 7. 

In [32] (Theorem 4.2), the authors established duality properties for ϵ-quasi-weakly efficient solutions between $\left(\mathrm{FP}\right)$ and its Mond Weir-type dual problem. Therefore, Theorem 2 encompasses [32] (Theorem 4.2), where the corresponding results were given in terms of the similar methods.

Similarly, we give robust strong duality properties for robust $ϵ$-quasi efficient solutions between $\left(\mathrm{UMP}\right)$ and $\left(\mathrm{UMD}\right)$.

Corollary 2. 

Let $ϵ\in {\mathbb{R}}_{+}^p\setminus \left\{0\right\}$. Assume that $\left(\mathrm{RSCQ}\right)$ holds at $\overline{x}\in \mathcal{F}$. Suppose that $(f,h_T)$ is generalized convex on ${\mathbb{R}}^n$ at $y\in {\mathbb{R}}^n$. If $\overline{x}$ is a robust ϵ-quasi-efficient solution of $\left(\mathrm{UMP}\right)$, then there exist ${\overline{\eta }}_T\in {\mathbb{R}}_{+}^{\left(T\right)}$ and ${\overline{v}}_T\in {\mathcal{V}}_T$, such that $(\overline{x},{\overline{\eta }}_T,{\overline{v}}_T)\in \overline{\mathcal{F}}$ is a robust $2ϵ$-quasi-efficient solution of $\left(\mathrm{UMD}\right)$.

Now, we give a robust converse-like duality property between $\left(\mathrm{UFP}\right)$ and $\left(\mathrm{UFD}\right)$.

Theorem 3. 

Let $\epsilon \in {\mathbb{R}}_{+}^p\setminus \left\{0\right\}$ and $(\overline{x},{\overline{\eta }}_T,{\overline{v}}_T)\in \widehat{\mathcal{F}}$. If $(f,-g,h_T)$ is generalized convex-inclusion on ${\mathbb{R}}^n$ at $\overline{x}\in \mathcal{F}$, then, $\overline{x}\in \mathcal{F}$ is a robust $2\epsilon$-quasi efficient solution of $\left(\mathrm{UMP}\right)$.

Proof. 

Sine $(\overline{x},{\overline{\eta }}_T,{\overline{v}}_T)\in \widehat{\mathcal{F}}$ and $(f,-g,h_T)$ is generalized convex-inclusion on ${\mathbb{R}}^n$ at $\overline{x}$, it follows from Theorem 1 that

$$\left(\frac{f_1\left(x\right)}{g_1\left(x\right)},\dots ,\frac{f_p\left(x\right)}{g_p\left(x\right)}\right)\cancel{⪯}\left(\frac{f_1\left(\overline{x}\right)}{g_1\left(\overline{x}\right)}-2{ϵ}_1\parallel x-\overline{x}\parallel ,\dots ,\frac{f_p\left(\overline{x}\right)}{g_p\left(\overline{x}\right)}-2{ϵ}_p\parallel x-\overline{x}|\right),\forall x\in \mathcal{F}.$$

Therefore, $\overline{x}\in \mathcal{F}$ is a robust $2\epsilon$-quasi efficient solution of $\left(\mathrm{UFP}\right)$ and the proof is complete. □

Remark 8. 

Note that the converse-like duality result obtained in Theorem 3 extends [32] (Theorem 4.4) from the deterministic (i.e., with singleton uncertainty sets) to the robust setting. Moreover, Theorem 3 extends [43] (Theorem 4.3) from the scalar case to the multi-objective setting.

Similarly, we have the following results for $\left(\mathrm{UMP}\right)$ and $\left(\mathrm{UMD}\right)$, which has been considered in [21] (Theorem 4.3).

Corollary 3. 

Let $\epsilon \in {\mathbb{R}}_{+}^p\setminus \left\{0\right\}$ and $(\overline{x},{\overline{\eta }}_T,{\overline{v}}_T)\in \overline{\mathcal{F}}$. If $(f,h_T)$ is generalized convex on ${\mathbb{R}}^n$ at $\overline{x}\in \mathcal{F}$, then, $\overline{x}\in \mathcal{F}$ is a robust ε-quasi efficient solution of $\left(\mathrm{UMP}\right)$.

4. Conclusions

In this paper, we consider robust $\epsilon$-quasi-efficient solutions for a class of uncertain fractional optimization problems. By employing robust optimization and the obtained optimality conditions, a Mond–Weir-type robust dual problem for the fractional optimization problem is established. Then, we give robust $\epsilon$-quasi-weak, strong and converse duality properties between them in terms of generalized convex-inclusion assumptions. We also show that the obtained results extend the corresponding results obtained in [21,32,37].

In the future, similar to [21,43], it is of interest to formulate Mixed-type robust approximate dual problem of uncertain fractional optimization problems, and study robust $\epsilon$-quasi-weak, strong, and converse duality properties between them.