New Class of K-G-Type Symmetric Second Order Vector Optimization Problem

Abstract

In this paper, we present meanings of K-$G_f$-bonvexity/K-$G_f$-pseudobonvexity and their generalization between the above-notice functions. We also construct various concrete non-trivial examples for existing these types of functions. We formulate K-$G_f$-Wolfe type multiobjective second-order symmetric duality model with cone objective as well as cone constraints and duality theorems have been established under these aforesaid conditions. Further, we have validates the weak duality theorem under those assumptions. Our results are more generalized than previous known results in the literature.

1. Introduction

The field of optimization theory has progressed far beyond anyone’s expectations. Due to its wide variety of uses, it has made its way into all disciplines of science and engineering. When approximations are utilized, one of the most important practical applications of duality is that it provides bounds on the value of the objective functions because there are more factors involved, second-order duality has a greater computational benefit than first-order duality. For intriguing applications and breakthroughs in multiobjective optimization, we refer to [1], and the references cited therein. Dorn [2] presented the primary symmetric duality definition for quadratic programming in 1965. Dantzig et al. [3] and Mond [4] proposed a pair of symmetric dual Duality plays a vital role in investigating nonlinear programming problem solutions. Several writers have proposed several duality models, such as Wolfe dual [5] and Mond-Weir dual [6]. Nanda and Das [7] introduced four different forms of duality models for the nonlinear programming problem with cone constraints. The work of Bazaraa and Goode [8] and Hanson and Mond [9] inspired these findings.

Mangasian [10] established the duality theorem in the context of a second-order dual problem in nonlinear programming, where none of the constraints imposed convexity restrictions on all functions. Mond [11] introduced second-order symmetric dual models and established second-order symmetric duality theorems under second-order convexity conditions for the first time. In mathematical programming, Hasnson [12] defined the second-order invexity of a differentiable function and studied it. In 1999, Mishra [13] proposed a pair of second-order vector symmetric dual multiobjective models for arbitrary cones based on the Wolfe and Mond-Weir types. In addition 2006, ref. [14] a couple of Mond–Weir type second-order symmetric duality multiobjective calculations for cone second-order pseudoinvex and emphatically cone second-order pseudoinvex algorithm were presented. A couple of Mond–Weir type second-order symmetric dual multiobjective projects over discretion cones is created under pseudoinvexity/$K˘F$-convexity assumptions by Gulati [15], which is as:

Primal(MP):

$$\begin{array}{cc}\hfill \mathrm{K-minimize}& \psi (\iota ,\kappa )\hfill \\ \hfill \mathrm{subject\; to}\\ \hfill & -\left({\nabla }_{\kappa }\left({\lambda }^T\psi \right)(\iota ,\kappa )+{\nabla }_{\kappa \kappa }\left(w^T\varphi \right)(\iota ,\kappa )p\right)\in C_2^{*},\hfill \\ \hfill & {\kappa }^T\left({\nabla }_{\kappa }\left({\lambda }^T\psi \right)(\iota ,\kappa )+{\nabla }_{\kappa \kappa }\left(w^T\varphi \right)(\iota ,\kappa )p\right)\geqq 0,\hfill \\ \hfill & \lambda \in int{K}^{*},\iota \in C_1\hfill \end{array}$$

Dual(MD):

$$\begin{array}{cc}\hfill \mathrm{K-maximize}& \psi (\mu ,\nu )\hfill \\ \hfill \mathrm{subject\; to}\\ \hfill & \left({\nabla }_{\iota }\left({\lambda }^T\psi \right)(\mu ,\nu )+{\nabla }_{\iota \iota }\left(w^T\varphi \right)(\mu ,\nu )p\right)\in C_1^{*},\hfill \\ \hfill & {\mu }^T\left({\nabla }_{\iota }\left({\lambda }^T\psi \right)(\mu ,\nu )+{\nabla }_{\iota \iota }\left(w^T\varphi \right)(\mu ,\nu )r\right)\leqq 0,\hfill \\ \hfill & \lambda \in int{K}^{*},\iota \in C_2,\hfill \end{array}$$

where,

(i)

$R_1\subseteq {\mathbb{R}}^n,R_2\subseteq {\mathbb{R}}^m$ are open sets,

(ii)

$\psi ,\varphi :R_1\times R_2\to {\mathbb{R}}^k$ is a twice differentiable function of $\iota$ and $\kappa$, is a differentiable function of $\iota$ and $\kappa$,

(iii)

$\lambda \in {\mathbb{R}}^k,w\in {\mathbb{R}}^q,p\in {\mathbb{R}}^{m}andr\in {\mathbb{R}}^n$,

(iv)

for i = 1,2, $C_i\subset S_i$ is a closed convex cone with non-empty interior and $C_i^{*}$ is its positive polar cone.

Aside from them, a number of other researchers are working in this field. For additional information, see [16,17,18,19,20].

In this paper be start by defining in Section 2, K-$G_f$-bonvexity as well as pseduobonvexity and construct non-trivial numerical examples for clear understanding the concept introduced by authors. We identify several examples lying exclusively K-$G_f$-bonvex and not in the class of K-invex function with respect to same $\eta$ already exist in the literature. We illustrate an example which is K-$G_f$-pseudobonvex but not K-$G_f$-bonvex with respect to same $\eta$. In the next section, we formulate a new pair of multiobjective symmetric second order K-$G_f$-primal-dual models over arbitrary cone and drive duality results under K-$G_f$-bonvex as well as K-$G_f$-pseudobonvex assumptions. We, also construct a non-trivial example for validate the weak duality theorem presented in the paper. we also introduced geometry figure for clear understanding the concept through figure.

2. Preliminaries and Definitions

In this paper, we used ${\mathbb{R}}^n$ for n-dimensional Euclidean space and ${\mathbb{R}}_{+}^n$ for semi-positive orthant. Also, here $C_1$ and $C_2$ used for closed convex cone ${\mathbb{R}}^n$ and ${\mathbb{R}}^m$ respectively, with non-void interiors. For a real-valued twice differentiable function $g(\phi ,\vartheta )$ described on an open set in ${\mathbb{R}}^n\times {\mathbb{R}}^m$, indicate by ${\nabla }_{\phi }g(\overline{\phi },\overline{\vartheta })$ the gradient vector of g with respect to a at $(\overline{\phi },\overline{\vartheta })$, ${\nabla }_{\phi \phi }g(\overline{\phi },\overline{\vartheta })$ the Hessian matrix with respect to $\phi$ an at $(\overline{\phi },\overline{\vartheta }).$

Throughout the paper $\tilde{N}=\{1,2,\dots ,k\}$, $\tilde{O}=\{1,2,\dots ,m\}$.

A differentiable function $f:X\times Y\to R^k,{\eta }_1:X\times Y\to R^k,{\eta }_2:X\times Y\to R^k,G_f=(G_{f_1},G_{f_2},\dots ,G_{f_k}):R\to R^k,G_{f_i}:I_{f_i}\left(X\right)\to R$ is range $f_i$ for $i=\tilde{N}$. Also, K is used for pointed convex cone with non-void interiors in ${\mathbb{R}}^k$, for $\vartheta ,z\in {\mathbb{R}}^k$ and we specify cone orders with respect to K as follows:

$$\vartheta \leqq z\iff z-\vartheta \in K;\vartheta \le z\iff z-\vartheta \in K\setminus \left\{0\right\};\vartheta <z\iff z-\vartheta \in intK.$$

Let $f:X\to {\mathbb{R}}^k$ be a differentiable function defined on open set $\varphi \ne X\subseteq {\mathbb{R}}^n$ and $I_{f_i}\left(X\right),i\in \tilde{N}$ be the range of $f_i$.

Consider the following multiobjective programming problem with cone objective as well as constraints as:

(MP)     K-min $f\left(\phi \right)$

$$ subject to

$$\phi \in X^0=\left\{\phi \in S:g\left(\phi \right)\in Q\right\}.$$

where $S\subseteq {\mathbb{R}}^n,f:S\to {\mathbb{R}}^k,g:S\to {\mathbb{R}}^m.$ Q is a closed convex cone with a non-empty interior in ${\mathbb{R}}^m$.

Definition 1

([21]). $\overline{\phi }\in X^0$ is a weak efficient solution of (MP),$\nexists$$\phi \in X$ such that

$$f\left(\overline{\phi }\right)-f\left(\phi \right)\in intK.$$

Definition 2

([21]). $\overline{\phi }\in X^0$ is an efficient solution of (MP),$\nexists$$\phi \in X$ such that

$$f\left(\overline{\phi }\right)-f\left(\phi \right)\in K\setminus \left\{0\right\}.$$

Now, we consider the following multiobjective programming with cone objective and cone constraints as:

(GMP)        $K-$ min $G_f\left(f\left(z\right)\right)$

$$subject to $z\in Z^0=\left\{z\in S:-G_g\left(g\left(z\right)\right)\in Q\right\}.$

Definition 3

([21]). $\overline{z}\in Z^0$ is a weak efficient solution of (GMP), $\nexists z\in Z^0$ s.t.

$$G_f\left(f\left(\overline{z}\right)\right)-G_f\left(f\left(z\right)\right)\in intK.$$

Definition 4

([21]). $\overline{z}\in Z^0$ is a efficient solution of (GMP), $\nexists z\in Z^0$ s.t. $G_f\left(f\left(\overline{z}\right)\right)-G_f\left(f\left(z\right)\right)\in K\setminus \left\{0\right\}.$

Definition 5

([21]). The positive polar cone $C_i^{*}$ of $C_i$ (i=1,2) is defined as $C_i^{*}=\left\{z:{\phi }^{T}z\geqq 0,\forall \phi \in C_1\right\}.$Suppose that $S_1\subseteq {\mathbb{R}}^n$ and $S_2\subseteq {\mathbb{R}}^m$ are open sets such that

$$C_1\times C_2\subset S_1\times S_2.$$

A differentiable function $f:X\to R^k$ and $G_f$ such that every component $G_{f_i}$ is strictly increasing on the range of $I_{f_i}$.

Definition 6.

If $\exists G_f$ and η such that $\forall \phi \in X$ and $p_i\in R^n$, we have

$$\begin{array}{c}\hfill \begin{array}{c}{\displaystyle \{G_{f_1}\left(f_1\left(\phi \right)\right)-G_{f_1}\left(f_1\left(\delta \right)\right)+\frac{1}{2}{p}_1^T\left[G_{f_1}^{″}\left(f_1\left(\delta \right)\right){\nabla }_{\phi }{f}_1\left(\delta \right){\left({\nabla }_{\phi }{f}_1\left(\delta \right)\right)}^T+G_{f_1}^{\prime }\left(f_1\left(\delta \right)\right){\nabla }_{\phi \phi }{f}_1\left(\delta \right)\right]p_1-{\eta }^T(\phi ,\delta )[G_{f_1}^{\prime }\left(f_1\left(\delta \right)\right){\nabla }_{\phi }{f}_1\left(\delta \right)}\hfill \\ +\left\{G_{f_1}^{″}\left(f_1\left(\delta \right)\right){\nabla }_{\phi }{f}_1\left(\delta \right){\left({\nabla }_{\phi }{f}_1\left(\delta \right)\right)}^T+G_{f_1}^{\prime }\left(f_1\left(\delta \right)\right){\nabla }_{\phi \phi }{f}_1\left(\delta \right)\right\}{p}_1],\dots ,G_{f_k}\left(f_k\left(\phi \right)\right)-G_{f_k}\left(f_k\left(\delta \right)\right)+\frac{1}{2}{p}_k^T[G_{f_k}^{″}\left(f_k\left(\delta \right)\right){\nabla }_{\phi }{f}_k\left(\delta \right){\left({\nabla }_{\phi }{f}_k\left(\delta \right)\right)}^T\hfill \\ +G_{f_k}^{\prime }\left(f_k\left(\delta \right)\right){\nabla }_{\phi \phi }{f}_k\left(\delta \right)]p_k-{\eta }^T(\phi ,\delta )\left[G_{f_k}^{\prime }\left(f_k\left(\delta \right)\right){\nabla }_{\phi }{f}_k\left(\delta \right)+\left\{G_{f_k}^{″}\left(f_k\left(\delta \right)\right){\nabla }_{\phi }{f}_k\left(\delta \right){\left({\nabla }_{\phi }{f}_k\left(\delta \right)\right)}^T+G_{f_k}^{\prime }\left(f_k\left(\delta \right)\right){\nabla }_{\phi \phi }{f}_k\left(\delta \right)\right\}{p}_k\right]\}\in K,\hfill \end{array}\end{array}$$

then f is K-$G_f$-bonvex at $\delta \in X$ with respect to η.

Definition 7.

If ∃$G_f$ and η such that $\forall \phi \in X$ and $p_i\in R^m$, we have

$$\begin{array}{c}\hfill \begin{array}{c}{\displaystyle \{G_{f_1}\left(f_1\left(\phi \right)\right)-G_{f_1}\left(f_1\left(\delta \right)\right)+\frac{1}{2}{p}_1^T\left[G_{f_1}^{″}\left(f_1\left(\delta \right)\right){\nabla }_{\phi }{f}_1\left(\delta \right){\left({\nabla }_{\phi }{f}_1\left(\delta \right)\right)}^T+G_{f_1}^{\prime }\left(f_1\left(\delta \right)\right){\nabla }_{\phi \phi }{f}_1\left(\delta \right)\right]p_1-{\eta }^T(\phi ,\delta )[G_{f_1}^{\prime }\left(f_1\left(\delta \right)\right){\nabla }_{\phi }{f}_1\left(\delta \right)}\hfill \\ {\displaystyle +\left\{G_{f_1}^{″}\left(f_1\left(\delta \right)\right){\nabla }_{\phi }{f}_1\left(\delta \right){\left({\nabla }_{\phi }{f}_1\left(\delta \right)\right)}^T+G_{f_1}^{\prime }\left(f_1\left(\delta \right)\right){\nabla }_{\phi \phi }{f}_1\left(\delta \right)\right\}{p}_1],\dots ,G_{f_k}\left(f_k\left(\phi \right)\right)-G_{f_k}\left(f_k\left(\delta \right)\right)+\frac{1}{2}{p}_k^T[G_{f_k}^{″}\left(f_k\left(\delta \right)\right){\nabla }_{\phi }{f}_k\left(\delta \right){\left({\nabla }_{\phi }{f}_k\left(\delta \right)\right)}^T}\hfill \\ +G_{f_k}^{\prime }\left(f_k\left(\delta \right)\right){\nabla }_{\phi \phi }{f}_k\left(\delta \right)]p_k-{\eta }^T(\phi ,\delta )\left[G_{f_k}^{\prime }\left(f_k\left(\delta \right)\right){\nabla }_{\phi }{f}_k\left(\delta \right)+\left\{G_{f_k}^{″}\left(f_k\left(\delta \right)\right){\nabla }_{\phi }{f}_k\left(\delta \right){\left({\nabla }_{\phi }{f}_k\left(\delta \right)\right)}^T+G_{f_k}^{\prime }\left(f_k\left(\delta \right)\right){\nabla }_{\phi \phi }{f}_k\left(\delta \right)\right\}{p}_k\right]\}\in -K,\hfill \end{array}\end{array}$$

then f is K-$G_f$-boncave at $\delta \in X$ with respect to η.

Generalized the above definitions on two variable, as follows,

Definition 8.

If ∃ and $G_f$ and ${\eta }_1$ such that $\forall \phi \in X$ and $q_i\in R^n$, we have

$$\begin{array}{c}\hfill \begin{array}{c}{\displaystyle \{G_{f_1}\left(f_1(\phi ,\ell )\right)-G_{f_1}\left(f_1(\delta ,\ell )\right)+\frac{1}{2}{q}_1^T\left[G_{f_1}^{″}\left(f_1(\delta ,\ell )\right){\nabla }_{\phi }{f}_1(\delta ,\ell ){\left({\nabla }_{\phi }{f}_1(\delta ,\ell )\right)}^T+G_{f_1}^{\prime }\left(f_1(\delta ,\ell )\right){\nabla }_{\phi \phi }{f}_1(\delta ,\ell )\right]q_1-{\eta }_1^T(\phi ,\delta )[G_{f_1}^{\prime }\left(f_1(\delta ,\ell )\right)}\hfill \\ {\nabla }_{\phi }{f}_1(\delta ,\ell )+\{G_{f_1}^{″}\left(f_1(\delta ,\ell )\right){\nabla }_{\phi }{f}_1(\delta ,\ell ){\left({\nabla }_{\phi }{f}_1(\delta ,\ell )\right)}^T+G_{f_1}^{\prime }\left(f_1(\delta ,\ell )\right){\nabla }_{\phi \phi }{f}_1(\delta ,\ell )\}{q}_1],\dots ,G_{f_k}\left(f_k(\phi ,\ell )\right)-G_{f_k}\left(f_k(\delta ,\ell )\right)\hfill \\ {\displaystyle +\frac{1}{2}{q}_k^T\left[G_{f_k}^{″}\left(f_k(\delta ,\ell )\right){\nabla }_{\phi }{f}_k(\delta ,\ell ){\left({\nabla }_{\phi }{f}_k(\delta ,\ell )\right)}^T+G_{f_k}^{\prime }\left(f_k(\delta ,\ell )\right){\nabla }_{\phi \phi }{f}_k(\delta ,\ell )\right]q_k}\hfill \\ -{\eta }_1^T(\phi ,\delta )\left[G_{f_k}^{\prime }\left(f_k\left(\delta \right)\right){\nabla }_{\phi }{f}_k(\delta ,\ell )+\left\{G_{f_k}^{″}\left(f_k(\delta ,\ell )\right){\nabla }_{\phi }{f}_k(\delta ,\ell ){\left({\nabla }_{\phi }{f}_k(\delta ,\ell )\right)}^T+G_{f_k}^{\prime }\left(f_k(\delta ,\ell )\right){\nabla }_{\phi \phi }{f}_k(\delta ,\ell )\right\}{q}_k\right]\}\in K,\hfill \end{array}\end{array}$$

then, f is K-$G_f$-bonvex in the first variable at $\delta \in X$ for fixed $\ell \in Y$ with ${\eta }_1$,

and

If ∃$G_f$${\eta }_2$ such that $\forall \vartheta \in Y$ and $p_i\in R^m$, we have

$$\begin{array}{c}\hfill \begin{array}{c}{\displaystyle \{G_{f_1}\left(f_1(\delta ,\vartheta )\right)-G_{f_1}\left(f_1(\delta ,\ell )\right)+\frac{1}{2}{p}_1^T\left[G_{f_1}^{″}\left(f_1(\delta ,\ell )\right){\nabla }_{\vartheta }{f}_1(\delta ,\ell ){\left({\nabla }_{\vartheta }{f}_1(\delta ,\ell )\right)}^T+G_{f_1}^{\prime }{f}_1(\delta ,\ell ){\nabla }_{\vartheta \vartheta }{f}_1(\delta ,\ell )\right]p_1-{\eta }_2^T(\ell ,\vartheta )[G_{f_1}^{\prime }\left(f_1(\delta ,\ell )\right)}\hfill \\ {\nabla }_{\vartheta }{f}_1(\delta ,\ell )+\{G_{f_1}^{″}\left(f_1(\delta ,\ell )\right){\nabla }_{\vartheta }{f}_1(\delta ,\ell ){\left({\nabla }_{\vartheta }{f}_1(\delta ,\ell )\right)}^T+G_{f_1}^{\prime }\left(f_1(\delta ,\ell )\right){\nabla }_{\vartheta \vartheta }{f}_1(\delta ,\ell )\}{p}_1],\dots ,G_{f_k}\left(f_k(\delta ,\vartheta )\right)-G_{f_k}\left(f_k(\delta ,\ell )\right)\hfill \\ {\displaystyle +\frac{1}{2}{p}_k^T\left[G_{f_k}^{″}\left(f_k(\delta ,\ell )\right){\nabla }_{\vartheta }{f}_k(\delta ,\ell ){\left({\nabla }_{\vartheta }{f}_k(\delta ,\ell )\right)}^T+G_{f_k}^{\prime }\left(f_k(\delta ,\ell )\right){\nabla }_{\vartheta \vartheta }{f}_k(\delta ,\ell )\right]p_k}\hfill \\ -{\eta }_2^T(\ell ,\vartheta )\left[G_{f_k}^{\prime }\left(f_k(\delta ,\ell )\right){\nabla }_{\vartheta }{f}_k(\delta ,\ell )+\left\{G_{f_k}^{″}\left(f_k(\delta ,\ell )\right){\nabla }_{\vartheta }{f}_k(\delta ,\ell ){\left({\nabla }_{\vartheta }{f}_k(\delta ,\ell )\right)}^T+G_{f_k}^{\prime }\left(f_k(\delta ,\ell )\right){\nabla }_{\vartheta \vartheta }{f}_k(\delta ,\ell )\right\}{p}_k\right]\}\in K,\hfill \end{array}\end{array}$$

then, f is K-$G_f$-bonvex in the second variable at $\ell \in Y$ for fixed $\delta \in X$ with ${\eta }_2$.

Definition 9.

If ∃$G_f$ and ${\eta }_1$ such that $\forall \phi \in X$ and $q_i\in R^n$, we have

$$\begin{array}{c}\hfill \begin{array}{c}{\displaystyle \{G_{f_1}\left(f_1(\phi ,\ell )\right)-G_{f_1}\left(f_1(\delta ,\ell )\right)+\frac{1}{2}{q}_1^T\left[G_{f_1}^{″}\left(f_1(\delta ,\ell )\right){\nabla }_{\phi }{f}_1(\delta ,\ell ){\left({\nabla }_{\phi }{f}_1(\delta ,\ell )\right)}^T+G_{f_1}^{\prime }\left(f_1(\delta ,\ell )\right){\nabla }_{\phi \phi }{f}_1(\delta ,\ell )\right]q_1-{\eta }_1^T(\phi ,\delta )[G_{f_1}^{\prime }\left(f_1(\delta ,\ell )\right)}\hfill \\ {\nabla }_{\phi }{f}_1(\delta ,\ell )+\{G_{f_1}^{″}\left(f_1(\delta ,\ell )\right){\nabla }_{\phi }{f}_1(\delta ,\ell ){\left({\nabla }_{\phi }{f}_1(\delta ,\ell )\right)}^T+G_{f_1}^{\prime }\left(f_1(\delta ,\ell )\right){\nabla }_{\phi \phi }{f}_1(\delta ,\ell )\}{q}_1],\dots ,G_{f_k}\left(f_k(\phi ,\ell )\right)-G_{f_k}\left(f_k(\delta ,\ell )\right)\hfill \\ {\displaystyle +\frac{1}{2}{q}_k^T\left[G_{f_k}^{″}\left(f_k(\delta ,\ell )\right){\nabla }_{\phi }{f}_k(\delta ,\ell ){\left({\nabla }_{\phi }{f}_k(\delta ,\ell )\right)}^T+G_{f_k}^{\prime }\left(f_k(\delta ,\ell )\right){\nabla }_{\phi \phi }{f}_k(\delta ,\ell )\right]q_k}\hfill \\ -{\eta }_1^T(\phi ,\delta )\left[G_{f_k}^{\prime }\left(f_k\left(\delta \right)\right){\nabla }_{\phi }{f}_k(\delta ,\ell )+\left\{G_{f_k}^{″}\left(f_k(\delta ,\ell )\right){\nabla }_{\phi }{f}_k(\delta ,\ell ){\left({\nabla }_{\phi }{f}_k(\delta ,\ell )\right)}^T+G_{f_k}^{\prime }\left(f_k(\delta ,\ell )\right){\nabla }_{\phi \phi }{f}_k(\delta ,\ell )\right\}{q}_k\right]\}\in -K,\hfill \end{array}\end{array}$$

then, f is K-$G_f$-boncave in the first variable at $\delta \in X$ for fixed $\ell \in Y$ with respect to ${\eta }_1$,

and

If ∃$G_f$ and ${\eta }_2$ such that $\forall \vartheta \in Y$ and $p_i\in R^m$, we have

$$\begin{array}{c}\hfill \begin{array}{c}{\displaystyle \{G_{f_1}\left(f_1(\delta ,\vartheta )\right)-G_{f_1}\left(f_1(\delta ,\ell )\right)+\frac{1}{2}{p}_1^T\left[G_{f_1}^{″}\left(f_1(\delta ,\ell )\right){\nabla }_{\vartheta }{f}_1(\delta ,\ell ){\left({\nabla }_{\vartheta }{f}_1(\delta ,\ell )\right)}^T+G_{f_1}^{\prime }\left(f_1(\delta ,\ell )\right){\nabla }_{\vartheta \vartheta }{f}_1(\delta ,\ell )\right]p_1-{\eta }_2^T(\ell ,\vartheta )[G_{f_1}^{\prime }\left(f_1(\delta ,\ell )\right)}\hfill \\ {\nabla }_{\vartheta }{f}_1(\delta ,\ell )+\{G_{f_1}^{″}\left(f_1(\delta ,\ell )\right){\nabla }_{\vartheta }{f}_1(\delta ,\ell ){\left({\nabla }_{\vartheta }{f}_1(\delta ,\ell )\right)}^T+G_{f_1}^{\prime }\left(f_1(\delta ,\ell )\right){\nabla }_{\vartheta \vartheta }{f}_1(\delta ,\ell )\}{p}_1],\dots ,G_{f_k}\left(f_k(\delta ,\vartheta )\right)-G_{f_k}\left(f_k(\delta ,\ell )\right)\hfill \\ {\displaystyle +\frac{1}{2}{p}_k^T[G_{f_k}^{"}\left(f_k(\delta ,\ell )\right){\nabla }_{\vartheta }{f}_k(\delta ,\ell ){\left({\nabla }_{\vartheta }{f}_k(\delta ,\ell )\right)}^T}\hfill \\ -{\eta }_2^T(\ell ,\vartheta )[G_{f_k}^{\prime }\left(f_k(\delta ,\ell )\right){\nabla }_{\vartheta }{f}_k(\delta ,\ell )+\left\{G_{f_k}^{″}\left(f_k(\delta ,\ell )\right){\nabla }_{\vartheta }{f}_k(\delta ,\ell ){\left({\nabla }_{\vartheta }{f}_k(\delta ,\ell )\right)}^T+G_{f_k}^{\prime }\left(f_k(\delta ,\ell )\right){\nabla }_{\vartheta \vartheta }{f}_k(\delta ,\ell )\right\}{p}_k\left]\right\}\in -K,\hfill \end{array}\end{array}$$

then function f is K-$G_f$-boncave in the second variable at $\ell \in Y$ for fixed $\delta \in X$ with respect to ${\eta }_2$.

Example 1.

Let $X=[1,2]\subseteq \mathbb{R}, n=m=1$ and $k=2$. Consider $f:X\to {\mathbb{R}}^2$ be defined by

$$f\left(\phi \right)=\left(f_1\left(\phi \right),f_2\left(\phi \right)\right),$$

where,

$$f_1\left(\phi \right)=\phi sin\left(\frac{1}{\phi }\right),f_2\left(\phi \right)=cos\phi .$$

Next, $G_f:(G_{f_1},G_{f_2}):\mathbb{R}\to {\mathbb{R}}^2$ defined by

$${}^{}{G}_{f_1}=t^2,G_{f_2}=t^4.$$

Let $K=\left\{(\phi ,\vartheta );\phi \geqq 0\mathrm{and}\vartheta \geqq 0\right\}$ and $\eta :X\times X\to \mathbb{R}$ be given by

$$\eta (\phi ,\delta )=(1-{\delta }^2).$$

Now, we have to claim that f is $K-G_f$-bonvex, for this, we have driven that the following expression as

$$\begin{array}{c}\hfill \{G_{f_1}\left(f_1\left(\phi \right)\right)-G_{f_1}\left(f_1\left(\delta \right)\right)+{\displaystyle \frac{1}{2}}{p}_1^T\left[G_{f_1}^{″}\left(f_1\left(\delta \right)\right){\nabla }_{\phi }{f}_1\left(\delta \right){\left({\nabla }_{\phi }{f}_1\left(\delta \right)\right)}^T+G_{f_1}^{\prime }\left(f_1\left(\delta \right)\right){\nabla }_{\phi \phi }{f}_1\left(\delta \right)\right]p_1-{\eta }^T(\phi ,\delta )[G_{f_1}^{\prime }\left(f_1\left(\delta \right)\right){\nabla }_{\phi }{f}_1\left(\delta \right)\\ \hfill +\{G_{f_1}^{″}\left(f_1\left(\delta \right)\right){\nabla }_{\phi }{f}_1\left(\delta \right){\left({\nabla }_{\phi }{f}_1\left(\delta \right)\right)}^T+G_{f_1}^{\prime }\left(f_1\left(\delta \right)\right){\nabla }_{\phi \phi }{f}_1\left(\delta \right)\}{p}_1],G_{f_2}\left(f_2\left(\phi \right)\right)-G_{f_2}\left(f_2\left(\delta \right)\right)+{\displaystyle \frac{1}{2}}{p}_2^T[G_{f_2}^{″}\left(f_2\left(\delta \right)\right){\nabla }_{\phi }{f}_2\left(\delta \right){\left({\nabla }_{\phi }{f}_2\left(\delta \right)\right)}^T\\ \hfill +G_{f_2}^{\prime }\left(f_2\left(\delta \right)\right){\nabla }_{\phi \phi }{f}_2\left(\delta \right)]p_2-{\eta }^T(\phi ,\delta )[G_{f_2}^{\prime }\left(f_2\left(\delta \right)\right){\nabla }_{\phi }{f}_2\left(\delta \right)+\{G_{f_2}^{″}\left(f_2\left(\delta \right)\right){\nabla }_{\phi }{f}_2\left(\delta \right){\left({\nabla }_{\phi }{f}_2\left(\delta \right)\right)}^T+G_{f_2}^{\prime }\left(f_2\left(\delta \right)\right){\nabla }_{\phi \phi }{f}_2\left(\delta \right)\}{p}_2]\}\in K.\end{array}$$

Let

$$\begin{array}{c}\hfill \begin{array}{c}\prod =\{G_{f_1}\left(f_1\left(\phi \right)\right)-G_{f_1}\left(f_1\left(\delta \right)\right)+{\displaystyle \frac{1}{2}}{p}_1^T\left[G_{f_1}^{″}\left(f_1\left(\delta \right)\right){\nabla }_{\phi }{f}_1\left(\delta \right){\left({\nabla }_{\phi }{f}_1\left(\delta \right)\right)}^T+G_{f_1}^{\prime }\left(f_1\left(\delta \right)\right){\nabla }_{\phi \phi }{f}_1\left(\delta \right)\right]p_1-{\eta }^T(\phi ,\delta )[G_{f_1}^{\prime }\left(f_1\left(\delta \right)\right){\nabla }_{\phi }{f}_1\left(\delta \right)\hfill \\ +\left\{G_{f_1}^{″}\left(f_1\left(\delta \right)\right){\nabla }_{\phi }{f}_1\left(\delta \right){\left({\nabla }_{\phi }{f}_1\left(\delta \right)\right)}^T+G_{f_1}^{\prime }\left(f_1\left(\delta \right)\right){\nabla }_{\phi \phi }{f}_1\left(\delta \right)\right\}{p}_1],G_{f_2}\left(f_2\left(\phi \right)\right)-G_{f_2}\left(f_2\left(\delta \right)\right)+{\displaystyle \frac{1}{2}}{p}_2^T[G_{f_2}^{″}\left(f_2\left(\delta \right)\right){\nabla }_{\phi }{f}_2\left(\delta \right){\left({\nabla }_{\phi }{f}_2\left(\delta \right)\right)}^T\hfill \\ +G_{f_2}^{\prime }\left(f_2\left(\delta \right)\right){\nabla }_{\phi \phi }{f}_2\left(\delta \right)]p_2-{\eta }^T(\phi ,\delta )\left[G_{f_2}^{\prime }\left(f_2\left(\delta \right)\right){\nabla }_{\phi }{f}_2\left(\delta \right)+\left\{G_{f_2}^{″}\left(f_2\left(\delta \right)\right){\nabla }_{\phi }{f}_2\left(\delta \right){\left({\nabla }_{\phi }{f}_2\left(\delta \right)\right)}^T+G_{f_2}^{\prime }\left(f_2\left(\delta \right)\right){\nabla }_{\phi \phi }{f}_2\left(\delta \right)\right\}{p}_2\right]\}.\hfill \end{array}\end{array}$$

Substituting the values of $f_1,f_2,G_{f_1},G_{f_2}$ and η, we obtain

$$\begin{array}{c}\hfill \begin{array}{c}{\displaystyle \prod =\{{\phi }^{2}si{n}^2\frac{1}{{\phi }^2}-{\delta }^{2}si{n}^2\frac{1}{{\delta }^2}+\frac{1}{2}{p}^2\left[2{\left(sin\frac{1}{\delta }-\frac{1}{\delta }cos\frac{1}{\delta }\right)}^2+2\delta sin\frac{1}{\delta }\left(-\frac{1}{{\delta }^3}sin\frac{1}{\delta }\right)\right]-(1-{\delta }^2)[2\delta sin\frac{1}{\delta }\left(sin\frac{1}{\delta }-\frac{1}{\delta }cos\frac{1}{\delta }\right)}\hfill \\ {\displaystyle +p\left[2{\left(sin\frac{1}{\delta }-\frac{1}{\delta }cos\frac{1}{\delta }\right)}^2+2\delta sin\frac{1}{\delta }\left(-\frac{1}{{\delta }^3}sin\frac{1}{\delta }\right)\right]],co{s}^4\phi -co{s}^4\delta +\frac{1}{2}{p}^2[12co{s}^2\delta {(-sin\delta )}^2}\hfill \\ +4co{s}^3\delta (-cos\delta )]-(1-{\delta }^2)[4co{s}^3\delta (-sin\delta )+p(12co{s}^2\delta {(-sin\delta )}^2+4co{s}^3\delta (-cos\delta ))]\}.\hfill \end{array}\end{array}$$

Now, we consider

${\displaystyle \Psi ={\phi }^{2}si{n}^2\frac{1}{{\phi }^2}-{\delta }^{2}si{n}^2\frac{1}{{\delta }^2}+\frac{1}{2}{p}^2\left[2{\left(sin\frac{1}{\delta }-\frac{1}{\delta }cos\frac{1}{\delta }\right)}^2+2\delta sin\frac{1}{\delta }\left(-\frac{1}{{\delta }^3}sin\frac{1}{\delta }\right)\right]}$

$$\begin{array}{c}\hfill {\displaystyle -(1-{\delta }^2)\left[2\delta sin\frac{1}{\delta }\left(sin\frac{1}{\delta }-\frac{1}{\delta }cos\frac{1}{\delta }\right)+p\left[2{\left(sin\frac{1}{\delta }-\frac{1}{\delta }cos\frac{1}{\delta }\right)}^2+2\delta sin\frac{1}{\delta }\left(-\frac{1}{{\delta }^3}sin\frac{1}{\delta }\right)\right]\right].}\end{array}$$

Let us apply the following ansatz:

$$\Psi ={\Psi }_1+{\Psi }_2\left(say\right),$$

consider

$$\Phi =\{co{s}^4\phi -co{s}^4\delta +\frac{1}{2}{p}^2\left[12co{s}^2\delta {(-sin\delta )}^2+4co{s}^3\delta (-cos\delta )\right]$$

$$-(1-{\delta }^2)\left[4co{s}^3\delta (-sin\delta )+p\left(12co{s}^2\delta {(-sin\delta )}^2+4co{s}^3\delta (-cos\delta )\right)\right]\}\in K.$$

The above expression breaks in ${\Phi }_1$ and ${\Phi }_2$ (say) as follows:

$$\Phi ={\Phi }_1+{\Phi }_2,$$

where

$${\displaystyle {\Psi }_1={\phi }^{2}si{n}^2\frac{1}{{\phi }^2}-{\delta }^{2}si{n}^2\frac{1}{{\delta }^2}-(1-{\delta }^2)\left[{\displaystyle 2\delta sin\frac{1}{\delta }\left({\displaystyle sin\frac{1}{\delta }-\frac{1}{\delta }cos\frac{1}{\delta }}\right)}\right].}$$

It is easily verified from Figure 1, we have

$$\begin{array}{c}\hfill {\Psi }_1\geqq 0,\forall \phi ,\delta \in X.\end{array}$$

$$\begin{array}{c}\hfill {\Psi }_2=\frac{1}{2}{p}^2\left[2{\left(sin\frac{1}{\delta }-\frac{1}{\delta }cos\frac{1}{\delta }\right)}^2+2\delta sin\frac{1}{\delta }\left(-\frac{1}{{\delta }^3}sin\frac{1}{\delta }\right)\right]+p\left[2{\left(sin\frac{1}{\delta }-\frac{1}{\delta }cos\frac{1}{\delta }\right)}^2+2\delta sin\frac{1}{\delta }\left(-\frac{1}{{\delta }^3}sin\frac{1}{\delta }\right)\right].\end{array}$$

It is clear from Figure 2, we obtain

$$\begin{array}{c}\hfill {\Psi }_2\geqq 0,\forall \delta \in X\mathrm{and}p\in \left[-\frac{1}{{10}^{10}},-1\right].\end{array}$$

Now,

$${\Phi }_1=co{s}^4\phi -co{s}^4\delta +-(1-{\delta }^2)\left[4co{s}^3\delta (-sin\delta )\right],$$

as can be seen from Figure 3.

$${\Phi }_1\geqq 0\forall \phi ,\delta \in X,$$

and

$${\Phi }_2=\frac{1}{2}{p}^2\left[12co{s}^2\delta {(-sin\delta )}^2+4co{s}^3\delta (-cos\delta )+p\left(12co{s}^2\delta {(-sin\delta )}^2+4co{s}^3\delta (-cos\delta )\right)\right].$$

As can be seen from Figure 4. ${\Phi }_2\geqq 0,\forall \delta \in X\mathrm{and}{p}_1,p_2\in [-\frac{1}{{10}^{10}},-1].$ (From Figure 4).

Hence, $\Psi \geqq 0$ and $\Phi \geqq 0$. This gives $\psi +\varphi \geqq 0$. Thus, we can find that $(\Psi ,\Phi )\in K.$

Hence, f is K-$G_f$-bonvex function at $(\Psi ,\Phi )$ w.r.t. $\eta$.

We will show that f is not invex. For this it is either

$$f_1\left(\phi \right)-f_1\left(\delta \right)-{\eta }^T(\phi ,\delta ){\nabla }_{\phi }{f}_1\left(\delta \right)\ngeqq 0$$

or

$$f_2\left(\phi \right)-f_2\left(\delta \right)-{\eta }^T(\phi ,\delta ){\nabla }_{\phi }{f}_2\left(\delta \right)\ngeqq 0.$$

Since $f_1\left(\phi \right)-f_1\left(\delta \right)-{\eta }^T(\phi ,\delta ){\nabla }_{\phi }{f}_1\left(\delta \right)=\phi sin\frac{1}{\phi }-\delta sin\frac{1}{\delta }-(1-{\delta }^2)sin\frac{1}{\delta }-\frac{1}{\delta }cos\frac{1}{\delta }\ngeqq 0$, is not $\forall \phi ,\delta \in X$ as can be seen from Figure 5. Also, $f_2\left(\phi \right)-f_1\left(\delta \right)-{\eta }^T(\phi ,\delta ){\nabla }_{\phi }{f}_2\left(\delta \right)=cos\phi -cos\delta +(1-{\delta }^2)sin\delta \ngeqq 0$, is not $\forall \phi ,\delta \in X$ as can be seen from Figure 6.

Therefore, from the above example, it shows that f is K-$G_f$-bonvex, but it is not invex with respect to same $\eta$.

Definition 10.

If ∃$G_f$ and η such that $\forall \phi \in X$ and $q_i\in R^n$, we have

$$\begin{array}{c}\hfill \begin{array}{c}{\eta }^T(\phi ,\delta )\{G_{f_1}^{\prime }\left(f_1\left(\delta \right)\right){\nabla }_{\phi }{f}_1\left(\delta \right)+q_1\left\{G_{f_1}^{″}\left(f_1\left(\delta \right)\right){\left({\nabla }_{\phi }{f}_1\left(\delta \right)\right)}^T+G_{f_1}^{\prime }\left(f_1\left(\delta \right)\right){\nabla }_{\phi \phi }{f}_1\left(\delta \right)\right\},\dots ,G_{f_k}^{\prime }\left(f_k\left(\delta \right)\right){\nabla }_{\phi }{f}_k\left(\delta \right)+q_k\{G_{f_k}^{″}\left(f_k\left(\delta \right)\right){\left({\nabla }_{\phi }{f}_k\left(\delta \right)\right)}^T\hfill \\ {\displaystyle +G_{f_k}^{\prime }\left(f_k\left(\delta \right)\right){\nabla }_{\phi \phi }{f}_k\left(\delta \right)\left\}\right\}\in K\Rightarrow [G_{f_1}\left(f_1\left(\phi \right)\right)-G_{f_1}\left(f_1\left(\delta \right)\right)+\frac{1}{2}{q}_1^T\left\{G_{f_1}^{″}\left(f_1\left(\delta \right)\right){\nabla }_{\phi }{f}_1\left(\delta \right){\left({\nabla }_{\phi }{f}_1\left(\delta \right)\right)}^T+G_{f_1}^{\prime }\left(f_1\left(\delta \right)\right){\nabla }_{\phi \phi }{f}_1\left(\delta \right)\right\}{q}_1}\hfill \\ ,\dots ,G_{f_k}\left(f_k\left(\phi \right)\right)-G_{f_k}\left(f_k\left(\delta \right)\right)+\frac{1}{2}{q}_k^T\left\{G_{f_k}^{″}\left(f_k\left(\delta \right)\right){\nabla }_{\phi }{f}_k\left(\delta \right){\left({\nabla }_{\phi }{f}_k\left(\delta \right)\right)}^T+G_{f_k}^{\prime }\left(f_k\left(\delta \right)\right){\nabla }_{\phi \phi }{f}_k\left(\delta \right)\right\}{q}_k]\in K,\hfill \end{array}\end{array}$$

then, f is $G_f$-pseudobonvex at $\delta \in X$ with η.

Definition 11.

If ∃$G_f$ and η such that $\forall \phi \in X$ and $q_1\in R^n$, we have

$$\begin{array}{c}\hfill \begin{array}{c}{\eta }^T(\phi ,\delta )\{G_{f_1}^{\prime }\left(f_1\left(\delta \right)\right){\nabla }_{\phi }{f}_1\left(\delta \right)+q_1\left\{G_{f_1}^{″}\left(f_1\left(\delta \right)\right){\left({\nabla }_{\phi }{f}_1\left(\delta \right)\right)}^T+G_{f_1}^{\prime }\left(f_1\left(\delta \right)\right){\nabla }_{\phi \phi }{f}_1\left(\delta \right)\right\},\dots ,G_{f_k}^{\prime }\left(f_k\left(\delta \right)\right){\nabla }_{\phi }{f}_k\left(\delta \right)+q_k\{G_{f_k}^{″}\left(f_k\left(\delta \right)\right){\left({\nabla }_{\phi }{f}_k\left(\delta \right)\right)}^T\hfill \\ {\displaystyle +G_{f_k}^{\prime }\left(f_k\left(\delta \right)\right){\nabla }_{\phi \phi }{f}_k\left(\delta \right)\left\}\right\}\in -K\Rightarrow [G_{f_1}\left(f_1\left(\phi \right)\right)-G_{f_1}\left(f_1\left(\delta \right)\right)+\frac{1}{2}{q}_1^T\left\{G_{f_1}^{″}\left(f_1\left(\delta \right)\right){\nabla }_{\phi }{f}_1\left(\delta \right){\left({\nabla }_{\phi }{f}_1\left(\delta \right)\right)}^T+G_{f_1}^{\prime }\left(f_1\left(\delta \right)\right){\nabla }_{\phi \phi }{f}_1\left(\delta \right)\right\}{q}_1}\hfill \\ ,\dots ,G_{f_k}\left(f_k\left(\phi \right)\right)-G_{f_k}\left(f_k\left(\delta \right)\right)+\frac{1}{2}{q}_k^T\left\{G_{f_k}^{″}\left(f_k\left(\delta \right)\right){\nabla }_{\phi }{f}_k\left(\delta \right){\left({\nabla }_{\phi }{f}_k\left(\delta \right)\right)}^T+G_{f_k}^{\prime }\left(f_k\left(\delta \right)\right){\nabla }_{\phi \phi }{f}_k\left(\delta \right)\right\}{q}_k]\in -K,\hfill \end{array}\end{array}$$

then f is $G_f$-pseudoboncave at $\delta \in X$ with respect to η.

We generalized the above definition as follows:

Definition 12.

If ∃$G_f$ and ${\eta }_1$ such that $\forall \phi \in X$ and $q_i\in R^n$, we have

$$\begin{array}{c}\hfill \begin{array}{c}{\displaystyle {\eta }_1^T(\phi ,\delta )\{G_{f_1}^{\prime }\left(f_1(\delta ,\ell )\right){\nabla }_{\phi }{f}_1(\delta ,\ell )+q_1\left\{G_{f_1}^{″}\left(f_1(\delta ,\ell )\right){\left({\nabla }_{\phi }{f}_1(\delta ,\ell )\right)}^T+G_{f_1}^{\prime }\left(f_1(\delta ,\ell )\right){\nabla }_{\phi \phi }{f}_1(\delta ,\ell )\right\},\dots ,G_{f_k}^{\prime }\left(f_k(\delta ,\ell )\right){\nabla }_{\phi }{f}_k(\delta ,\ell )}\hfill \\ +q_k\left\{G_{f_k}^{″}\left(f_k(\delta ,\ell )\right){\left({\nabla }_{\phi }{f}_k(\delta ,\ell )\right)}^T+G_{f_k}^{\prime }\left(f_k(\delta ,\ell )\right){\nabla }_{\phi \phi }{f}_k(\delta ,\ell )\right\}\}\in K\hfill \\ {\displaystyle \Rightarrow [G_{f_1}\left(f_1(\phi ,\ell )\right)-G_{f_1}\left(f_1(\delta ,\ell )\right)+\frac{1}{2}{q}_1^T\left\{G_{f_1}^{″}\left(f_1(\delta ,\ell )\right){\nabla }_{\phi }{f}_1(\delta ,\ell ){\left({\nabla }_{\phi }{f}_1(\delta ,\ell )\right)}^T+G_{f_1}^{\prime }\left(f_1(\delta ,\ell )\right){\nabla }_{\phi \phi }{f}_1(\delta ,\ell )\right\}{q}_1,\dots ,G_{f_k}\left(f_k(\phi ,\ell )\right)}\hfill \\ {\displaystyle -G_{f_k}\left(f_k(\delta ,\ell )\right)+\frac{1}{2}{q}_k^T\left\{G_{f_k}^{″}\left(f_k(\delta ,\ell )\right){\nabla }_{\phi }{f}_k(\delta ,\ell ){\left({\nabla }_{\phi }{f}_k(\delta ,\ell )\right)}^T+G_{f_k}^{\prime }\left(f_k(\delta ,\ell )\right){\nabla }_{\phi \phi }{f}_k(\delta ,\ell )\right\}{q}_k]\in K,}\hfill \end{array}\end{array}$$

then f is K-$G_f$-bonvex in the first variable at $\delta \in X$ for fixed $\ell \in Y$ with ${\eta }_1$,

and

if ∃$G_f$ and ${\eta }_2$ such that $\forall \vartheta \in Y$ and $p_i\in R^m$, we have

$$\begin{array}{c}\hfill \begin{array}{c}{\eta }_2^T(\delta ,\vartheta )\{G_{f_1}^{\prime }\left(f_1(\delta ,\vartheta )\right){\nabla }_{\vartheta }{f}_1(\delta ,\ell )+\{G_{f_1}^{″}\left(f_1(\delta ,\ell )\right){\left({\nabla }_{\vartheta }{f}_1(\delta ,\ell )\right)}^T+G_{f_1}^{\prime }\left(f_1(\delta ,\ell )\right){\nabla }_{\vartheta \vartheta }{f}_1(\delta ,\ell )\ell \}{p}_1,\dots ,G_{f_k}^{\prime }\left(f_k(\delta ,\ell )\right){\nabla }_{\vartheta }{f}_k(\delta ,\ell )\hfill \\ +p_k\left\{G_{f_k}^{″}\left(f_k(\delta ,\ell )\right){\left({\nabla }_{\vartheta }{f}_k(\delta ,\ell )\right)}^T+G_{f_k}^{\prime }\left(f_k(\delta ,\ell )\right){\nabla }_{\vartheta \vartheta }{f}_k(\delta ,\ell )\right\}\}\in K\hfill \\ {\displaystyle \Rightarrow [G_{f_1}\left(f_1(\delta ,\vartheta )\right)-G_{f_1}\left(f_1(\delta ,\ell )\right)+\frac{1}{2}{p}_1^T\left\{G_{f_1}^{″}\left(f_1(\delta ,\ell )\right){\nabla }_{\vartheta }{f}_1(\delta ,\ell ){\left({\nabla }_{\vartheta }{f}_1(\delta ,\ell )\right)}^T+G_{f_1}^{\prime }\left(f_1(\delta ,\ell )\right){\nabla }_{\vartheta \vartheta }{f}_1(\delta ,\ell )\right\}{p}_1,\dots ,G_{f_k}\left(f_k(\delta ,\vartheta )\right)}\hfill \\ {\displaystyle -G_{f_k}\left(f_k(\delta ,\ell )\right)+\frac{1}{2}{p}_k^T\left\{G_{f_k}^{″}\left(f_k(\delta ,\ell )\right){\nabla }_{\vartheta }{f}_k(\delta ,\ell ){\left({\nabla }_{\vartheta }{f}_k(\delta ,\ell )\right)}^T+G_{f_k}^{\prime }\left(f_k(\delta ,\ell )\right){\nabla }_{\vartheta \vartheta }{f}_k(\delta ,\ell )\right\}{p}_k]\in K,}\hfill \end{array}\end{array}$$

then f is K-$G_f$-bonvex in the second variable at $\ell \in Y$ for fixed $\delta \in X$ with ${\eta }_2$.

Definition 13.

If ∃$G_f$ and ${\eta }_1$ such that $\forall \phi \in X$ and $q_i\in R^n$, we have

$$\begin{array}{c}\hfill \begin{array}{c}{\eta }_1^T(\phi ,\delta )\{G_{f_1}^{\prime }\left(f_1(\delta ,\ell )\right){\nabla }_{\phi }{f}_1(\delta ,\ell )+q_1\left\{G_{f_1}^{″}\left(f_1(\delta ,\ell )\right){\left({\nabla }_{\phi }{f}_1(\delta ,\ell )\right)}^T+G_{f_1}^{\prime }\left(f_1(\delta ,\ell )\right){\nabla }_{\phi \phi }{f}_1(\delta ,\ell )\right\},\dots ,G_{f_k}^{\prime }\left(f_k(\delta ,\ell )\right){\nabla }_{\phi }{f}_k(\delta ,\ell )\hfill \\ +q_k\left\{G_{f_k}^{″}\left(f_k(\delta ,\ell )\right){\left({\nabla }_{\phi }{f}_k(\delta ,\ell )\right)}^T+G_{f_k}^{\prime }\left(f_k(\delta ,\ell )\right){\nabla }_{\phi \phi }{f}_k(\delta ,\ell )\right\}\}\in -K\hfill \\ {\displaystyle \Rightarrow [G_{f_1}\left(f_1(\phi ,\ell )\right)-G_{f_1}\left(f_1(\delta ,\ell )\right)+\frac{1}{2}{q}_1^T\left\{G_{f_1}^{″}\left(f_1(\delta ,\ell )\right){\nabla }_{\phi }{f}_1(\delta ,\ell ){\left({\nabla }_{\phi }{f}_1(\delta ,\ell )\right)}^T+G_{f_1}^{\prime }\left(f_1(\delta ,\ell )\right){\nabla }_{\phi \phi }{f}_1(\delta ,\ell )\right\}{q}_1,\dots ,G_{f_k}\left(f_k(\phi ,\ell )\right)}\hfill \\ {\displaystyle -G_{f_k}\left(f_k(\delta ,\ell )\right)+\frac{1}{2}{q}_k^T\left\{G_{f_k}^{″}\left(f_k(\delta ,\ell )\right){\nabla }_{\phi }{f}_k(\delta ,\ell ){\left({\nabla }_{\phi }{f}_k(\delta ,\ell )\right)}^T+G_{f_k}^{\prime }\left(f_k(\delta ,\ell )\right){\nabla }_{\phi \phi }{f}_k(\delta ,\ell )\right\}{q}_k]\in -K,}\hfill \end{array}\end{array}$$

then f is K-$G_f$-bonvex in the first variable at $\delta \in X$ for fixed $\ell \in Y$ with ${\eta }_1$,

and

If ∃$G_f$ and ${\eta }_2$ such that $\forall \vartheta \in Y$ and $p_i\in R^m$, we have

$$\begin{array}{c}\hfill \begin{array}{c}{\eta }_2^T(\delta ,\vartheta )\{G_{f_1}^{\prime }\left(f_1(\delta ,\vartheta )\right){\nabla }_{\vartheta }{f}_1(\delta ,\ell )+\left\{G_{f_1}^{″}\left(f_1(\delta ,\ell )\right){\left({\nabla }_{\vartheta }{f}_1(\delta ,\ell )\right)}^T+G_{f_1}^{\prime }\left(f_1(\delta ,\ell )\right){\nabla }_{\vartheta \vartheta }{f}_1(\delta ,\ell )\right\}{p}_1,\dots ,G_{f_k}^{\prime }\left(f_k(\delta ,\ell )\right){\nabla }_{\vartheta }{f}_k(\delta ,\ell )\hfill \\ +p_k\left\{G_{f_k}^{″}\left(f_k(\delta ,\ell )\right){\left({\nabla }_{\vartheta }{f}_k(\delta ,\ell )\right)}^T+G_{f_k}^{\prime }\left(f_k(\delta ,\ell )\right){\nabla }_{\vartheta \vartheta }{f}_k(\delta ,\ell )\right\}\}\in -K\hfill \\ {\displaystyle \Rightarrow [G_{f_1}\left(f_1(\delta ,\vartheta )\right)-G_{f_1}\left(f_1(\delta ,\ell )\right)+\frac{1}{2}{p}_1^T\left\{G_{f_1}^{″}\left(f_1(\delta ,\ell )\right){\nabla }_{\vartheta }{f}_1(\delta ,\ell ){\left({\nabla }_{\vartheta }{f}_1(\delta ,\ell )\right)}^T+G_{f_1}^{\prime }\left(f_1(\delta ,\ell )\right){\nabla }_{\vartheta \vartheta }{f}_1(\delta ,\ell )\right\}{p}_1,\dots ,G_{f_k}\left(f_k(\delta ,\vartheta )\right)}\hfill \\ {\displaystyle -G_{f_k}\left(f_k(\delta ,\ell )\right)+\frac{1}{2}{p}_k^T\left\{G_{f_k}^{″}\left(f_k(\delta ,\ell )\right){\nabla }_{\vartheta }{f}_k(\delta ,\ell ){\left({\nabla }_{\vartheta }{f}_k(\delta ,\ell )\right)}^T+G_{f_k}^{\prime }\left(f_k(\delta ,\ell )\right){\nabla }_{\vartheta \vartheta }{f}_k(\delta ,\ell )\right\}{p}_k]\in -K.}\hfill \end{array}\end{array}$$

then f is K-$G_f$-boncave in the second variable at $\ell \in Y$ for fixed $\delta \in X$ with respect to ${\eta }_2$.

Remark 1.

If $G_f\left(t\right)=t$, then above definition reduces in $K-\eta$-pseudo bonvex w.r.t. η,

${\eta }^T(\phi ,\delta )\left[{\nabla }_{\phi }{f}_1\left(\delta \right)+{\nabla }_{\phi \phi }{f}_1\left(\delta \right)q_1,\dots \dots ,{\nabla }_{\phi }{f}_k\left(\delta \right)+{\nabla }_{\phi \phi }{f}_k\left(\delta \right)q_k\right]\in K$

$$\Rightarrow \left[f_1\left(\phi \right)-f_1\left(\delta \right)+\frac{1}{2}{q}_1^T{\nabla }_{\phi \phi }{f}_1\left(\delta \right)q_1,\dots ,f_k\left(\phi \right)-f_k\left(\delta \right)+\frac{1}{2}{q}^T{\nabla }_{\phi \phi }{f}_k\left(\delta \right)q_k\right]\in K.$$

Example 2.

Let $X=[-10,10]$ and $K=\left\{(\phi ,\vartheta ):\phi \geqq 0,\phi \leqq \vartheta \right\}$. Consider the function $f:X\to {\mathbb{R}}^2$ defined by

$$f\left(\phi \right)=(f_1\left(\phi \right),f_2\left(\phi \right)),$$

where

$$f_1\left(\phi \right)=sin\phi ,f_2\left(\phi \right)=e^{\phi }$$

Define $G_f=(G_{f_1},G_{f_2}):R^2\to R$ given by

$$G_{f_1}=t^2,G_{f_2}=t^3,\eta ={\phi }^2-{\delta }^2,and{q}_1=q_2\in [2,\infty ].$$

We have to claim that function f is K-$G_f$-pseudobonvex at point δ, i.e.,

$$\begin{array}{c}\hfill \begin{array}{c}{\eta }^T(\phi ,\delta )\{G_{f_1}^{\prime }\left(f_1\left(\delta \right)\right){\nabla }_{\phi }{f}_1\left(\delta \right)+q_1\left\{G_{f_1}^{″}\left(f_1\left(\delta \right)\right){\left({\nabla }_{\phi }{f}_1\left(\delta \right)\right)}^T+G_{f_1}^{\prime }\left(f_1\left(\delta \right)\right){\nabla }_{\phi \phi }{f}_1\left(\delta \right)\right\},G_{f_2}^{\prime }\left(f_2\left(\delta \right)\right){\nabla }_{\phi }{f}_2\left(\delta \right)+q_2\{G_{f_2}^{″}\left(f_2\left(\delta \right)\right){\left({\nabla }_{\phi }{f}_2\left(\delta \right)\right)}^T\hfill \\ {\displaystyle +G_{f_2}^{\prime }\left(f_2\left(\delta \right)\right){\nabla }_{\phi \phi }{f}_2\left(\delta \right)\left\}\right\}\in K\Rightarrow \{G_{f_1}\left(f_1\left(\phi \right)\right)-G_{f_1}\left(f_1\left(\delta \right)\right)+\frac{1}{2}{q}_1^T\left\{G_{f_1}^{″}\left(f_1\left(\delta \right)\right){\nabla }_{\phi }{f}_1\left(\delta \right){\left({\nabla }_{\phi }{f}_1\left(\delta \right)\right)}^T+G_{f_1}^{\prime }\left(f_1\left(\delta \right)\right){\nabla }_{\phi \phi }{f}_1\left(\delta \right)\right\}{q}_1,}\hfill \\ G_{f_2}\left(f_2\left(\phi \right)\right)-G_{f_2}\left(f_2\left(\delta \right)\right)+\frac{1}{2}{q}_2^T\left\{G_{f_2}^{″}\left(f_2\left(\delta \right)\right){\nabla }_{\phi }{f}_2\left(\delta \right){\left({\nabla }_{\phi }{f}_2\left(\delta \right)\right)}^T+G_{f_2}^{\prime }\left(f_2\left(\delta \right)\right){\nabla }_{\phi \phi }{f}_2\left(\delta \right)\right\}{q}_2\}\in K.\hfill \end{array}\end{array}$$

Consider

$$\begin{array}{c}\hfill \begin{array}{c}\tau ={\eta }^T(\phi ,\delta )\{G_{f_1}^{\prime }\left(f_1\left(\delta \right)\right){\nabla }_{\phi }{f}_1\left(\delta \right)+q_1\left\{G_{f_1}^{″}\left(f_1\left(\delta \right)\right){\left({\nabla }_{\phi }{f}_1\left(\delta \right)\right)}^T+G_{f_1}^{\prime }\left(f_1\left(\delta \right)\right){\nabla }_{\phi \phi }{f}_1\left(\delta \right)\right\},G_{f_2}^{\prime }\left(f_2\left(\delta \right)\right){\nabla }_{\phi }{f}_2\left(\delta \right)\hfill \\ +q_2\left\{G_{f_2}^{″}\left(f_2\left(\delta \right)\right){\left({\nabla }_{\phi }{f}_2\left(\delta \right)\right)}^T+G_{f_2}^{\prime }\left(f_2\left(\delta \right)\right){\nabla }_{\phi \phi }{f}_2\left(\delta \right)\right\}\}.\hfill \end{array}\end{array}$$

Putting the values of $f_1,f_2,G_{f_1},G_{f_2}$ and η, we have

$$\tau =({\phi }^2-{\delta }^2)\left(sin2\delta +2q_1(cos\delta -si{n}^2\delta ),3e^{3\delta }+9e^{2\delta }{q}_2\right).$$

At the point $\delta =0$, the value of above expression becomes

$$\tau =\left\{2{\phi }^2{q}_1,3{\phi }^2(1+3q_2)\right\},\forall q_1=q_2\in [2,\infty )$$

Obviously,

$$\tau =\left\{2{\phi }^2{q}_1,3{\phi }^2(1+3q_2)\right\}\in K.$$

Next, consider

$$\begin{array}{c}\hfill \begin{array}{c}{\displaystyle \Psi =\{G_{f_1}\left(f_1\left(\phi \right)\right)-G_{f_1}\left(f_1\left(\delta \right)\right)+\frac{1}{2}{q}_1^T\{G_{f_1}^{″}\left(f_1\left(\delta \right)\right){\nabla }_{\phi }{f}_1\left(\delta \right){\left({\nabla }_{\phi }{f}_1\left(\delta \right)\right)}^T+G_{f_1}^{\prime }\left(f_1\left(\delta \right)\right){\nabla }_{\phi \phi }{f}_1\left(\delta \right)\}{q}_1,G_{f_2}\left(f_2\left(\phi \right)\right)}\hfill \\ {\displaystyle -G_{f_2}\left(f_2\left(\delta \right)\right)+\frac{1}{2}{q}_2^T\{G_{f_2}^{″}\left(f_2\left(\delta \right)\right){\nabla }_{\phi }{f}_2\left(\delta \right){\left({\nabla }_{\phi }{f}_2\left(\delta \right)\right)}^T+G_{f_2}^{\prime }\left(f_2\left(\delta \right)\right){\nabla }_{\phi \phi }{f}_2\left(\delta \right)\}{q}_2\}.}\hfill \end{array}\end{array}$$

Putting the values of $f_1,f_2,G_{f_1},G_{f_2}$ and η, we have

$$\Psi =\left\{si{n}^2\phi -si{n}^2\delta +\frac{1}{2}{q}_1^2(2co{s}^2\delta -2si{n}^2\delta ),e^{3\phi }-e^{3\delta }+\frac{9}{2}{q}_2^2{e}^{3\delta }\right\}.$$

The value of above expression at the point $\delta =0$, we get

$$\Psi =\left\{si{n}^2\phi +q_1^2,e^{3\phi }+\frac{9}{2}{q}_2^2-1\right\}\in K.$$

From the Figure 7. We can easily observe that the value of φ-coordinate always less than ϑ-coordinate in $K,$ so $\phi \in K.$

Hence, f is K-$G_f$-pseudobonvex at the point $\delta =0$ with respect to $\eta .$

Next,

$$\begin{array}{c}\hfill \begin{array}{c}\{G_{f_1}\left(f_1\left(\phi \right)\right)-G_{f_1}\left(f_1\left(\delta \right)\right)+{\displaystyle \frac{1}{2}}{p}_1^T\left[G_{f_1}^{″}\left(f_1\left(\delta \right)\right){\nabla }_{\phi }{f}_1\left(\delta \right){\left({\nabla }_{\phi }{f}_1\left(\delta \right)\right)}^T+G_{f_1}^{\prime }\left(f_1\left(\delta \right)\right){\nabla }_{\phi \phi }{f}_1\left(\delta \right)\right]p_1-{\eta }^T(\phi ,\delta )[G_{f_1}^{\prime }\left(f_1\left(\delta \right)\right){\nabla }_{\phi }{f}_1\left(\delta \right)\hfill \\ +\left\{G_{f_1}^{″}\left(f_1\left(\delta \right)\right){\nabla }_{\phi }{f}_1\left(\delta \right){\left({\nabla }_{\phi }{f}_1\left(\delta \right)\right)}^T+G_{f_1}^{\prime }\left(f_1\left(\delta \right)\right){\nabla }_{\phi \phi }{f}_1\left(\delta \right)\right\}{p}_1],\hfill \\ G_{f_2}\left(f_2\left(\phi \right)\right)-G_{f_2}\left(f_2\left(\delta \right)\right)+{\displaystyle \frac{1}{2}}{p}_2^T[G_{f_2}^{″}\left(f_2\left(\delta \right)\right){\nabla }_{\phi }{f}_2\left(\delta \right){\left({\nabla }_{\phi }{f}_2\left(\delta \right)\right)}^T\hfill \\ +G_{f_2}^{\prime }{f}_2\left(\delta \right){\nabla }_{\phi \phi }{f}_2\left(\delta \right)]p_2-{\eta }^T(\phi ,\delta )\left[G_{f_2}^{\prime }{f}_2\left(\delta \right){\nabla }_{\phi }{f}_2\left(\delta \right)+\{G_{f_2}^{″}\left(f_2\left(\delta \right)\right){\nabla }_{\phi }{f}_2\left(\delta \right){\left({\nabla }_{\phi }{f}_2\left(\delta \right)\right)}^T+G_{f_2}^{\prime }{f}_2\left(\delta \right){\nabla }_{\phi \phi }{f}_2\left(\delta \right)\}{p}_2\right]\}\notin K.\hfill \end{array}\end{array}$$

Let

$$\begin{array}{c}\hfill \begin{array}{c}{\displaystyle \Psi =\{G_{f_1}\left(f_1\left(\phi \right)\right)-G_{f_1}\left(f_1\left(\delta \right)\right)+{\displaystyle \frac{1}{2}}{p}_1^T\left[G_{f_1}^{″}\left(f_1\left(\delta \right)\right){\nabla }_{\phi }{f}_1\left(\delta \right){\left({\nabla }_{\phi }{f}_1\left(\delta \right)\right)}^T+G_{f_1}^{\prime }\left(f_1\left(\delta \right)\right){\nabla }_{\phi \phi }{f}_1\left(\delta \right)\right]p_1-{\eta }^T(\phi ,\delta )[G_{f_1}^{\prime }\left(f_1\left(\delta \right)\right){\nabla }_{\phi }{f}_1\left(\delta \right)}\hfill \\ +\{G_{f_1}^{″}\left(f_1\left(\delta \right)\right){\nabla }_{\phi }{f}_1\left(\delta \right){\left({\nabla }_{\phi }{f}_1\left(\delta \right)\right)}^T+G_{f_1}^{\prime }\left(f_1\left(\delta \right)\right){\nabla }_{\phi \phi }{f}_1\left(\delta \right)\}{p}_1],G_{f_2}\left(f_2\left(\phi \right)\right)-G_{f_2}\left(f_2\left(\delta \right)\right)+{\displaystyle \frac{1}{2}}{p}_2^T[G_{f_2}^{″}\left(f_2\left(\delta \right)\right){\nabla }_{\phi }{f}_2\left(\delta \right){\left({\nabla }_{\phi }{f}_2\left(\delta \right)\right)}^T\hfill \\ +G_{f_2}^{\prime }{f}_2\left(\delta \right){\nabla }_{\phi \phi }{f}_2\left(\delta \right)]p_2-{\eta }^T(\phi ,\delta )\left[G_{f_2}^{\prime }{f}_2\left(\delta \right){\nabla }_{\phi }{f}_2\left(\delta \right)+\left\{G_{f_2}^{″}\left(f_2\left(\delta \right)\right){\nabla }_{\phi }{f}_2\left(\delta \right){\left({\nabla }_{\phi }{f}_2\left(\delta \right)\right)}^T+G_{f_2}^{\prime }{f}_2\left(\delta \right){\nabla }_{\phi \phi }{f}_2\left(\delta \right)\right\}{p}_2\right]\}.\hfill \end{array}\end{array}$$

Substituting the values of $f_1,f_2,G_{f_1},G_{f_2}$ and η, we obtain

$$\begin{array}{c}\hfill \Psi =\left\{si{n}^2\phi -si{n}^2\delta +p_1^2(co{s}^2\delta -si{n}^2\delta )-({\phi }^2-{\delta }^2)(sin2\delta +2p_1(cos\delta -si{n}^2\delta )),e^{3\phi }-e^{3\delta }+\frac{9}{2}{p}_2^2{e}^{3\delta }-({\phi }^2-{\delta }^2)(3e^{3\delta }+9e^{2\delta }{p}_2)\right\}.\end{array}$$

At the point $\delta =0$, it follows that

$$\Psi =\left\{si{n}^2\phi +p_1^2-2p_1{\phi }^2,e^{3\phi }+\frac{9}{2}{p}_2^2-1-{\phi }^2(3+9p_2)\right\},p_1=p_2\in [2,\infty ).$$

Take particular point $\phi =-\frac{\pi }{2}$ and $p_1=p_2=2\in [2,\infty ),$ we obtain,

$$\Psi =(-4.86,-34.80)\notin K.$$

Hence, f is K-$G_f$-pseudobonvex, but it is not K-$G_f$-bonvex at $\delta =0$ with respect to η.

In the following example, we showed that the function f is K-$G_f$-pseudobonvex, but it is not K-$G_f$-bonvex function with same $\eta$.

Example 3.

Let $X=\left[0,\frac{\pi }{2}\right]$ and $K=\left\{\right(\phi ,\vartheta ):\phi \geqq 0,\vartheta \geqq \phi \}$. Consider $G_f=(G_{f_1},G_{f_2}):R^2\to R$ and $f:X\to {\mathbb{R}}^2$ given by

$$f\left(\phi \right)=(f_1\left(\phi \right),f_2\left(\phi \right)),$$

where

$$f_1\left(\phi \right)=sin\phi ,f_2\left(\phi \right)=\phi ,$$

$$G_{f_1}=t,G_{f_2}=t^2.$$

Define $\eta :X\times X\to R^n$ given by

$$\eta (\phi ,\delta )=\phi -\delta and{q}_1,q_2\in [1,\infty ].$$

Solution: In this example, we will try to derive that f is K-$G_f$-pseudobonvex i.e.,

$$\begin{array}{c}\hfill \begin{array}{c}{\displaystyle {\eta }^T(\phi ,\delta )\{G_{f_1}^{\prime }\left(f_1\left(\delta \right)\right){\nabla }_{\phi }{f}_1\left(\delta \right)+q_1\left\{G_{f_1}^{″}\left(f_1\left(\delta \right)\right){\left({\nabla }_{\phi }{f}_1\left(\delta \right)\right)}^T+G_{f_1}^{\prime }\left(f_1\left(\delta \right)\right){\nabla }_{\phi \phi }{f}_1\left(\delta \right)\right\},G_{f_2}^{\prime }\left(f_2\left(\delta \right)\right){\nabla }_{\phi }{f}_2\left(\delta \right)+q_2\{G_{f_2}^{″}\left(f_2\left(\delta \right)\right){\left({\nabla }_{\phi }{f}_2\left(\delta \right)\right)}^T}\hfill \\ {\displaystyle +G_{f_2}^{\prime }\left(f_2\left(\delta \right)\right){\nabla }_{\phi \phi }{f}_2\left(\delta \right)\left\}\right\}\in K\Rightarrow \{G_{f_1}\left(f_1\left(\phi \right)\right)-G_{f_1}\left(f_1\left(\delta \right)\right)+\frac{1}{2}{q}_1^T\left\{G_{f_1}^{″}\left(f_1\left(\delta \right)\right){\nabla }_{\phi }{f}_1\left(\delta \right){\left({\nabla }_{\phi }{f}_1\left(\delta \right)\right)}^T+G_{f_1}^{\prime }\left(f_1\left(\delta \right)\right){\nabla }_{\phi \phi }{f}_1\left(\delta \right)\right\}{q}_1,}\hfill \\ G_{f_2}\left(f_2\left(\phi \right)\right)-G_{f_2}\left(f_2\left(\delta \right)\right)+\frac{1}{2}{q}_2^T\left\{G_{f_2}^{″}\left(f_2\left(\delta \right)\right){\nabla }_{\phi }{f}_2\left(\delta \right){\left({\nabla }_{\phi }{f}_2\left(\delta \right)\right)}^T+G_{f_2}^{\prime }\left(f_2\left(\delta \right)\right){\nabla }_{\phi \phi }{f}_2\left(\delta \right)\right\}{q}_2\}\in K.\hfill \end{array}\end{array}$$

Consider

$$\begin{array}{c}\hfill \begin{array}{c}{\Pi }_1={\eta }^T(\phi ,\delta )\{G_{f_1}^{\prime }\left(f_1\left(\delta \right)\right){\nabla }_{\phi }{f}_1\left(\delta \right)+q_1\left\{G_{f_1}^{″}\left(f_1\left(\delta \right)\right){\left({\nabla }_{\phi }{f}_1\left(\delta \right)\right)}^T+G_{f_1}^{\prime }\left(f_1\left(\delta \right)\right){\nabla }_{\phi \phi }{f}_1\left(\delta \right)\right\},G_{f_2}^{\prime }\left(f_2\left(\delta \right)\right){\nabla }_{\phi }{f}_2\left(\delta \right)\hfill \\ +q_2\left\{G_{f_2}^{″}\left(f_2\left(\delta \right)\right){\left({\nabla }_{\phi }{f}_2\left(\delta \right)\right)}^T+G_{f_2}^{\prime }\left(f_2\left(\delta \right)\right){\nabla }_{\phi \phi }{f}_2\left(\delta \right)\right\}\}.\hfill \end{array}\end{array}$$

Putting the values of $f_1,f_2,G_{f_1},G_{f_2}$ and $\eta$, we have

$${\Pi }_1=\left\{(\phi -\delta )cos\delta ,(\phi -\delta )(2\delta +2q_2)\right\}.$$

The value of above expression at the point $\delta =0$, we get

$${\Pi }_1=\left\{\phi ,2\delta q_2\right\}\in K.$$

Next, let

$$\begin{array}{c}\hfill \begin{array}{c}{\Pi }_2=\{G_{f_1}\left(f_1\left(\phi \right)\right)-G_{f_1}\left(f_1\left(\delta \right)\right)+\frac{1}{2}{q}_1^T\left\{G_{f_1}^{″}\left(f_1\left(\delta \right)\right){\nabla }_{\phi }{f}_1\left(\delta \right){\left({\nabla }_{\phi }{f}_1\left(\delta \right)\right)}^T+G_{f_1}^{\prime }\left(f_1\left(\delta \right)\right){\nabla }_{\phi \phi }{f}_1\left(\delta \right)\right\}{q}_1,G_{f_2}\left(f_2\left(\phi \right)\right)\hfill \\ -G_{f_2}\left(f_2\left(\delta \right)\right)+\frac{1}{2}{q}_2^T\left\{G_{f_2}^{″}\left(f_2\left(\delta \right)\right){\nabla }_{\phi }{f}_2\left(\delta \right){\left({\nabla }_{\phi }{f}_2\left(\delta \right)\right)}^T+G_{f_2}^{\prime }\left(f_2\left(\delta \right)\right){\nabla }_{\phi \phi }{f}_2\left(\delta \right)\right\}{q}_2\}.\hfill \end{array}\end{array}$$

Putting the values of $f_1,f_2,G_{f_1},G_{f_2}$ and $\eta$, we have

$${\Pi }_2=\left\{sin\phi -sin\delta +\frac{1}{2}{q}_1^2(-sin\delta ),\phi -\delta +q_2^2\right\}.$$

After simplifying and the value at $\delta =0$, it follows that

$${\Pi }_2=\left\{sin\phi ,\phi +q_2^2\right\}\in K.$$

Hence, f is K-$G_f$-pseudobonvex at the point $\delta =0$ with respect to $\eta .$

Next,

$$\begin{array}{c}\hfill \begin{array}{c}\{G_{f_1}\left(f_1\left(\phi \right)\right)-G_{f_1}\left(f_1\left(\delta \right)\right)+{\displaystyle \frac{1}{2}}{p}_1^T\left[G_{f_1}^{″}\left(f_1\left(\delta \right)\right){\nabla }_{\phi }{f}_1\left(\delta \right){\left({\nabla }_{\phi }{f}_1\left(\delta \right)\right)}^T+G_{f_1}^{\prime }\left(f_1\left(\delta \right)\right){\nabla }_{\phi \phi }{f}_1\left(\delta \right)\right]p_1-{\eta }^T(\phi ,\delta )[G_{f_1}^{\prime }\left(f_1\left(\delta \right)\right){\nabla }_{\phi }{f}_1\left(\delta \right)\hfill \\ +\left\{G_{f_1}^{″}\left(f_1\left(\delta \right)\right){\nabla }_{\phi }{f}_1\left(\delta \right){\left({\nabla }_{\phi }{f}_1\left(\delta \right)\right)}^T+G_{f_1}^{\prime }\left(f_1\left(\delta \right)\right){\nabla }_{\phi \phi }{f}_1\left(\delta \right)\right\}{p}_1],\hfill \\ G_{f_2}\left(f_2\left(\phi \right)\right)-G_{f_2}\left(f_2\left(\delta \right)\right)+{\displaystyle \frac{1}{2}}{p}_2^T[G_{f_2}^{″}\left(f_2\left(\delta \right)\right){\nabla }_{\phi }{f}_2\left(\delta \right){\left({\nabla }_{\phi }{f}_2\left(\delta \right)\right)}^T\hfill \\ +G_{f_2}^{\prime }{f}_2\left(\delta \right){\nabla }_{\phi \phi }{f}_2\left(\delta \right)]p_2-{\eta }^T(\phi ,\delta )\left[G_{f_2}^{\prime }{f}_2\left(\delta \right){\nabla }_{\phi }{f}_2\left(\delta \right)+\{G_{f_2}^{″}\left(f_2\left(\delta \right)\right){\nabla }_{\phi }{f}_2\left(\delta \right){\left({\nabla }_{\phi }{f}_2\left(\delta \right)\right)}^T+G_{f_2}^{\prime }{f}_2\left(\delta \right){\nabla }_{\phi \phi }{f}_2\left(\delta \right)\}{p}_2\right]\}\notin K.\hfill \end{array}\end{array}$$

Let

$$\begin{array}{c}\hfill \begin{array}{c}\Psi =\{G_{f_1}\left(f_1\left(\phi \right)\right)-G_{f_1}\left(f_1\left(\delta \right)\right)+{\displaystyle \frac{1}{2}}{p}_1^T\left[G_{f_1}^{″}\left(f_1\left(\delta \right)\right){\nabla }_{\phi }{f}_1\left(\delta \right){\left({\nabla }_{\phi }{f}_1\left(\delta \right)\right)}^T+G_{f_1}^{\prime }\left(f_1\left(\delta \right)\right){\nabla }_{\phi \phi }{f}_1\left(\delta \right)\right]p_1-{\eta }^T(\phi ,\delta )[G_{f_1}^{\prime }\left(f_1\left(\delta \right)\right){\nabla }_{\phi }{f}_1\left(\delta \right)\hfill \\ +\{G_{f_1}^{″}\left(f_1\left(\delta \right)\right){\nabla }_{\phi }{f}_1\left(\delta \right){\left({\nabla }_{\phi }{f}_1\left(\delta \right)\right)}^T+G_{f_1}^{\prime }\left(f_1\left(\delta \right)\right){\nabla }_{\phi \phi }{f}_1\left(\delta \right)\}{p}_1],G_{f_2}\left(f_2\left(\phi \right)\right)-G_{f_2}\left(f_2\left(\delta \right)\right)+{\displaystyle \frac{1}{2}}{p}_2^T[G_{f_2}^{″}\left(f_2\left(\delta \right)\right){\nabla }_{\phi }{f}_2\left(\delta \right){\left({\nabla }_{\phi }{f}_2\left(\delta \right)\right)}^T\hfill \\ +G_{f_2}^{\prime }{f}_2\left(\delta \right){\nabla }_{\phi \phi }{f}_2\left(\delta \right)]p_2-{\eta }^T(\phi ,\delta )\left[G_{f_2}^{\prime }{f}_2\left(\delta \right){\nabla }_{\phi }{f}_2\left(\delta \right)+\left\{G_{f_2}^{″}\left(f_2\left(\delta \right)\right){\nabla }_{\phi }{f}_2\left(\delta \right){\left({\nabla }_{\phi }{f}_2\left(\delta \right)\right)}^T+G_{f_2}^{\prime }{f}_2\left(\delta \right){\nabla }_{\phi \phi }{f}_2\left(\delta \right)\right\}{p}_2\right]\}.\hfill \end{array}\end{array}$$

Substituting the values of $f_1,f_2,G_{f_1},G_{f_2}$ and $\eta$, we obtain

$$\Psi =\left\{sin\phi -sin\delta +\frac{1}{2}{p}_1^2(-sin\delta )-(\phi -\delta )p_{1}cos\delta ,{\phi }^2+p_2^2-2(\phi -\delta )p_2\right\}.$$

At the point $\delta =0$, it follows that

$$\Psi =\left\{sin\phi -p_1\phi ,{(\phi -p_2)}^2\right\}\notin K.$$

Hence, f is K-$G_f$-pseudobonvex, but it is not K-$G_f$-bonvex at $\delta =0$ with respect to $\eta$.

3. ${K}$-${{G}}_{{f}}$-Wolfe Type Second-Order Symmetric Primal-Dual Pair with Cones

The study of second-order duality is more significant due to computational advantage over first order duality as it provides tighter bounds for the objective functions, when approximation is used.

The motivated by [21,22,23,24,25,26,27] several researches in this area, we formulated a new type K-$G_f$-Wolfe type primal dual pair, with cone objectives as well as cone constraint as follows:

Primal Problem (GWPP):

K-min$L(\phi ,\vartheta ,\lambda ,p)=\left\{L_1(\phi ,\vartheta ,\lambda ,p),L_2(\phi ,\vartheta ,\lambda ,p),L_3(\phi ,\vartheta ,\lambda ,p),\dots ,L_k(\phi ,\vartheta ,\lambda ,p)\right\},$where

$$\begin{array}{c}\hfill \begin{array}{c}{\displaystyle L_i(\phi ,\vartheta ,\lambda ,p)=G_{f_i}\left(f_i(\phi ,\vartheta )\right)-{\vartheta }^T\sum _{i=1}^k{\lambda }_i[G_{f_i}^{\prime }\left(f_i(\phi ,\vartheta )\right){\nabla }_{\vartheta }{f}_i(\phi ,\vartheta )+\{G_{f_i}^{″}\left(f_i(\phi ,\vartheta )\right){\nabla }_{\vartheta }{f}_i(\phi ,\vartheta ){\left({\nabla }_{\vartheta }{f}_i(\phi ,\vartheta )\right)}^T}\hfill \\ +G_{f_i}^{\prime }\left(f_i(\phi ,\vartheta )\right){\nabla }_{\phi ,\vartheta }{f}_i(\phi ,\vartheta )\left\}{p}_i\right]-\frac{1}{2}\sum _{i=1}^k{\lambda }_i{p}_i\left\{G_{f_i}^{″}\left(f_i(\phi ,\vartheta )\right){\nabla }_{\vartheta }{f}_i(\phi ,\vartheta ){\left({\nabla }_{\vartheta }{f}_i(\phi ,\vartheta )\right)}^T+G_{f_i}^{\prime }\left(f_i(\phi ,\vartheta )\right){\nabla }_{\vartheta \vartheta }{f}_i(\phi ,\vartheta )\right\}{p}_i,\hfill \end{array}\end{array}$$

subject to

$$\begin{array}{cc}\hfill & -\sum _{i=1}^k{\lambda }_i\left[G_{f_i}^{\prime }\left(f_i(\phi ,\vartheta )\right){\nabla }_{\vartheta }{f}_i(\phi ,\vartheta )+\left\{G_{f_i}^{″}\left(f_i(\phi ,\vartheta )\right){\nabla }_{\vartheta }{f}_i(\phi ,\vartheta ){\left({\nabla }_{\vartheta }{f}_i(\phi ,\vartheta )\right)}^T+G_{f_i}^{\prime }\left(f_i(\phi ,\vartheta )\right){\nabla }_{\vartheta \vartheta }{f}_i(\phi ,\vartheta )\right\}{p}_i\right]\in C_2^{*},\hfill \end{array}$$

(1)

$${\lambda }^T{e}_k=1,\lambda \in \mathrm{int}{K}^{*},\phi \in C_1.$$

(2)

Dual Problem (GWDP):

K-max $M(\delta ,\ell ,\lambda ,q)=\left\{M_1(\delta ,\ell ,q),M_2(\delta ,\ell ,\lambda ,q),M_3(\delta ,\ell ,\lambda ,q),\dots ,M_k(\delta ,\ell ,\lambda ,q)\right\},$where

$$\begin{array}{c}\hfill \begin{array}{c}{\displaystyle M_i(\delta ,\ell ,\lambda ,q)=G_{f_i}\left(f_i(\delta ,\ell )\right)-{\delta }^T\sum _{i=1}^k{\lambda }_i[G_{f_i}^{\prime }\left(f_i(\delta ,\ell )\right){\nabla }_{\phi }{f}_i(\delta ,\ell )+\{G_{f_i}^{″}\left(f_i(\delta ,\ell )\right){\nabla }_{\phi }{f}_i(\delta ,\ell ){\left({\nabla }_{\phi }{f}_i(\delta ,\ell )\right)}^T}\hfill \\ +G_{f_i}^{\prime }\left(f_i(\delta ,\ell )\right){\nabla }_{\phi \phi }{f}_i(\delta ,\ell )\left\}{q}_i\right]-\frac{1}{2}\sum _{i=1}^k{\lambda }_i{q}_i\left[G_{f_i}^{"}\left(f_i(\delta ,\ell )\right){\nabla }_{\phi }{f}_i(\delta ,\ell ){\left({\nabla }_{\phi }{f}_i(\delta ,\ell )\right)}^T+G_{f_i}^{\prime }\left(f_i(\delta ,\ell )\right){\nabla }_{\phi \phi }{f}_i(\delta ,\ell )\right]q_i,\hfill \end{array}\end{array}$$

subject to

$$\begin{array}{cc}\hfill & \sum _{i=1}^k{\lambda }_i\left[G_{f_i}^{\prime }\left(f_i(\delta ,\ell )\right){\nabla }_{\phi }{f}_i(\delta ,\ell )+\left\{G_{f_i}^{″}\left(f_i(\delta ,\ell )\right){\nabla }_{\phi }{f}_i(\delta ,\ell ){\left({\nabla }_{\phi }{f}_i(\delta ,\ell )\right)}^T+G_{f_i}^{\prime }\left(f_i(\delta ,\ell )\right){\nabla }_{\phi \phi }{f}_i(\delta ,\ell )\right\}{q}_i\right]\in C_1^{*},\hfill \end{array}$$

(3)

$${\lambda }^T{e}_k=1,\lambda \in \mathrm{int}{K}^{*},\delta \in C_2,$$

(4)

where, for $i\in \tilde{Q},$

• $f_i:R_1\times R_2\to R,$ is a differential function of $\phi$ and $\vartheta ,$$e_k={(1,1,\dots ,1)}^T\in R^k$,
• $q_i$ and $p_i$ are vectors in $R^n$ and $R^m$, respectively and $\lambda \in R^k.$

Let $V^{*}$ and $W^{*}$ be the sets of feasible solutions of (GWPP) and (GWDP) respectively.

Theorem 1

(Weak duality). Let $(\phi ,\vartheta ,\lambda ,p)\in V^{*}$ and $(\delta ,\ell ,\lambda ,q)\in W^{*}$. Let, for $i\in \tilde{N}$

(i) 

$\left\{f_1(.,\ell ),f_2(.,\ell ),\dots ,f_k(.,\ell )\right\}$ be K-$G_{f_i}$-bonvex at δ w.r.t. ${\eta }_1$,

(ii) 

$\left\{f_1(\phi ,.),f_2(\phi ,.),\dots ,f_k(\phi ,.)\right\}$ be K-$G_{f_i}$-boncave in ϑ w.r.t. ${\eta }_2$,

(iii) 

${\eta }_1(\phi ,\delta )+\delta \in C_1$, ∀$(\phi ,\delta )\in C_1\times C_2$,

(iv) 

${\eta }_2(\ell ,\vartheta )+\vartheta \in C_2$, ∀$(\ell ,\vartheta )\in C_1\times C_2$,

Then, $L(\phi ,\vartheta ,\lambda ,p)-M(\delta ,\ell ,\lambda ,q)\notin -K\setminus \left\{0\right\}.$

Proof. 

If possible, then suppose

$$L(\phi ,\vartheta ,\lambda ,p)-M(\delta ,\ell ,\lambda ,q)\in -K\setminus \left\{0\right\},$$

or

$$\begin{array}{c}\hfill \begin{array}{c}{\displaystyle \{G_{f_1}\left(f_1(\phi ,\vartheta )\right)-{\vartheta }^T\sum _{i=1}^k{\lambda }_i\left(G_{f_i}^{\prime }\left(f_i(\phi ,\vartheta )\right){\nabla }_{\vartheta }{f}_i(\phi ,\vartheta )+\left\{G_{f_i}^{″}\left(f_i(\phi ,\vartheta )\right)\left({\nabla }_{\vartheta }{f}_i(\phi ,\vartheta )\right){\left({\nabla }_{\vartheta }{f}_i(\phi ,\vartheta )\right)}^T+G_{f_i}^{\prime }\left(f_i(\phi ,\vartheta )\right){\nabla }_{\vartheta \vartheta }{f}_i(\phi ,\vartheta )\right\}{p}_i\right)}\hfill \\ -\frac{1}{2}\sum _{i=1}^k{\lambda }_i{p}_i^T\left\{G_{f_i}^{″}\left(f_i(\phi ,\vartheta )\right)\left({\nabla }_{\vartheta }{f}_i(\phi ,\vartheta )\right){\left({\nabla }_{\vartheta }{f}_i(\phi ,\vartheta )\right)}^T+G_{f_i}^{\prime }\left(f_i(\phi ,\vartheta )\right){\nabla }_{\vartheta \vartheta }{f}_i(\phi ,\vartheta )\right\}{p}_i,\dots ,G_{f_k}\left(f_k(\phi ,\vartheta )\right)\hfill \\ -{\vartheta }^T\sum _{i=1}^k{\lambda }_i\left(G_{f_i}^{\prime }\left(f_i(\phi ,\vartheta )\right){\nabla }_{\vartheta }{f}_i(\phi ,\vartheta )+\left\{G_{f_i}^{″}\left(f_i(\phi ,\vartheta )\right)\left({\nabla }_{\vartheta }{f}_i(\phi ,\vartheta )\right){\left({\nabla }_{\vartheta }{f}_i(\phi ,\vartheta )\right)}^T+G_{f_i}^{\prime }\left(f_i(\phi ,\vartheta )\right){\nabla }_{\vartheta \vartheta }{f}_i(\phi ,\vartheta )\right\}{p}_i\right)\hfill \\ -\frac{1}{2}\sum _{i=1}^k{\lambda }_i{p}_i^T\left\{G_{f_i}^{″}\left(f_i(\phi ,\vartheta )\right)\left({\nabla }_{\vartheta }{f}_i(\phi ,\vartheta )\right){\left({\nabla }_{\vartheta }{f}_i(\phi ,\vartheta )\right)}^T+G_{f_i}^{\prime }\left(f_i(\phi ,\vartheta )\right){\nabla }_{\vartheta \vartheta }{f}_i(\phi ,\vartheta )\right\}{p}_i-G_{f_1}\left(f_1(\delta ,\ell )\right)-{\delta }^T\sum _{i=1}^k{\lambda }_i(G_{f_i}^{\prime }\left(f_i(\delta ,\ell )\right)\hfill \\ {\nabla }_{\phi }{f}_i(\delta ,\ell )+\left\{G_{f_i}^{″}\left(f_i(\delta ,\ell )\right)\left({\nabla }_{\phi }{f}_i(\delta ,\ell )\right){\left({\nabla }_{\phi }{f}_i(\delta ,\ell )\right)}^T+G_{f_i}^{\prime }\left(f_i(\delta ,\ell )\right){\nabla }_{\phi \phi }{f}_i(\delta ,\ell )\right\}{q}_i)-\frac{1}{2}\sum _{i=1}^k{\lambda }_i{q}_i^T\{G_{f_i}^{″}\left(f_i(\delta ,\ell )\right)\left({\nabla }_{\phi }{f}_i(\delta ,\ell )\right){\left({\nabla }_{\phi }{f}_i(\delta ,\ell )\right)}^T\hfill \\ +G_{f_i}^{\prime }\left(f(\delta ,\ell )\right){\nabla }_{\phi \phi }{f}_i(\delta ,\ell )\}{q}_i,\dots ,G_{f_k}\left(f_k(\delta ,\ell )\right)-{\delta }^T\sum _{i=1}^k{\lambda }_i(G_{f_i}^{\prime }\left(f_i(\delta ,\ell )\right){\nabla }_{\phi }{f}_i(\delta ,\ell )+\{G_{f_i}^{″}\left(f_i(\delta ,\ell )\right)\left({\nabla }_{\phi }{f}_i(\delta ,\ell )\right){\left({\nabla }_{\phi }{f}_i(\delta ,\ell )\right)}^T\hfill \\ +G_{f_i}^{\prime }\left(f_i(\delta ,\ell )\right){\nabla }_{\phi \phi }{f}_i(\delta ,\ell )\}{q}_i)-\frac{1}{2}\sum _{i=1}^k{\lambda }_i{q}_i^T\left\{G_{f_i}^{″}\left(f_i(\delta ,\ell )\right)\left({\nabla }_{\phi }{f}_i(\delta ,\ell )\right){\left({\nabla }_{\phi }{f}_i(\delta ,\ell )\right)}^T+G_{f_i}^{\prime }\left(f(\delta ,\ell )\right){\nabla }_{\phi \phi }{f}_i(\delta ,\ell )\right\}{q}_i\}\in -K\setminus \left\{0\right\}.\hfill \end{array}\end{array}$$

Since $\lambda \in \mathrm{int}{K}^{*}$, we get

$$\begin{array}{c}{\displaystyle \sum _{i=1}^k{\lambda }_i\{G_{f_i}\left(f_i(\phi ,\vartheta )\right)-{\vartheta }^T\sum _{i=1}^k{\lambda }_i[G_{f_i}^{\prime }\left(f_i(\phi ,\vartheta )\right){\nabla }_{\vartheta }{f}_i(\phi ,\vartheta )+\{G_{f_i}^{″}\left(f_i(\phi ,\vartheta )\right){\nabla }_{\vartheta }{f}_i(\phi ,\vartheta ){\left({\nabla }_{\vartheta }{f}_i(\phi ,\vartheta )\right)}^T}\hfill \\ +G_{f_i}^{\prime }\left(f_i(\phi ,\vartheta )\right){\nabla }_{\vartheta \vartheta }{f}_i(\phi ,\vartheta )\left\}{p}_i\right]-{\delta }^T\sum _{i=1}^k{\lambda }_i[G_{f_i}^{\prime }\left(f_i(\delta ,\ell )\right){\nabla }_{\phi }{f}_i(\delta ,\ell )+\{G_{f_i}^{″}\left(f_i(\delta ,\ell )\right){\nabla }_{\phi }{f}_i(\delta ,\ell )\hfill \\ {\left({\nabla }_{\phi }{f}_i(\delta ,\ell )\right)}^T\left\}\right]-\frac{1}{2}\sum _{i=1}^k{\lambda }_i{p}_i^T\left\{G_{f_i}^{″}\left(f_i(\phi ,\vartheta )\right){\nabla }_{\vartheta }{f}_i(\phi ,\vartheta ){\left({\nabla }_{\vartheta }{f}_i(\phi ,\vartheta )\right)}^T+G_{f_i}^{\prime }\left(f_i(\phi ,\vartheta )\right){\nabla }_{\vartheta \vartheta }{f}_i(\phi ,\vartheta )\right\}-\{G_{f_i}\left(f_i(\delta ,\ell )\right)\hfill \\ +G_{f_i}^{\prime }\left(f_i(\delta ,\ell )\right){\nabla }_{\phi \phi }{f}_i(\delta ,\ell )\}-\frac{1}{2}\sum _{i=1}^k{\lambda }_i{q}_i^T\left\{G_{f_i}^{″}\left(f_i(\delta ,\ell )\right)\left({\nabla }_{\phi }{f}_i(\delta ,\ell )\right){\left({\nabla }_{\phi }{f}_i(\delta ,\ell )\right)}^T+G_{f_i}^{\prime }\left(f_i(\delta ,\ell )\right){\nabla }_{\phi \phi }{f}_i(\phi ,\vartheta )\right\}{q}_i\}<0.\hfill \end{array}$$

(5)

By hypothesis $\left(i\right)$ and using $\lambda \in \mathrm{int}{K}^{*}$, we get

$$\begin{array}{c}\hfill \begin{array}{c}{\displaystyle \sum _{i=1}^k{\lambda }_i\{G_{f_i}\left(f_i(\phi ,\ell )\right)-G_{f_i}\left(f_i(\delta ,\ell )\right)+\frac{1}{2}{q}_i^T\left\{G_{f_i}^{″}\left(f_i(\delta ,\ell )\right)\left({\nabla }_{\phi }{f}_i(\delta ,\ell )\right){\nabla }_{\phi }{f}_i{(\delta ,\ell )}^T+G_{f_i}^{\prime }\left(f_i(\delta ,\ell )\right){\nabla }_{\phi \phi }{f}_i(\delta ,\ell )\right\}{q}_i}\hfill \\ -{\eta }_1^T(\phi ,\delta )\left[G_{f_i}^{\prime }\left(f_i(\delta ,\ell )\right){\nabla }_{\phi }{f}_i(\delta ,\ell )+\left\{G_{f_i}^{″}\left(f_i(\delta ,\ell )\right)\left({\nabla }_{\phi }{f}_i(\delta ,\ell )\right){\nabla }_{\phi }{f}_i{(\delta ,\ell )}^T+G_{f_i}^{\prime }\left(f_i(\delta ,\ell )\right){\nabla }_{\phi \phi }{f}_i(\delta ,\ell )\right\}{q}_i\right]\}\geqq 0,\hfill \end{array}\end{array}$$

Using feasibility of dual problem (GWDP) & using dual constraints with assumption $\left(iii\right)$, it yields

$$\begin{array}{c}\hfill {\displaystyle {\left({\eta }_1(\phi ,\delta )+\delta \right)}^T\sum _{i=1}^k{\lambda }_i\left[G_{f_i}^{\prime }\left(f_i(\delta ,\ell )\right){\nabla }_{\phi }{f}_i(\delta ,\ell )+\left\{G_{f_i}^{″}\left(f_i(\delta ,\ell )\right)\left({\nabla }_{\phi }{f}_i(\delta ,\ell )\right){\left({\nabla }_{\phi }{f}_i(\delta ,\ell )\right)}^T+G_{f_i}^{\prime }\left(f_i(\delta ,\ell )\right){\nabla }_{\phi }{f}_i(\delta ,\ell )\right\}{q}_i\right]\geqq 0,}\end{array}$$

it implies that

$$\begin{array}{c}\hfill \begin{array}{c}{\displaystyle \sum _{i=1}^k{\lambda }_i\left[G_{f_i}\left(f_i(\phi ,\ell )\right)-G_{f_i}\left(f_i(\delta ,\ell )\right)+\frac{1}{2}{q}_i^T\left\{G_{f_i}^{″}\left(f_i(\delta ,\ell )\right)\left({\nabla }_{\phi }{f}_i(\delta ,\ell )\right){\left({\nabla }_{\phi }{f}_i(\delta ,\ell )\right)}^T+G_{f_i}^{\prime }\left(f_i(\delta ,\ell )\right){\nabla }_{\phi \phi }{f}_i(\delta ,\ell )\right\}{q}_i\right]}\hfill \end{array}\end{array}$$

$$\geqq -{\delta }^T\sum _{i=1}^k{\lambda }_i\left[G_{f_i}^{\prime }\left(f_i(\delta ,\ell )\right){\nabla }_{\phi }\left(f_i(\delta ,\ell )\right)+\left\{G_{f_i}^{″}\left(f_i(\delta ,\ell )\right)\left({\nabla }_{\phi }{f}_i(\delta ,\ell )\right){\left({\nabla }_{\phi }{f}_i(\delta ,\ell )\right)}^T+G_{f_i}^{\prime }\left(f_i(\delta ,\ell )\right){\nabla }_{\phi \phi }{f}_i(\delta ,\ell )\right\}{q}_i\right].$$

(6)

Similarly, using hypotheses $\left(ii\right),\left(iv\right)$, feasible conditions of primal problem (GWPP), dual constraint and $\lambda \in int{K}^{*},$

we get

$$\begin{array}{c}\hfill \begin{array}{c}{\displaystyle \sum _{i=1}^k{\lambda }_i\left[G_{f_i}\left(f_i(\phi ,\vartheta )\right)-G_{f_i}\left(f_i(\phi ,\ell )\right)+\frac{1}{2}{p}_i^T\left\{G_{f_i}^{″}\left(f_i(\delta ,\ell )\right)\left({\nabla }_{\vartheta }{f}_i(\delta ,\ell )\right){\left({\nabla }_{\vartheta }{f}_i(\delta ,\ell )\right)}^T+G_{f_i}^{\prime }\left(f_i(\delta ,\ell )\right){\nabla }_{\phi \phi }{f}_i(\delta ,\ell )\right\}{p}_i\right]}\hfill \end{array}\end{array}$$

$$\geqq {\vartheta }^T\sum _{i=1}^k{\lambda }_i\left[G_{f_i}^{\prime }\left(f_i(\delta ,\ell )\right){\nabla }_{\vartheta }\left(f_i(\delta ,\ell )\right)+\left\{G_{f_i}^{″}\left(f_i(\delta ,\ell )\right)\left({\nabla }_{\phi }{f}_i(\delta ,\ell )\right){\left({\nabla }_{\phi }{f}_i(\delta ,\ell )\right)}^T+G_{f_i}^{\prime }\left(f_i(\delta ,\ell )\right){\nabla }_{\phi \phi }{f}_i(\delta ,\ell )\right\}{p}_i\right].$$

(7)

Now, from inequalities (6), (7) and using the fact that ${\lambda }^T{e}_k=1$, we find that

$$\begin{array}{c}\hfill \begin{array}{c}{\displaystyle \sum _{i=1}^k{\lambda }_i[G_{f_i}\left(f_i(\phi ,\vartheta )\right)-{\vartheta }^T\sum _{i=1}^k{\lambda }_i\left[G_{f_i}^{\prime }\left(f_i(\phi ,\vartheta )\right){\nabla }_{\vartheta }\left(f_i(\phi ,\vartheta )\right)+\left\{G_{f_i}^{″}\left(f_i(\phi ,\vartheta )\right)\left({\nabla }_{\vartheta }{f}_i(\delta ,\ell )\right){\left({\nabla }_{\vartheta }{f}_i(\delta ,\ell )\right)}^T+G_{f_i}^{\prime }\left(f_i(\phi ,\vartheta )\right){\nabla }_{\vartheta \vartheta }{f}_i(\phi ,\vartheta )\right\}{p}_i\right]}\hfill \\ -\frac{1}{2}\sum _{i=1}^k{\lambda }_i{p}_i^T\left\{G_{f_i}^{″}\left(f_i(\phi ,\vartheta )\right)\left({\nabla }_{\vartheta }{f}_i(\phi ,\vartheta )\right){\left({\nabla }_{\vartheta }{f}_i(\phi ,\vartheta )\right)}^T+G_{f_i}^{\prime }\left(f_i(\phi ,\vartheta )\right){\nabla }_{\vartheta \vartheta }{f}_i(\phi ,\vartheta )\right\}-G_{f_i}\left(f_i(\delta ,\ell )\right)\hfill \\ -{\delta }^T\sum _{i=1}^k{\lambda }_i\left[G_{f_i}^{\prime }\left(f_i(\delta ,\ell )\right){\nabla }_{\phi }\left(f_i(\delta ,\ell )\right)+\left\{G_{f_i}^{″}\left(f_i(\delta ,\ell )\right)\left({\nabla }_{\phi }{f}_i(\delta ,\ell )\right){\left({\nabla }_{\phi }{f}_i(\delta ,\ell )\right)}^T+G_{f_i}^{\prime }\left(f_i(\delta ,\ell )\right){\nabla }_{\phi \phi }{f}_i(\delta ,\ell )\right\}\right]\hfill \\ -\frac{1}{2}{\delta }^T\sum _{i=1}^k{\lambda }_i{q}_i^T\left\{G_{f_i}^{″}\left(f_i(\delta ,\ell )\right)\left({\nabla }_{\phi }{f}_i(\delta ,\ell )\right){\left({\nabla }_{\phi }{f}_i(\delta ,\ell )\right)}^T+G_{f_i}^{\prime }\left(f_i(\delta ,\ell )\right)\left({\nabla }_{\phi }{f}_i(\delta ,\ell )\right)q_i\right\}]\geqq 0,\hfill \end{array}\end{array}$$

we arrive at contradiction. □

Through following example, we validate the Weak duality theorem as:

Example 4.

Let n = m = 1, $k=2$, $X=[1,2]$, $p\in [2^2,2^{10}]$, $q\in [{10}^{-19},{10}^{19}]$, $K=\left\{(\phi ,\vartheta );\phi \geqq 0,\phi \geqq \vartheta \right\}$ and

$-K=\left\{(\phi ,\vartheta );\phi \le 0,\phi \le \vartheta \right\}$, $R_1=R_2=R_{+}.$ Let $f_i:R_1\times R_2\to R$ and $G_{f_i}$ for $i=1,2.$ be defined as

$$f_1(\phi ,\vartheta )=\phi +cos\vartheta ,f_2(\phi ,\vartheta )=sin\vartheta ,G_{f_1}\left(t\right)=t^2,G_{f_2}\left(t\right)=t.$$

Further, let

$${\eta }_1(\phi ,\delta )=\phi \delta ,{\eta }_2(\ell ,\vartheta )=\ell -\vartheta .$$

Assume that $C_1=C_2=C_1^{*}=C_2^{*}=R_{+}.$

(GWPP)$K$-minimize $L(\phi ,\vartheta ,\lambda ,p)=\left\{L_1(\phi ,\vartheta ,\lambda ,p),L_2(\phi ,\vartheta ,\lambda ,p)\right\}$

$$Subject to constraints

$${\lambda }_1\left[2(\phi +cos\vartheta )(-sin\vartheta )+\left\{2si{n}^2\vartheta +2(\phi +cos\vartheta )(-cos\vartheta )\right\}{p}_1\right]+{\lambda }_2\left[cos\vartheta -p_{2}sin\vartheta \right]\leqq 0,$$

(8)

$${\lambda }_1+{\lambda }_2=1,{\lambda }_i\in \mathrm{int}{K}^{*},\phi \in C_1,i=1,2.$$

(9)

(GWDP) K-maximize $M(\delta ,\ell ,\lambda ,q)=\left\{M_1(\delta ,\ell ,\lambda ,q),M_2(\delta ,\ell ,\lambda ,q)\right\}$

$$Subject to constraints

$${\lambda }_1\left[2(\phi +cos\vartheta )+2q_1\right]\geqq 0,$$

(10)

$${\lambda }_1+{\lambda }_2=1,{\lambda }_i\in \mathrm{int}{K}^{*},\phi \in C_2,i=1,2.$$

(11)

(A1).$\left\{f_1(.,\ell ),f_2(.,\ell )\right\}$ is K-$G_f$-bonvex at $\delta =0$ w.r.t. ${\eta }_1$,∀$\phi \in S_1$, i.e.,

$$\begin{array}{c}\hfill \begin{array}{c}{\displaystyle \{G_{f_1}\left(f_1(\phi ,\ell )\right)-G_{f_1}\left(f_1(\delta ,\ell )\right)+{\displaystyle \frac{1}{2}}{p}_1^T\left[G_{f_1}^{″}\left(f_1(\delta ,\ell )\right){\nabla }_{\phi }{f}_1(\delta ,\ell ){\left({\nabla }_{\phi }{f}_1(\delta ,\ell )\right)}^T+G_{f_1}^{\prime }\left(f_1(\delta ,\ell )\right){\nabla }_{\phi \phi }{f}_1(\delta ,\ell )\right]p_1}\hfill \\ -{\eta }^T(\phi ,\delta )\left[G_{f_1}^{\prime }\left(f_1(\delta ,\ell )\right){\nabla }_{\phi }{f}_1(\delta ,\ell )+\left\{G_{f_1}^{″}\left(f_1(\delta ,\ell )\right){\nabla }_{\phi }{f}_1(\delta ,\ell ){\left({\nabla }_{\phi }{f}_1(\delta ,\ell )\right)}^T+G_{f_1}^{\prime }\left(f_1(\delta ,\ell )\right){\nabla }_{\phi \phi }{f}_1(\delta ,\ell )\right\}{p}_1\right],\hfill \\ G_{f_2}\left(f_2(\phi ,\ell )\right)-G_{f_2}\left(f_2(\delta ,\ell )\right)+{\displaystyle \frac{1}{2}}{p}_2^T\left[G_{f_2}^{″}\left(f_2(\delta ,\ell )\right){\nabla }_{\phi }{f}_2(\delta ,\ell ){\left({\nabla }_{\phi }{f}_2(\delta ,\ell )\right)}^T+G_{f_2}^{\prime }\left(f_2(\delta ,\ell )\right){\nabla }_{\phi \phi }{f}_2(\delta ,\ell )\right]p_2\hfill \end{array}\end{array}$$

$$\begin{array}{c}\hfill -{\eta }^T(\phi ,\delta )\left[G_{f_2}^{\prime }\left(f_2(\delta ,\ell )\right){\nabla }_{\phi }{f}_2(\delta ,\ell )+\left\{G_{f_2}^{″}\left(f_2(\delta ,\ell )\right){\nabla }_{\phi }{f}_2(\delta ,\ell ){\left({\nabla }_{\phi }{f}_2(\delta ,\ell )\right)}^T+G_{f_2}^{\prime }\left(f_2(\delta ,\ell )\right){\nabla }_{\phi \phi }{f}_2(\delta ,\ell )\right\}{p}_2\right]\}\in K.\end{array}$$

(12)

Consider

$$\begin{array}{c}\hfill \begin{array}{c}\Psi =\{G_{f_1}\left(f_1(\phi ,\ell )\right)-G_{f_1}\left(f_1(\delta ,\ell )\right)+{\displaystyle \frac{1}{2}}{p}_1^T\left[G_{f_1}^{″}\left(f_1(\delta ,\ell )\right){\nabla }_{\phi }{f}_1(\delta ,\ell ){\left({\nabla }_{\phi }{f}_1(\delta ,\ell )\right)}^T+G_{f_1}^{\prime }\left(f_1(\delta ,\ell )\right){\nabla }_{\phi \phi }{f}_1(\delta ,\ell )\right]p_1\hfill \\ -{\eta }^T(\phi ,\delta )\left[G_{f_1}^{\prime }\left(f_1(\delta ,\ell )\right){\nabla }_{\phi }{f}_1(\delta ,\ell )+\left\{G_{f_1}^{″}\left(f_1(\delta ,\ell )\right){\nabla }_{\phi }{f}_1(\delta ,\ell ){\left({\nabla }_{\phi }{f}_1(\delta ,\ell )\right)}^T+G_{f_1}^{\prime }\left(f_1(\delta ,\ell )\right){\nabla }_{\phi \phi }{f}_1(\delta ,\ell )\right\}{p}_1\right],\hfill \\ G_{f_2}\left(f_2(\phi ,\ell )\right)-G_{f_2}\left(f_2(\delta ,\ell )\right)+{\displaystyle \frac{1}{2}}{p}_2^T\left[G_{f_2}^{″}\left(f_2(\delta ,\ell )\right){\nabla }_{\phi }{f}_2(\delta ,\ell ){\left({\nabla }_{\phi }{f}_2(\delta ,\ell )\right)}^T+G_{f_2}^{\prime }\left(f_2(\delta ,\ell )\right){\nabla }_{\phi \phi }{f}_2(\delta ,\ell )\right]p_2\hfill \end{array}\end{array}$$

$$\begin{array}{c}\hfill -{\eta }^T(\phi ,\delta )\left[G_{f_2}^{\prime }\left(f_2(\delta ,\ell )\right){\nabla }_{\phi }{f}_2(\delta ,\ell )+\left\{G_{f_2}^{″}\left(f_2(\delta ,\ell )\right){\nabla }_{\phi }{f}_2(\delta ,\ell ){\left({\nabla }_{\phi }{f}_2(\delta ,\ell )\right)}^T+G_{f_2}^{\prime }\left(f_2(\delta ,\ell )\right){\nabla }_{\phi \phi }{f}_2(\delta ,\ell )\right\}{p}_2\right]\}.\end{array}$$

(13)

Putting the values of $f_1,f_2,G_{f_1},G_{f_2}$ and ${\eta }_1$ at the point $\delta =0$, and simplifying, we get

$$\Psi =\left({\phi }^2+2\phi cos\ell +p^2,0\right).$$

It is clear that

$$\Psi =\left({\phi }^2+2\phi cos\ell +p^2,0\right)\in K.$$

(A2).$\left\{f_1(\phi ,.),f_2(\phi ,.)\right\}$ is K-$G_f$-boncave at $\vartheta =0$ w.r.t. ${\eta }_2$, $\ell \in S_2$,

$$\begin{array}{c}\hfill \begin{array}{c}\{G_{f_1}\left(f_1(\phi ,\ell )\right)-G_{f_1}\left(f_1(\phi ,\vartheta )\right)+{\displaystyle \frac{1}{2}}{p}_1^T\left[G_{f_1}^{″}\left(f_1(\phi ,\vartheta )\right){\nabla }_{\vartheta }{f}_1(\phi ,\vartheta ){\left({\nabla }_{\vartheta }{f}_1(\phi ,\vartheta )\right)}^T+G_{f_1}^{\prime }\left(f_1(\phi ,\vartheta )\right){\nabla }_{\vartheta \vartheta }{f}_1(\phi ,\vartheta )\right]p_1\hfill \\ -{\eta }^T(\ell ,\vartheta )\left[G_{f_1}^{\prime }\left(f_1(\phi ,\vartheta )\right){\nabla }_{\vartheta }{f}_1(\phi ,\vartheta )+\left\{G_{f_1}^{″}\left(f_1(\phi ,\vartheta )\right){\nabla }_{\vartheta }{f}_1(\phi ,\vartheta ){\left({\nabla }_{\vartheta }{f}_1(\phi ,\vartheta )\right)}^T+G_{f_1}^{\prime }\left(f_1(\phi ,\vartheta )\right){\nabla }_{\vartheta \vartheta }{f}_1(\phi ,\vartheta )\right\}{p}_1\right],\hfill \\ G_{f_2}\left(f_2(\phi ,\ell )\right)-G_{f_2}\left(f_2(\phi ,\vartheta )\right)+{\displaystyle \frac{1}{2}}{p}_2^T\left[G_{f_2}^{″}\left(f_2(\phi ,\vartheta )\right){\nabla }_{\vartheta }{f}_2(\phi ,\vartheta ){\left({\nabla }_{\vartheta }{f}_2(\phi ,\vartheta )\right)}^T+G_{f_2}^{\prime }\left(f_2(\phi ,\vartheta )\right){\nabla }_{\vartheta \vartheta }{f}_2(\phi ,\vartheta )\right]p_2\hfill \end{array}\end{array}$$

$$\begin{array}{c}\hfill -{\eta }^T(\ell ,\vartheta )\left[G_{f_2}^{\prime }\left(f_2(\phi ,\vartheta )\right){\nabla }_{\vartheta }{f}_2(\phi ,\vartheta )+\left\{G_{f_2}^{″}\left(f_2(\phi ,\vartheta )\right){\nabla }_{\vartheta }{f}_2(\phi ,\vartheta ){\left({\nabla }_{\vartheta }{f}_2(\phi ,\vartheta )\right)}^T+G_{f_2}^{\prime }\left(f_2(\phi ,\vartheta )\right){\nabla }_{\vartheta \vartheta }{f}_2(\phi ,\vartheta )\right\}{p}_2\right]\}\in -K.\end{array}$$

(14)

$$\begin{array}{c}\hfill \begin{array}{c}\mathrm{Let}{\Psi }_1=\{G_{f_1}\left(f_1(\phi ,\ell )\right)-G_{f_1}\left(f_1(\phi ,\vartheta )\right)+{\displaystyle \frac{1}{2}}{p}_1^T\left[G_{f_1}^{″}\left(f_1(\phi ,\vartheta )\right){\nabla }_{\vartheta }{f}_1(\phi ,\vartheta ){\left({\nabla }_{\vartheta }{f}_1(\phi ,\vartheta )\right)}^T+G_{f_1}^{\prime }\left(f_1(\phi ,\vartheta )\right){\nabla }_{\vartheta \vartheta }{f}_1(\phi ,\vartheta )\right]p_1\hfill \\ -{\eta }^T(\ell ,\vartheta )\left[G_{f_1}^{\prime }\left(f_1(\phi ,\vartheta )\right){\nabla }_{\vartheta }{f}_1(\phi ,\vartheta )+\left\{G_{f_1}^{″}\left(f_1(\phi ,\vartheta )\right){\nabla }_{\vartheta }{f}_1(\phi ,\vartheta ){\left({\nabla }_{\vartheta }{f}_1(\phi ,\vartheta )\right)}^T+G_{f_1}^{\prime }\left(f_1(\phi ,\vartheta )\right){\nabla }_{\vartheta \vartheta }{f}_1(\phi ,\vartheta )\right\}{p}_1\right],\hfill \\ G_{f_2}\left(f_2(\phi ,\ell )\right)-G_{f_2}\left(f_2(\phi ,\vartheta )\right)+{\displaystyle \frac{1}{2}}{p}_2^T\left[G_{f_2}^{″}\left(f_2(\phi ,\vartheta )\right){\nabla }_{\vartheta }{f}_2(\phi ,\vartheta ){\left({\nabla }_{\vartheta }{f}_2(\phi ,\vartheta )\right)}^T+G_{f_2}^{\prime }\left(f_2(\phi ,\vartheta )\right){\nabla }_{\vartheta \vartheta }{f}_2(\phi ,\vartheta )\right]p_2\hfill \end{array}\end{array}$$

$$\begin{array}{c}\hfill -{\eta }^T(\ell ,\vartheta )\left[G_{f_2}^{\prime }(f_2(\phi ,\vartheta ){\nabla }_{\vartheta }{f}_2(\phi ,\vartheta )+\left\{G_{f_2}^{″}\left(f_2(\phi ,\vartheta )\right){\nabla }_{\vartheta }{f}_2(\phi ,\vartheta ){\left({\nabla }_{\vartheta }{f}_2(\phi ,\vartheta )\right)}^T+G_{f_2}^{\prime }\left(f_2(\phi ,\vartheta )\right){\nabla }_{\vartheta \vartheta }{f}_2(\phi ,\vartheta )\right\}{p}_2\right]\}.\end{array}$$

(15)

Putting the values of $f_1,f_2,G_{f_1},G_{f_2}\mathrm{and}{\eta }_2$ at $\vartheta =0$, we obtain

$${\Psi }_1=\left({(\phi +cos\ell )}^2-{(\phi +1)}^2-p_1^2(\phi +1)+2\ell (\phi +1),sin\ell -\ell \right).$$

$${\Psi }_1=\left({(\phi +cos\ell )}^2-{(\phi +1)}^2-p_1^2(\phi +1)+2\ell (\phi +1),sin\ell -\ell \right)\in -K.$$

(A3).${\eta }_1(\phi ,\delta )+\delta \in C_1$, $\forall$$\phi \in C_1$.

(A4).${\eta }_2(\ell ,\vartheta )+\vartheta \in C_2$, $\forall$$\ell \in C_2$.

Validation: To validate Weak duality theorem it is enough to claim that any point $(\phi ,0,{\lambda }_1,{\lambda }_2,p)$ such that $\phi \geqq 0$, ${\lambda }_1+{\lambda }_2=1$ are feasible to $\left(GWPP\right)$. Also, the points $(0,\ell ,{\lambda }_1,{\lambda }_2,q)$ such that $\ell \geqq 0$, ${\lambda }_1+{\lambda }_2=1$ are feasible to $\left(GWDP\right)$. Now, at these feasible points,

$$L=(L_1,L_2)=\left({(\phi +1)}^2+{\lambda }_1{p}_1^2(\phi +1),{\lambda }_1{p}_1^2(\phi +1)\right),$$

and

$$M=(M_1,M_2)=\left(co{s}^2\ell -{\lambda }_1{q}_1^2,sin\ell -{\lambda }_1{q}_1^2\right).$$

Now, calculate the value at above feasible points, we have

$$L(\phi ,\vartheta ,\lambda ,p)-M(\delta ,\ell ,\lambda ,q)=\left({(\phi +1)}^2+{\lambda }_1{p}_1^2(\phi +1)-co{s}^2\ell +{\lambda }_1{q}_1^2,{\lambda }_1{p}_1^2(\phi +1)-sin\ell +{\lambda }_1{q}_1^2\right)\notin K\setminus \left\{0\right\}.$$

(16)

In particular, the points ${\displaystyle \left(\phi ,\vartheta ,{\lambda }_1,{\lambda }_2,p\right)=\left(1,0,\frac{1}{2},\frac{1}{2},4\right)}$ and ${\displaystyle \left(\delta ,\ell ,{\lambda }_1,{\lambda }_2,q\right)}$$=\left(0,\frac{22}{14},\frac{1}{2},\frac{1}{2},2\right)$ are feasible solutions for $\left(GWPP\right)$ and $\left(GWDP\right)$, respectively. Also

$$L(\phi ,\vartheta ,\lambda ,p)-M(\delta ,\ell ,\lambda ,q)=\left(22,17\right)\notin -K\setminus \left\{0\right\}.$$

(17)

Hence, this validate the results.

Remark 2.

Every pseudoconvex function is convex function. On the same pattern we can proof that K-$G_f$-pseudobonvex is K-$G_f$-bonvex with respect to same η. So, above proof of Weak duality 3.2 follows on same pattern as Theorem 1.

Theorem 2

(Weak duality). Let $(\phi ,\vartheta ,\lambda ,p)\in V^{*}$ and $(\delta ,\ell ,\lambda ,q)\in W^{*}$. Let, For $i\in \tilde{N}$

(i) 

$\left\{f_1(.,\ell ),f_2(.,\ell ),\dots ,f_k(.,\ell )\right\}$ be K-$G_f$-pseudobonvex at ℓ w.r.t. ${\eta }_1$,

(ii) 

$\left\{f_1(\phi ,.),f_2(\phi ,.),\dots ,f_k(\phi ,.)\right\}$ be K-$G_f$-pseudoboncave at ϑ, w.r.t. ${\eta }_2$,

(iii) 

${\eta }_1(\phi ,\delta )+\delta \in C_1$, ∀$(\phi ,\delta )\in C_1\times C_2$,

(iv) 

${\eta }_2(\ell ,\vartheta )+\vartheta \in C_2$, ∀$(\ell ,\vartheta )\in C_1\times C_2$,

Then, $L(\phi ,\vartheta ,\lambda ,p)-M(\delta ,\ell ,\lambda ,q)\notin -K\setminus \left\{0\right\}.$

Proof. 

Proof follows on same lines as Weak Duality Theorem 1. □

Example 5.

For n = m = 1, $k=2$,$X=[2,3],p\in [0,1],q\in [2,2^{10}],$$K=\left\{(\phi ,\vartheta );\phi \leqq 0,\vartheta \geqq 0,\left|\phi \right|\geqq \vartheta \right\}$,

$R_1=R_2=R_{+}.$ Let $f_i:R_1\times R_2\to R$ be given as

$$f_1(\phi ,\vartheta )=\phi +{\vartheta }^2,f_2(\phi ,\vartheta )=1-\vartheta ,G_{f_1}\left(t\right)=t^2,G_{f_2}\left(t\right)=t.$$

Further, Let

$${\eta }_1(\phi ,\delta )=\phi \delta ,{\eta }_2(\ell ,\vartheta )=\ell -\vartheta .$$

Assume that $C_1=C_2=C_1^{*}=C_2^{*}=R_{+}.$

(GWPP)$K$-minimize $L(\phi ,\vartheta ,\lambda ,p)=\left\{L_1(\phi ,\vartheta ,\lambda ,p),L_2(\phi ,\vartheta ,\lambda ,p)\right\}$

$$Subject to constraints

$${\lambda }_1\left[4\vartheta (\phi +{\vartheta }^2)+p_1\{8{\vartheta }^2+4(\phi +{\vartheta }^2)\}\right]-{\lambda }_2\leqq 0,$$

(18)

$${\lambda }_1+{\lambda }_2=1,{\lambda }_i\in \mathrm{int}{K}^{*},\phi \in C_1,i=1,2.$$

(19)

(GWDP)$K$-maximize $M(\delta ,\ell ,\lambda ,q)=\left\{M_1(\delta ,\ell ,\lambda ,q),M_2(\delta ,\ell ,\lambda ,q)\right\}$

$$Subject to constraints

$${\lambda }_1\left[2(\delta +{\ell }^2+q)\right]\geqq 0,$$

(20)

$${\lambda }_1+{\lambda }_2=1,{\lambda }_i\in \mathrm{int}{K}^{*},\delta \in C_2,i=1,2.$$

(21)

(A1).$\left\{f_1(.,\ell ),f_2(.,\ell )\right\}$ is K-$G_f$-pseudobonvex at $\delta$ with respect to ${\eta }_1$, $\phi \in R_1$, so that

$$\begin{array}{c}\hfill {\eta }_1^T(\phi ,\delta )\{G_{f_1}^{\prime }\left(f_1(\delta ,\ell )\right){\nabla }_{\phi }{f}_1(\delta ,\ell )+p\left\{G_{f_1}^{″}\left(f_1(\delta ,\ell )\right){\left({\nabla }_{\phi }{f}_1(\delta ,\ell )\right)}^T+G_{f_1}^{\prime }\left(f_1(\delta ,\ell )\right){\nabla }_{\phi \phi }{f}_1(\delta ,\ell )\right\},\end{array}$$

$$\begin{array}{c}\hfill G_{f_2}^{\prime }\left(f_2(\delta ,\ell )\right){\nabla }_{\phi }{f}_2(\delta ,\ell )+p\left\{G_{f_2}^{″}\left(f_2(\delta ,\ell )\right){\left({\nabla }_{\phi }{f}_2(\delta ,\ell )\right)}^T+G_{f_2}^{\prime }\left(f_2(\delta ,\ell )\right){\nabla }_{\phi \phi }{f}_2(\delta ,\ell )\right\}\}\in K.\end{array}$$

(22)

Let

$$\begin{array}{c}\hfill {\Pi }_1={\eta }_1^T(\phi ,\delta )\{G_{f_1}^{\prime }\left(f_1(\delta ,\ell )\right){\nabla }_{\phi }{f}_1(\delta ,\ell )+p\left\{G_{f_1}^{″}\left(f_1(\delta ,\ell )\right){\left({\nabla }_{\phi }{f}_1(\delta ,\ell )\right)}^T+G_{f_1}^{\prime }\left(f_1(\delta ,\ell )\right){\nabla }_{\phi \phi }{f}_1(\delta ,\ell )\right\},\end{array}$$

$$\begin{array}{c}\hfill G_{f_2}^{\prime }\left(f_2(\delta ,\ell )\right){\nabla }_{\phi }{f}_2(\delta ,\ell )+p\left\{G_{f_2}^{″}\left(f_2(\delta ,\ell )\right){\left({\nabla }_{\phi }{f}_2(\delta ,\ell )\right)}^T+G_{f_2}^{\prime }\left(f_2(\delta ,\ell )\right){\nabla }_{\phi \phi }{f}_2(\delta ,\ell )\right\}\}.\end{array}$$

(23)

Next, let

$$\begin{array}{c}\hfill {\Pi }_2=[G_{f_1}\left(f_1(\phi ,\ell )\right)-G_{f_1}\left(f_1(\delta ,\ell )\right)+\frac{1}{2}{p}^T\left\{G_{f_1}^{″}\left(f_1(\delta ,\ell )\right){\nabla }_{\phi }{f}_1(\delta ,\ell ){\left({\nabla }_{\phi }{f}_1(\delta ,\ell )\right)}^T+G_{f_1}^{\prime }\left(f_1(\delta ,\ell )\right){\nabla }_{\phi \phi }{f}_1(\delta ,\ell )\right\}p,\end{array}$$

$$\begin{array}{c}\hfill G_{f_2}\left(f_2(\phi ,\ell )\right)-G_{f_2}\left(f_2(\delta ,\ell )\right)+\frac{1}{2}{p}^T\left\{G_{f_2}^{″}\left(f_2(\delta ,\ell )\right){\nabla }_{\phi }{f}_2(\delta ,\ell ){\left({\nabla }_{\phi }{f}_2(\delta ,\ell )\right)}^T+G_{f_2}^{\prime }\left(f_2(\delta ,\ell )\right){\nabla }_{\phi \phi }{f}_2(\delta ,\ell )\right\}p].\end{array}$$

(24)

After simplification, substituting the value of $f_1,f_2,G_{f_1},G_{f_2}$ and ${\eta }_1$ at $\delta =0$, we get

$${\Pi }_1=(0,0)\in K\Rightarrow {\Pi }_2=({\phi }^2-2\phi {\ell }^2+p^2,0)\in K.$$

(A2).$\left\{f_1(\phi ,.),f_2(\phi ,.)\right\}$ is K-$G_f$-pseudoboncave at $\vartheta$ with respect to ${\eta }_2$ for fixed $\phi$ for all $\ell \in S_2$, i.e.,

$$\begin{array}{c}\hfill \begin{array}{c}{\eta }_2^T(\phi ,\delta )\{G_{f_1}^{\prime }\left(f_1(\phi ,\vartheta )\right){\nabla }_{\vartheta }{f}_1(\phi ,\vartheta )+q\left\{G_{f_1}^{″}\left(f_1(\phi ,\vartheta )\right){\left({\nabla }_{\vartheta }{f}_1(\phi ,\vartheta )\right)}^T+G_{f_1}^{\prime }\left(f_1(\phi ,\vartheta )\right){\nabla }_{\vartheta \vartheta }{f}_1(\phi ,\vartheta )\right\},G_{f_2}^{\prime }\left(f_2(\phi ,\vartheta )\right){\nabla }_{\vartheta }{f}_2(\phi ,\vartheta )\hfill \\ +q\left\{G_{f_2}^{″}\left(f_2(\phi ,\vartheta )\right){\left({\nabla }_{\vartheta }{f}_2(\phi ,\vartheta )\right)}^T+G_{f_2}^{\prime }\left(f_2(\phi ,\vartheta )\right){\nabla }_{\vartheta \vartheta }{f}_2(\phi ,\vartheta )\right\}\}\in K\hfill \\ {\displaystyle \Rightarrow [G_{f_1}\left(f_1(\phi ,\ell )\right)-G_{f_1}\left(f_1(\phi ,\vartheta )\right)+\frac{1}{2}{q}^T\left\{G_{f_1}^{″}\left(f_1(\phi ,\vartheta )\right){\nabla }_{\vartheta }{f}_1(\phi ,\vartheta ){\left({\nabla }_{\vartheta }{f}_1(\phi ,\vartheta )\right)}^T+G_{f_1}^{\prime }\left(f_1(\phi ,\vartheta )\right){\nabla }_{\vartheta \vartheta }{f}_1(\phi ,\vartheta )\right\}q,}\hfill \end{array}\end{array}$$

$$\begin{array}{c}\hfill G_{f_2}\left(f_2(\phi ,\ell )\right)-G_{f_2}\left(f_2(\phi ,\vartheta )\right)+\frac{1}{2}{q}^T\left\{G_{f_2}^{″}\left(f_2(\phi ,\vartheta )\right){\nabla }_{\vartheta }{f}_2(\phi ,\vartheta ){\left({\nabla }_{\vartheta }{f}_2(\phi ,\vartheta )\right)}^T+G_{f_2}^{\prime }\left(f_2(\phi ,\vartheta )\right){\nabla }_{\vartheta \vartheta }{f}_2(\phi ,\vartheta )\right\}q]\in -K.\end{array}$$

(25)

Let ${\Pi }_3={\eta }_2^T(\phi ,\delta )\{G_{f_1}^{\prime }\left(f_1(\phi ,\vartheta )\right){\nabla }_{\vartheta }{f}_1(\phi ,\vartheta )+q\left\{G_{f_1}^{″}\left(f_1(\phi ,\vartheta )\right){\left({\nabla }_{\vartheta }{f}_1(\phi ,\vartheta )\right)}^T+G_{f_1}^{\prime }\left(f_1(\phi ,\vartheta )\right){\nabla }_{\vartheta \vartheta }{f}_1(\phi ,\vartheta )\right\},$

$$\begin{array}{c}\hfill G_{f_2}^{\prime }\left(f_2(\phi ,\vartheta )\right){\nabla }_{\vartheta }{f}_2(\phi ,\vartheta )+q\left\{G_{f_2}^{″}\left(f_2(\phi ,\vartheta )\right){\left({\nabla }_{\vartheta }{f}_2(\phi ,\vartheta )\right)}^T+G_{f_2}^{\prime }\left(f_2(\phi ,\vartheta )\right){\nabla }_{\vartheta \vartheta }{f}_2(\phi ,\vartheta )\right\}\},\end{array}$$

(26)

and

$$\begin{array}{c}\hfill \begin{array}{c}{\displaystyle {\Pi }_4=[G_{f_1}\left(f_1(\phi ,\ell )\right)-G_{f_1}\left(f_1(\phi ,\vartheta )\right)+\frac{1}{2}{q}^T\left\{G_{f_1}^{″}\left(f_1(\phi ,\vartheta )\right){\nabla }_{\vartheta }{f}_1(\phi ,\vartheta ){\left({\nabla }_{\vartheta }{f}_1(\phi ,\vartheta )\right)}^T+G_{f_1}^{\prime }\left(f_1(\phi ,\vartheta )\right){\nabla }_{\vartheta \vartheta }{f}_1(\phi ,\vartheta )\right\}q,}\hfill \end{array}\end{array}$$

$$\begin{array}{c}\hfill G_{f_2}\left(f_2(\phi ,\ell )\right)-G_{f_2}\left(f_2(\phi ,\vartheta )\right)+\frac{1}{2}{q}^T\left\{G_{f_2}^{″}\left(f_2(\phi ,\vartheta )\right){\nabla }_{\vartheta }{f}_2(\phi ,\vartheta ){\left({\nabla }_{\vartheta }{f}_2(\phi ,\vartheta )\right)}^T+G_{f_2}^{\prime }\left(f_2(\phi ,\vartheta )\right){\nabla }_{\vartheta \vartheta }{f}_2(\phi ,\vartheta )\right\}q].\end{array}$$

(27)

Substituting the value of $f_1,f_2,G_{f_1},G_{f_2}$ and ${\eta }_2$ at the point $\delta =0$ and simplify, we get

$${\Pi }_3=\left(4vq\phi ,-1\right)\in -K\Rightarrow {\Pi }_4=\left({\ell }^4+2\phi {\ell }^2,-\ell \right)\in -K.$$

(A3).${\eta }_1(\phi ,\delta )+\delta \in C_1$, ∀$\phi \in C_1$.

(A4).${\eta }_2(\ell ,\vartheta )+\vartheta \in C_2$, ∀$\ell \in C_2$.

Validation: To prove our result its enough to prove that any point $\left(\phi ,0,{\lambda }_1,{\lambda }_2,p\right)$ such that $\phi \geqq 0$, ${\lambda }_1+{\lambda }_2=1$ are feasible to $\left(GWPP\right)$. Also, the points $\left(0,\ell ,{\lambda }_1,{\lambda }_2,q\right)$ such that $\ell \geqq 0$, ${\lambda }_1+{\lambda }_2=1$ are feasible to $\left(GWDP\right)$. Now, at these feasible points,

$$L=\left(L_1,L_2\right)=\left({\phi }^2-2\phi {\lambda }_1{p}^2,1-2\phi {\lambda }_1{p}^2\right)$$

and

$$M=\left(M_1,M_2\right)=\left({\ell }^4-{\lambda }_1{q}^2,1-\ell -{\lambda }_1{q}^2\right).$$

Now at above feasible condition

$$L-M=\left({\phi }^2-2\phi {\lambda }_1{p}^2-{\ell }^4+{\lambda }_1{q}^2,\ell -2\phi {\lambda }_1{p}^2+{\lambda }_1{q}^2\right)\notin K\setminus \left\{0\right\}.$$

(28)

In particular, the points ${\displaystyle \left(\phi ,\vartheta ,{\lambda }_1,{\lambda }_2,p\right)=\left(2,0,\frac{1}{2},\frac{1}{2},1\right)}$ and ${\displaystyle \left(\delta ,\ell ,{\lambda }_1,{\lambda }_2,q\right)=\left(0,2,\frac{1}{2},\frac{1}{2},2\right)}$ are

feasible for $\left(GWPP\right)$ and $\left(GWDP\right)$ respectively,

Now, calculate

$$L(\phi ,\vartheta ,\lambda ,p)-M(\delta ,\ell ,\lambda ,q)=\left(-12,2\right)\notin K\setminus \left\{0\right\}.$$

(29)

Hence, this validate the Weak duality Theorem 2.

Theorem 3

(Strong duality). Let $(\overline{\phi },\overline{\vartheta },\overline{\lambda },{\overline{p}}_1={\overline{p}}_2={\overline{p}}_3=\dots ={\overline{p}}_k)$ is an efficient solution of $\left(GWPP\right)$; fix $\lambda =\overline{\lambda }$ in $\left(GWDP\right)$ such that

(i) 

for all $i\in \tilde{N},\left[G_{f_i}^{″}\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\vartheta }{f}_i(\overline{\phi },\overline{\vartheta }){\left({\nabla }_{\vartheta }{f}_i(\overline{\phi },\overline{\vartheta })\right)}^T+G_{f_i}^{\prime }\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\vartheta \vartheta }{f}_i(\overline{\phi },\overline{\vartheta })\right]$ is nonsingular,

(ii) 

the vector ${\displaystyle \sum _{i=1}^k{\overline{\lambda }}_i{\nabla }_{\vartheta }\left[{\overline{p}}_i\{G_{f_i}^{″}\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\vartheta }{f}_i(\overline{\phi },\overline{\vartheta }){\left({\nabla }_{\vartheta }{f}_i(\overline{\phi },\overline{\vartheta })\right)}^T+G_{f_i}^{\prime }\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\vartheta \vartheta }{f}_i(\overline{\phi },\overline{\vartheta })\}{\overline{p}}_i\right]}$∉span $\left\{G_{f_1}^{\prime }\left(f_1(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\vartheta }{f}_1(\overline{\phi },\overline{\vartheta }),G_{f_2}^{\prime }\left(f_2(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\vartheta }{f}_2(\overline{\phi },\overline{\vartheta }),\dots ,G_{f_i}^{\prime }\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\vartheta }{f}_i(\overline{\phi },\overline{\vartheta })\right\},$

(iii) 

the set of vectors $\left\{G_{f_1}^{\prime }\left(f_1(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\vartheta }{f}_1(\overline{\phi },\overline{\vartheta }),G_{f_2}^{\prime }\left(f_2(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\vartheta }{f}_2(\overline{\phi },\overline{\vartheta }),\dots ,G_{f_k}^{\prime }\left(f_k(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\vartheta }{f}_k(\overline{\phi },\overline{\vartheta })\right\}$ are linearly independent,

(iv) 

${\displaystyle \sum _{i=1}^k{\overline{\lambda }}_i{\nabla }_{\vartheta }\left[{\overline{p}}_i\{G_{f_i}^{″}\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\vartheta }{f}_i(\overline{\phi },\overline{\vartheta }){\left({\nabla }_{\vartheta }{f}_i(\overline{\phi },\overline{\vartheta })\right)}^T+G_{f_i}^{\prime }\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\vartheta \vartheta }{f}_i(\overline{\phi },\overline{\vartheta })\}{\overline{p}}_i\right]=0\Rightarrow {\overline{p}}_i=0,\forall i,}$ and

(v) 

K is closed convex pointed cone with $R_{+}^k\subseteq K.$

Then, $(\overline{\phi },\overline{\vartheta },\overline{\lambda },{\overline{q}}_1={\overline{q}}_2={\overline{q}}_3=\dots ={\overline{q}}_k=0)\in W^{*}$ and $L(\overline{\phi },\overline{\vartheta },\overline{p})=M(\overline{\phi },\overline{\vartheta },\overline{q})$. Also, if the hypotheses of Theorem 1 or Theorem 2 are satisfied for all feasible solutions for $\left(GWPP\right)$ and $\left(GWDP\right)$, then $(\overline{\phi },\overline{\vartheta },\overline{\lambda },\overline{p})$ and $(\overline{\phi },\overline{\vartheta },\overline{\lambda },\overline{q})$ is an efficient solution for $\left(GWPP\right)$ and $\left(GWDP\right)$, respectively.

Proof. 

Since $(\overline{\phi },\overline{\vartheta },\overline{\lambda },{\overline{p}}_1,{\overline{p}}_2,{\overline{p}}_3,\dots ,{\overline{p}}_k),$ is an efficient solution of $\left(GWPP\right)$, there exist $\alpha \in K^{*},\beta \in C_2$ and $\overline{\eta }\in R$ such that the following Fritz -John optimality condition stated by [28] are satisfied at $(\overline{\phi },\overline{\vartheta },\overline{\lambda },{\overline{p}}_1,{\overline{p}}_2,{\overline{p}}_3,\dots ,{\overline{p}}_k):$

$$\begin{array}{c}\hfill \begin{array}{c}{\displaystyle (\phi -\overline{\phi }{)}^T[\sum _{i=1}^k{\alpha }_i\left[G_{f_i}^{\prime }\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\phi }{f}_i(\overline{\phi },\overline{\vartheta })\right]+\sum _{i=1}^k{\overline{\lambda }}_i\left[G_{f_i}^{″}\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\vartheta }{f}_i(\overline{\phi },\overline{\vartheta }){\nabla }_{\vartheta }{f}_i(\overline{\phi },\overline{\vartheta })+G_{f_i}^{\prime }\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\phi \vartheta }{f}_i(\overline{\phi },\overline{\vartheta })\right]\left[\beta -\left({\overline{\alpha }}^T{e}_k\right)\overline{\vartheta }\right]}\hfill \end{array}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle +\sum _{i=1}^k{\overline{\lambda }}_i{\nabla }_{\phi }\left[(G_{f_i}^{″}\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\vartheta }{f}_i(\overline{\phi },\overline{\vartheta }){\left({\nabla }_{\vartheta }{f}_i(\overline{\phi },\overline{\vartheta })\right)}^T+G_{f_i}^{\prime }\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\vartheta \vartheta }{f}_i(\overline{\phi },\overline{\vartheta })){\overline{p}}_i\right]\left(\beta -\left({\overline{\alpha }}^T{e}_k\right)\left(\overline{\vartheta }+\frac{1}{2}{\overline{p}}_i\right)\right)]\geqq 0,\forall \phi \in C_1,}\end{array}$$

(30)

$$\begin{array}{c}\hfill \begin{array}{c}{\displaystyle {\left(\vartheta -\overline{\vartheta }\right)}^T\{\sum _{i=1}^k{\alpha }_i\left[G_{f_i}^{\prime }\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\vartheta }{f}_i(\overline{\phi },\overline{\vartheta })\right]+\sum _{i=1}^k{\overline{\lambda }}_i\left[G_{f_i}^{″}\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\vartheta }{f}_i(\overline{\phi },\overline{\vartheta }){\left({\nabla }_{\vartheta }{f}_i(\overline{\phi },\overline{\vartheta })\right)}^T+G_{f_i}^{\prime }\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\vartheta \vartheta }{f}_i(\overline{\phi },\overline{\vartheta })\right]}\hfill \\ {\displaystyle \left(\overline{\beta }-\left({\overline{\alpha }}^T{e}_k\right)\overline{\vartheta }\right)+\sum _{i=1}^k{\overline{\lambda }}_i{\nabla }_{\vartheta }\left[(G_{f_i}^{″}\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\vartheta }{f}_i(\overline{\phi },\overline{\vartheta }){\left({\nabla }_{\vartheta }{f}_i(\overline{\phi },\overline{\vartheta })\right)}^T+G_{f_i}^{\prime }\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\vartheta \vartheta }{f}_i(\overline{\phi },\overline{\vartheta })){\overline{p}}_i\right]}\hfill \\ {\displaystyle \left[\overline{\beta }-\left({\overline{\alpha }}^T{e}_k\right)\left(\overline{\vartheta }+\frac{1}{2}{\overline{p}}_i\right)\right]-\sum _{i=1}^k{\overline{\lambda }}_i[G_{f_i}^{\prime }\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\vartheta }{f}_i(\overline{\phi },\overline{\vartheta })}\hfill \end{array}\end{array}$$

$$\begin{array}{c}\hfill +(G_{f_i}^{″}\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\vartheta }{f}_i(\overline{\phi },\overline{\vartheta }){\left({\nabla }_{\vartheta }{f}_i(\overline{\phi },\overline{\vartheta })\right)}^T+G_{f_i}^{\prime }\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\vartheta \vartheta }{f}_i(\overline{\phi },\overline{\vartheta })){\overline{p}}_i\left]\left({\overline{\alpha }}^T{e}_k\right)\right\}\geqq 0,\forall \vartheta \in R^m,\end{array}$$

(31)

$$\begin{array}{c}\hfill \begin{array}{c}{\displaystyle G_{f_i}^{\prime }\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\vartheta }{f}_i(\overline{\phi },\overline{\vartheta })\left(\overline{\beta }-\left({\overline{\alpha }}^T{e}_k\right)\overline{\vartheta }\right)+\overline{\eta }{e}_k+\left\{{\left\{\beta -\left({\overline{\alpha }}^T{e}_k\right)\left(\overline{\vartheta }+\frac{1}{2}{\overline{p}}_1\right)\right\}}^T\right(G_{f_i}^{″}\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\vartheta }{f}_i(\overline{\phi },\overline{\vartheta }){\left({\nabla }_{\vartheta }{f}_i(\overline{\phi },\overline{\vartheta })\right)}^T}\hfill \\ {\displaystyle +G_{f_i}^{\prime }\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\vartheta \vartheta }{f}_i(\overline{\phi },\overline{\vartheta })){\overline{p}}_1,\{{\left\{\beta -\left({\overline{\alpha }}^T{e}_k\right)\left(\overline{\vartheta }+\frac{1}{2}{\overline{p}}_2\right)\right\}}^T(G_{f_i}^{″}\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\vartheta }{f}_i(\overline{\phi },\overline{\vartheta }){\left({\nabla }_{\vartheta }{f}_i(\overline{\phi },\overline{\vartheta })\right)}^T}\hfill \\ {\displaystyle \{{\left\{\beta -\left({\overline{\alpha }}^T{e}_k\right)\left(\overline{\vartheta }+\frac{1}{2}{\overline{p}}_3\right)\right\}}^T\left(G_{f_i}^{″}\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\vartheta }{f}_i(\overline{\phi },\overline{\vartheta }){\left({\nabla }_{\vartheta }{f}_i(\overline{\phi },\overline{\vartheta })\right)}^T+G_{f_i}^{\prime }\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\vartheta \vartheta }{f}_i(\overline{\phi },\overline{\vartheta })\right){\overline{p}}_3,\cdots ,}\hfill \end{array}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle \left\{{\left\{\beta -\left({\overline{\alpha }}^T{e}_k\right)\left(\overline{\vartheta }+\frac{1}{2}{\overline{p}}_3\right)\right\}}^T\left(G_{f_i}^{″}\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\vartheta }{f}_i(\overline{\phi },\overline{\vartheta }){\left({\nabla }_{\vartheta }{f}_i(\overline{\phi },\overline{\vartheta })\right)}^T+G_{f_i}^{\prime }\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\vartheta \vartheta }{f}_i(\overline{\phi },\overline{\vartheta })\right){\overline{p}}_k\right\}=0,}\end{array}$$

(32)

$$\begin{array}{c}\hfill \left[G_{f_i}^{″}\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\vartheta }{f}_i(\overline{\phi },\overline{\vartheta }){\left({\nabla }_{\vartheta }{f}_i(\overline{\phi },\overline{\vartheta })\right)}^T+G_{f_i}^{\prime }\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\vartheta \vartheta }{f}_i(\overline{\phi },\overline{\vartheta })\right]\left((\overline{\beta }-\left({\overline{\alpha }}^T{e}_k\right)({\overline{p}}_i+\overline{\vartheta })){\overline{\lambda }}_i\right)=0,i\in \tilde{N},\end{array}$$

(33)

$$\begin{array}{c}\hfill {\displaystyle {\overline{\beta }}^T\sum _{i=1}^k{\overline{\lambda }}_i\left[G_{f_i}^{\prime }\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\vartheta }{f}_i(\overline{\phi },\overline{\vartheta })+\left\{G_{f_i}^{″}\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\phi }{f}_i(\overline{\phi },\overline{\vartheta }){\left({\nabla }_{\vartheta }{f}_i(\overline{\phi },\overline{\vartheta })\right)}^T+G_{f_i}^{\prime }\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\vartheta \vartheta }{f}_i(\overline{\phi },\overline{\vartheta })\right\}{\overline{p}}_i\right]=0,}\end{array}$$

(34)

$$\begin{array}{c}\hfill {\overline{\eta }}^T\left[{\overline{\lambda }}^T{e}_k-1\right]=0,\end{array}$$

(35)

$$\begin{array}{c}\hfill \left(\overline{\alpha },\overline{\beta },\overline{\eta }\right)\geqq 0,\left(\overline{\alpha },\overline{\beta },\overline{\eta }\right)\ne 0.\end{array}$$

(36)

Inequalities (31) and (32) can be rewritten in the following expressions:

$$\begin{array}{c}\hfill \begin{array}{c}{\displaystyle \sum _{i=1}^k{\alpha }_i\left[G_{f_i}^{\prime }\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\vartheta }{f}_i(\overline{\phi },\overline{\vartheta })\right]+\sum _{i=1}^k{\overline{\lambda }}_i\left[G_{f_i}^{″}\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\vartheta }{f}_i(\overline{\phi },\overline{\vartheta }){\left({\nabla }_{\vartheta }{f}_i(\overline{\phi },\overline{\vartheta })\right)}^T+G_{f_i}^{\prime }\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\vartheta \vartheta }{f}_i(\overline{\phi },\overline{\vartheta })\right]}\hfill \\ {\displaystyle \left(\overline{\beta }-\left({\overline{\alpha }}^T{e}_k\right)\overline{\vartheta }\right)+\sum _{i=1}^k{\overline{\lambda }}_i{\nabla }_{\vartheta }\left[(G_{f_i}^{″}\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\vartheta }{f}_i(\overline{\phi },\overline{\vartheta }){\left({\nabla }_{\vartheta }{f}_i(\overline{\phi },\overline{\vartheta })\right)}^T+G_{f_i}^{\prime }\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\vartheta \vartheta }{f}_i(\overline{\phi },\overline{\vartheta })){\overline{p}}_i\right]}\hfill \\ {\displaystyle \left[\overline{\beta }-\left({\overline{\alpha }}^T{e}_k\right)\left(\overline{\vartheta }+\frac{1}{2}{\overline{p}}_i\right)\right]-\sum _{i=1}^k{\overline{\lambda }}_i[G_{f_i}^{\prime }\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\vartheta }{f}_i(\overline{\phi },\overline{\vartheta })}\hfill \end{array}\end{array}$$

$$\begin{array}{c}\hfill +\left(G_{f_i}^{″}\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\vartheta }{f}_i(\overline{\phi },\overline{\vartheta }){\left({\nabla }_{\vartheta }{f}_i(\overline{\phi },\overline{\vartheta })\right)}^T+G_{f_i}^{\prime }\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\vartheta \vartheta }{f}_i(\overline{\phi },\overline{\vartheta })\right){\overline{p}}_i]\left({\overline{\alpha }}^T{e}_k\right)=0.\end{array}$$

(37)

${\displaystyle G_{f_i}^{\prime }\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\vartheta }{f}_i(\overline{\phi },\overline{\vartheta })\left(\overline{\beta }-\left({\overline{\alpha }}^T{e}_k\right)\overline{\vartheta }\right)+\{{\left\{\beta -\left({\overline{\alpha }}^T{e}_k\right)\left(\overline{\vartheta }+\frac{1}{2}{\overline{p}}_i\right)\right\}}^T}$

$$\begin{array}{c}\hfill \left(G_{f_i}^{″}\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\vartheta }{f}_i(\overline{\phi },\overline{\vartheta }){\left({\nabla }_{\vartheta }{f}_i(\overline{\phi },\overline{\vartheta })\right)}^T+G_{f_i}^{\prime }\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\vartheta \vartheta }{f}_i(\overline{\phi },\overline{\vartheta })\right){\overline{p}}_i\}+\overline{\eta }=0,i\in \tilde{N}.\end{array}$$

(38)

Now, from hypothesis $\left(iv\right)$, it is given that $R_{+}^k\subseteq K$⇒$int{K}^{*}\subseteq int{R}_{+}^k.$

Obviously, $\overline{\lambda }>0$ because $\overline{\lambda }\in \mathrm{int}{K}^{*}$.

By hypothesis $\left(i\right),$ (33) gives

$$\begin{array}{c}\hfill \beta =\left({\overline{\alpha }}^T{e}_k\right)({\overline{p}}_i+\overline{\vartheta }),i\in \tilde{N}.\end{array}$$

(39)

Suppose $\overline{\alpha }=0$, then (39) yields $\overline{\beta }=0$. Further, from (38) gives $\overline{\eta }=0$. Now, we reach at contradiction (36). Hence, $\overline{\alpha }\ne 0.$ Further, $\overline{\alpha }\in K^{*}\subseteq R_{+}^k$ implies

$$\begin{array}{c}\hfill {\overline{\alpha }}^T{e}_k>0.\end{array}$$

(40)

Now, we have to claim that ${\overline{p}}_i=0,i\in \tilde{N}.$ Using (39) and (40) in (38), we get

${\displaystyle \sum _{i=1}^k{\overline{\lambda }}_i\left[{\nabla }_{\vartheta }\left\{\frac{1}{2}{\overline{p}}_i(G_{f_i}^{″}\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\vartheta }{f}_i(\overline{\phi },\overline{\vartheta }){\left({\nabla }_{\vartheta }{f}_i(\overline{\phi },\overline{\vartheta })\right)}^T+G_{f_i}^{\prime }\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\vartheta \vartheta }{f}_i(\overline{\phi },\overline{\vartheta })){\overline{p}}_i\right\}\right]}$

$$\begin{array}{c}\hfill {\displaystyle =-\frac{1}{\mu }\sum _{i=1}^k\left({\alpha }_i-\mu {\overline{\lambda }}_i\right)\left[G_{f_i}^{\prime }\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\vartheta }{f}_i(\overline{\phi },\overline{\vartheta })\right],}\end{array}$$

(41)

By hypothesis (ii), we get

$$\begin{array}{c}\hfill {\displaystyle \sum _{i=1}^k{\overline{\lambda }}_i\left[{\nabla }_{\vartheta }\left\{{\overline{p}}_i(G_{f_i}^{″}\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\vartheta }{f}_i(\overline{\phi },\overline{\vartheta }){\left({\nabla }_{\vartheta }{f}_i(\overline{\phi },\overline{\vartheta })\right)}^T+G_{f_i}^{\prime }\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\vartheta \vartheta }{f}_i(\overline{\phi },\overline{\vartheta })){\overline{p}}_i\right\}\right]=0.}\end{array}$$

(42)

Again, from hypothesis (iv), we have

$$\begin{array}{c}\hfill {\overline{p}}_i=0,\forall i\in \tilde{N}.\end{array}$$

(43)

From (39) implies

$$\begin{array}{c}\hfill \overline{\beta }=\left({\overline{\alpha }}^T{e}_k\right)\overline{\vartheta }.\end{array}$$

(44)

Using (42) and (43) in (37), we obtain

$$\begin{array}{c}\hfill {\displaystyle \sum _{i=1}^k\left({\alpha }_i-\left({\overline{\alpha }}^T{e}_k\right){\overline{\lambda }}_i\right)\left[G_{f_i}^{\prime }\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\vartheta }{f}_i(\overline{\phi },\overline{\vartheta })\right]=0.}\end{array}$$

(45)

From hypothesis (iii), it yields

$$\begin{array}{c}\hfill {\alpha }_i=\left({\overline{\alpha }}^T{e}_k\right){\overline{\lambda }}_i,i\in \tilde{N}.\end{array}$$

(46)

Using (43) and (44) in (30), we get

$$\begin{array}{c}\hfill {\displaystyle {(\phi -\overline{\phi })}^T\sum _{i=1}^k{\alpha }_i\left[G_{f_i}^{\prime }\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\phi }{f}_i(\overline{\phi },\overline{\vartheta })]\right]\geqq 0.}\end{array}$$

Using (40), (43), (44) and (46) in (30), we find that

$$\begin{array}{c}\hfill {\displaystyle {(\phi -\overline{\phi })}^T\sum _{i=1}^k{\overline{\lambda }}_i\left[G_{f_i}^{\prime }\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\phi }{f}_i(\overline{\phi },\overline{\vartheta })]\right]\geqq 0,\forall \phi \in C_1.}\end{array}$$

(47)

Let $\phi \in C_1.$ Then, $\phi +\overline{\phi }\in C_1$ and inequality (47) gives that

$$\begin{array}{c}\hfill {\displaystyle {\overline{\phi }}^T\sum _{i=1}^k{\overline{\lambda }}_i\left[G_{f_i}^{\prime }\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\phi }{f}_i(\overline{\phi },\overline{\vartheta })]\right]\geqq 0,\forall \phi \in C_1.}\end{array}$$

(48)

Therefore,

$$\begin{array}{c}\hfill {\displaystyle \sum _{i=1}^k{\overline{\lambda }}_i\left[G_{f_i}^{\prime }\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\phi }{f}_i(\overline{\phi },\overline{\vartheta })]\right]\in C_1^{*}.}\end{array}$$

(49)

Also, from (44), we obtain

$$\begin{array}{c}\hfill {\displaystyle \overline{\vartheta }=\frac{\overline{\beta }}{{\overline{\alpha }}^T{e}_k}\in C_2.}\end{array}$$

(50)

Therefore, $(\overline{\phi },\overline{\vartheta },\overline{\lambda },{\overline{q}}_1={\overline{q}}_2={\overline{q}}_3=\dots ={\overline{q}}_k=0)$ satisfies the constraint of (GWDP) and is therefore a feasible solution for the dual problem (GWDP).

Now, letting $\phi =0$ and $\phi =2\overline{\phi }$ in (47), we obtain

$$\begin{array}{c}\hfill {\displaystyle {\overline{\phi }}^T\sum _{i=1}^k{\overline{\lambda }}_i\left[G_{f_i}^{\prime }\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\phi }{f}_i(\overline{\phi },\overline{\vartheta })\right]=0.}\end{array}$$

(51)

Further, from (34), (40), (43) and (44), we get

$$\begin{array}{c}\hfill {\displaystyle {\overline{\vartheta }}^T\sum _{i=1}^k{\overline{\lambda }}_i\left[G_{f_i}^{\prime }\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\vartheta }{f}_i(\overline{\phi },\overline{\vartheta })\right]=0.}\end{array}$$

(52)

Therefore, using (43), (51) and (52), we obtain

$$\begin{array}{c}\hfill \begin{array}{c}{\displaystyle (G_{f_1}\left(f_1(\overline{\phi },\overline{\vartheta })\right)-{\overline{\vartheta }}^T\sum _{i=1}^k{\overline{\lambda }}_i\left[G_{f_i}^{\prime }\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\vartheta }{f}_i(\overline{\phi },\overline{\vartheta })+\left\{G_{f_i}^{″}\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\vartheta }{f}_i(\overline{\phi },\overline{\vartheta }){\left({\nabla }_{\vartheta }{f}_i(\overline{\phi },\overline{\vartheta })\right)}^T+G_{f_i}^{\prime }\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\vartheta \vartheta }{f}_i(\overline{\phi },\overline{\vartheta })\right\}{\overline{p}}_i\right]}\hfill \\ {\displaystyle -\frac{1}{2}\sum _{i=1}^k{\lambda }_i{\overline{p}}_i\left\{G_{f_i}^{″}\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\vartheta }{f}_i(\overline{\phi },\overline{\vartheta }){\left({\nabla }_{\vartheta }{f}_i(\overline{\phi },\overline{\vartheta })\right)}^T+G_{f_i}^{\prime }\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\vartheta \vartheta }{f}_i(\overline{\phi },\overline{\vartheta })\right\}{\overline{p}}_i],\dots ,G_{f_k}\left(f_k(\overline{\phi },\overline{\vartheta })\right)-{\overline{\vartheta }}^T\sum _{i=1}^k{\overline{\lambda }}_i[G_{f_i}^{\prime }\left(f_i(\overline{\phi },\overline{\vartheta })\right)}\hfill \\ {\displaystyle {\nabla }_{\vartheta }{f}_i(\overline{\phi },\overline{\vartheta })+\left\{G_{f_i}^{″}\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\vartheta }{f}_i(\overline{\phi },\overline{\vartheta }){\left({\nabla }_{\vartheta }{f}_i(\overline{\phi },\overline{\vartheta })\right)}^T+G_{f_i}^{\prime }\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\vartheta \vartheta }{f}_i(\overline{\phi },\overline{\vartheta })\right\}{\overline{p}}_i]-\frac{1}{2}\sum _{i=1}^k{\lambda }_i{\overline{p}}_i\{G_{f_i}^{″}\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\vartheta }{f}_i(\overline{\phi },\overline{\vartheta }){\left({\nabla }_{\vartheta }{f}_i(\overline{\phi },\overline{\vartheta })\right)}^T}\hfill \\ +G_{f_i}^{\prime }\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\vartheta \vartheta }{f}_i(\overline{\phi },\overline{\vartheta })\left\}{\overline{p}}_i\right])\hfill \\ {\displaystyle =(G_{f_1}\left(f_1(\overline{\phi },\overline{\vartheta })\right)-{\overline{\phi }}^T\sum _{i=1}^k{\overline{\lambda }}_i\left[G_{f_i}^{\prime }\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\phi }{f}_i(\overline{\phi },\overline{\vartheta })+\left\{G_{f_i}^{″}\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\phi }{f}_i(\overline{\phi },\overline{\vartheta }){\left({\nabla }_{\phi }{f}_i(\overline{\phi },\overline{\vartheta })\right)}^T+G_{f_i}^{\prime }\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\phi \phi }{f}_i(\overline{\phi },\overline{\vartheta })\right\}{\overline{q}}_i\right]}\hfill \\ {\displaystyle -\frac{1}{2}\sum _{i=1}^k{\lambda }_i{\overline{q}}_i\left\{G_{f_i}^{″}\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\phi }{f}_i(\overline{\phi },\overline{\vartheta }){\left({\nabla }_{\phi }{f}_i(\overline{\phi },\overline{\vartheta })\right)}^T+G_{f_i}^{\prime }\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\phi \phi }{f}_i(\overline{\phi },\overline{\vartheta })\right\}{\overline{q}}_i],\dots ,G_{f_k}\left(f_k(\overline{\phi },\overline{\vartheta })\right)-{\overline{\phi }}^T\sum _{i=1}^k{\overline{\lambda }}_i[G_{f_i}^{\prime }\left(f_i(\overline{\phi },\overline{\vartheta })\right)}\hfill \\ {\displaystyle {\nabla }_{\phi }{f}_i(\overline{\phi },\overline{\vartheta })+\left\{G_{f_i}^{″}\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\phi }{f}_i(\overline{\phi },\overline{\vartheta }){\left({\nabla }_{\phi }{f}_i(\overline{\phi },\overline{\vartheta })\right)}^T+G_{f_i}^{\prime }\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\phi \phi }{f}_i(\overline{\phi },\overline{\vartheta })\right\}{\overline{q}}_i]-\frac{1}{2}\sum _{i=1}^k{\lambda }_i{\overline{q}}_i\{G_{f_i}^{″}\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\phi }{f}_i(\overline{\phi },\overline{\vartheta }){\left({\nabla }_{\phi }{f}_i(\overline{\phi },\overline{\vartheta })\right)}^T}\hfill \\ +G_{f_i}^{\prime }\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\phi \phi }{f}_i(\overline{\phi },\overline{\vartheta })\left\}{\overline{q}}_i\right]).\hfill \end{array}\end{array}$$

This shows that the objective values are equal.

Finally, we have to claim that $(\overline{\phi },\overline{\vartheta },\overline{\lambda },{\overline{q}}_1={\overline{q}}_2={\overline{q}}_3=\dots ={\overline{q}}_k=0)$ is an efficient solution of $\left(GWDP\right)$.

If possible, then suppose that $(\overline{\phi },\overline{\vartheta },\overline{\lambda },{\overline{q}}_1={\overline{q}}_2={\overline{q}}_3=\dots ={\overline{q}}_k=0)$ is not an efficient solution of $\left(GWDP\right)$, then there exist $(\overline{\delta },\overline{\ell },\overline{\lambda },{\overline{q}}_1={\overline{q}}_2={\overline{q}}_3=\dots ={\overline{q}}_k=0)$ is efficient solution of $\left(GWDP\right)$ such that

$$\begin{array}{c}\hfill \begin{array}{c}{\displaystyle (G_{f_1}\left(f_1(\overline{\phi },\overline{\vartheta })\right)-{\overline{\phi }}^T\sum _{i=1}^k{\overline{\lambda }}_i\left[G_{f_i}^{\prime }\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\phi }{f}_i(\overline{\phi },\overline{\vartheta })+\left\{G_{f_i}^{″}\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\phi }{f}_i(\overline{\phi },\overline{\vartheta }){\left({\nabla }_{\phi }{f}_i(\overline{\phi },\overline{\vartheta })\right)}^T+G_{f_i}^{\prime }\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\phi \phi }{f}_i(\overline{\phi },\overline{\vartheta })\right\}{\overline{q}}_i\right]}\hfill \\ {\displaystyle -\frac{1}{2}\sum _{i=1}^k{\lambda }_i{\overline{q}}_i\left\{G_{f_i}^{″}\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\phi }{f}_i(\overline{\phi },\overline{\vartheta }){\left({\nabla }_{\phi }{f}_i(\overline{\phi },\overline{\vartheta })\right)}^T+G_{f_i}^{\prime }\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\phi \phi }{f}_i(\overline{\phi },\overline{\vartheta })\right\}{\overline{q}}_i],\dots ,G_{f_k}\left(f_k(\overline{\phi },\overline{\vartheta })\right)-{\overline{\phi }}^T\sum _{i=1}^k{\overline{\lambda }}_i[G_{f_i}^{\prime }\left(f_i(\overline{\phi },\overline{\vartheta })\right)}\hfill \\ {\displaystyle {\nabla }_{\phi }{f}_i(\overline{\phi },\overline{\vartheta })+\left\{G_{f_i}^{″}\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\phi }{f}_i(\overline{\phi },\overline{\vartheta }){\left({\nabla }_{\phi }{f}_i(\overline{\phi },\overline{\vartheta })\right)}^T+G_{f_i}^{\prime }\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\phi \phi }{f}_i(\overline{\phi },\overline{\vartheta })\right\}{\overline{q}}_i]-\frac{1}{2}\sum _{i=1}^k{\lambda }_i{\overline{q}}_i\{G_{f_i}^{″}\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\phi }{f}_i(\overline{\phi },\overline{\vartheta }){\left({\nabla }_{\phi }{f}_i(\overline{\phi },\overline{\vartheta })\right)}^T}\hfill \\ {\displaystyle +G_{f_i}^{\prime }\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\phi \phi }{f}_i(\overline{\phi },\overline{\vartheta })\left\}{\overline{q}}_i\right]-G_{f_1}\left(f_1(\overline{\delta },\overline{\ell })\right)-{\overline{\delta }}^T\sum _{i=1}^k{\overline{\lambda }}_i[G_{f_i}^{\prime }\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\phi }{f}_i(\overline{\delta },\overline{\ell })+\{G_{f_i}^{″}\left(f_i(\overline{\delta },\overline{\ell })\right){\nabla }_{\phi }{f}_i(\overline{\delta },\overline{\ell }){\left({\nabla }_{\phi }{f}_i(\overline{\delta },\overline{\ell })\right)}^T}\hfill \\ +G_{f_i}^{\prime }\left(f_i(\overline{\delta },\overline{\ell })\right){\nabla }_{\phi \phi }{f}_i(\overline{\delta },\overline{\ell })\left\}{\overline{q}}_i\right]\hfill \\ {\displaystyle -\frac{1}{2}\sum _{i=1}^k{\lambda }_i{\overline{q}}_i\left\{G_{f_i}^{″}\left(f_i(\overline{\delta },\overline{\ell })\right){\nabla }_{\phi }{f}_i(\overline{\delta },\overline{\ell }){\left({\nabla }_{\phi }{f}_i(\overline{\delta },\overline{\ell })\right)}^T+G_{f_i}^{\prime }\left(f_i(\overline{\delta },\overline{\ell })\right){\nabla }_{\phi \phi }{f}_i(\overline{\delta },\overline{\ell })\right\}{\overline{q}}_i],\dots ,G_{f_k}\left(f_k(\overline{\delta },\overline{\ell })\right)-{\overline{\delta }}^T\sum _{i=1}^k{\overline{\lambda }}_i[G_{f_i}^{\prime }\left(f_i(\overline{\delta },\overline{\ell })\right)}\hfill \\ {\displaystyle {\nabla }_{\phi }{f}_i(\overline{\delta },\overline{\ell })+\left\{G_{f_i}^{″}\left(f_i(\overline{\delta },\overline{\ell })\right){\nabla }_{\phi }{f}_i(\overline{\delta },\overline{\ell }){\left({\nabla }_{\phi }{f}_i(\overline{\delta },\overline{\ell })\right)}^T+G_{f_i}^{\prime }\left(f_i(\overline{\delta },\overline{\ell })\right){\nabla }_{\phi \phi }{f}_i(\overline{\delta },\overline{\ell })\right\}{\overline{q}}_i]-\frac{1}{2}\sum _{i=1}^k{\lambda }_i{\overline{q}}_i\{G_{f_i}^{″}\left(f_i(\overline{\delta },\overline{\ell })\right){\nabla }_{\phi }{f}_i(\overline{\delta },\overline{\ell }){\left({\nabla }_{\phi }{f}_i(\overline{\delta },\overline{\ell })\right)}^T}\hfill \\ +G_{f_i}^{\prime }\left(f_i(\overline{\delta },\overline{\ell })\right){\nabla }_{\phi \phi }{f}_i(\overline{\delta },\overline{\ell })\}{\overline{q}}_i\left]\right)\in -K\setminus \left\{0\right\}.\hfill \end{array}\end{array}$$

As

$$\begin{array}{c}\hfill \begin{array}{c}{\displaystyle {\overline{\phi }}^T\sum _{i=1}^k{\overline{\lambda }}_i{G}_{f_i}^{\prime }\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\phi }{f}_i(\overline{\phi },\overline{\vartheta })={\overline{\vartheta }}^T\sum _{i=1}^k{\overline{\lambda }}_i{G}_{f_i}^{\prime }\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\vartheta }{f}_i(\overline{\phi },\overline{\vartheta })and{\overline{p}}_i=0,i\in \tilde{N},}\hfill \\ {\displaystyle (G_{f_1}\left(f_1(\overline{\phi },\overline{\vartheta })\right)-{\overline{\vartheta }}^T\sum _{i=1}^k{\overline{\lambda }}_i\left[G_{f_i}^{\prime }\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\vartheta }{f}_i(\overline{\phi },\overline{\vartheta })+\left\{G_{f_i}^{″}\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\phi }{f}_i(\overline{\phi },\overline{\vartheta }){\left({\nabla }_{\phi }{f}_i(\overline{\phi },\overline{\vartheta })\right)}^T+G_{f_i}^{\prime }\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\phi \phi }{f}_i(\overline{\phi },\overline{\vartheta })\right\}{\overline{q}}_i\right]}\hfill \\ {\displaystyle -\frac{1}{2}\sum _{i=1}^k{\lambda }_i{\overline{q}}_i\left\{G_{f_i}^{″}\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\phi }{f}_i(\overline{\phi },\overline{\vartheta }){\left({\nabla }_{\phi }{f}_i(\overline{\phi },\overline{\vartheta })\right)}^T+G_{f_i}^{\prime }\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\phi \phi }{f}_i(\overline{\phi },\overline{\vartheta })\right\}{\overline{q}}_i],\dots ,G_{f_k}\left(f_k(\overline{\phi },\overline{\vartheta })\right)-{\overline{\phi }}^T\sum _{i=1}^k{\overline{\lambda }}_i[G_{f_i}^{\prime }\left(f_i(\overline{\phi },\overline{\vartheta })\right)}\hfill \\ {\displaystyle {\nabla }_{\phi }{f}_i(\overline{\phi },\overline{\vartheta })+\left\{G_{f_i}^{″}\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\phi }{f}_i(\overline{\phi },\overline{\vartheta }){\left({\nabla }_{\phi }{f}_i(\overline{\phi },\overline{\vartheta })\right)}^T+G_{f_i}^{\prime }\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\phi \phi }{f}_i(\overline{\phi },\overline{\vartheta })\right\}{\overline{q}}_i]-\frac{1}{2}\sum _{i=1}^k{\lambda }_i{\overline{q}}_i\{G_{f_i}^{″}\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\phi }{f}_i(\overline{\phi },\overline{\vartheta }){\left({\nabla }_{\phi }{f}_i(\overline{\phi },\overline{\vartheta })\right)}^T}\hfill \\ {\displaystyle +G_{f_i}^{\prime }\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\phi \phi }{f}_i(\overline{\phi },\overline{\vartheta })\left\}{\overline{q}}_i\right]-G_{f_1}\left(f_1(\overline{\delta },\overline{\ell })\right)-{\overline{\delta }}^T\sum _{i=1}^k{\overline{\lambda }}_i[G_{f_i}^{\prime }\left(f_i(\overline{\phi },\overline{\vartheta })\right){\nabla }_{\phi }{f}_i(\overline{\delta },\overline{\ell })+}\hfill \\ \left\{G_{f_i}^{″}\left(f_i(\overline{\delta },\overline{\ell })\right){\nabla }_{\phi }{f}_i(\overline{\delta },\overline{\ell }){\left({\nabla }_{\phi }{f}_i(\overline{\delta },\overline{\ell })\right)}^T+G_{f_i}^{\prime }\left(f_i(\overline{\delta },\overline{\ell })\right){\nabla }_{\phi \phi }{f}_i(\overline{\delta },\overline{\ell })\right\}{\overline{q}}_i]\hfill \\ {\displaystyle -\frac{1}{2}\sum _{i=1}^k{\lambda }_i{\overline{q}}_i\left\{G_{f_i}^{″}\left(f_i(\overline{\delta },\overline{\ell })\right){\nabla }_{\phi }{f}_i(\overline{\delta },\overline{\ell }){\left({\nabla }_{\phi }{f}_i(\overline{\delta },\overline{\ell })\right)}^T+G_{f_i}^{\prime }\left(f_i(\overline{\delta },\overline{\ell })\right){\nabla }_{\phi \phi }{f}_i(\overline{\delta },\overline{\ell })\right\}{\overline{q}}_i],\dots ,G_{f_k}\left(f_k(\overline{\delta },\overline{\ell })\right)-{\overline{\delta }}^T\sum _{i=1}^k{\overline{\lambda }}_i[G_{f_i}^{\prime }\left(f_i(\overline{\delta },\overline{\ell })\right)}\hfill \\ {\displaystyle {\nabla }_{\phi }{f}_i(\overline{\delta },\overline{\ell })+\left\{G_{f_i}^{″}\left(f_i(\overline{\delta },\overline{\ell })\right){\nabla }_{\phi }{f}_i(\overline{\delta },\overline{\ell }){\left({\nabla }_{\phi }{f}_i(\overline{\delta },\overline{\ell })\right)}^T+G_{f_i}^{\prime }\left(f_i(\overline{\delta },\overline{\ell })\right){\nabla }_{\phi \phi }{f}_i(\overline{\delta },\overline{\ell })\right\}{\overline{q}}_i]-\frac{1}{2}\sum _{i=1}^k{\lambda }_i{\overline{q}}_i\{G_{f_i}^{″}\left(f_i(\overline{\delta },\overline{\ell })\right){\nabla }_{\phi }{f}_i(\overline{\delta },\overline{\ell }){\left({\nabla }_{\phi }{f}_i(\overline{\delta },\overline{\ell })\right)}^T}\hfill \\ +G_{f_i}^{\prime }\left(f_i(\overline{\delta },\overline{\ell })\right){\nabla }_{\phi \phi }{f}_i(\overline{\delta },\overline{\ell })\}{\overline{q}}_i\left]\right)\in -K\setminus \left\{0\right\},\hfill \end{array}\end{array}$$

which contradicts the Weak duality Theorem 1 or Theorem 2. Hence, completes the proof. □

Theorem 4

(Converse duality). Let $(\overline{\delta },\overline{\ell },\overline{\lambda },\overline{q})$ is an efficient solution of $\left(GWDP\right)$; fix $\lambda =\overline{\lambda }$ in $\left(GWPP\right)$ such that

(i) 

for all $i\in \{1,2,\dots ,k\},\left[G_{f_i}^{″}\left(f_i(\overline{\delta },\overline{\ell })\right){\nabla }_{\phi }{f}_i(\overline{\delta },\overline{\ell }){\left({\nabla }_{\phi }{f}_i(\overline{\delta },\overline{\ell })\right)}^T+G_{f_i}^{\prime }\left(f_i(\overline{\delta },\overline{\ell })\right){\nabla }_{\phi \phi }{f}_i(\overline{\delta },\overline{\ell })\right]$ is non singular,

(ii) 

${\displaystyle \sum _{i=1}^k{\overline{\lambda }}_i{\nabla }_{\phi }\left[{\overline{q}}_i\{G_{f_i}^{″}\left(f_i(\overline{\delta },\overline{\ell })\right){\nabla }_{\phi }{f}_i(\overline{\delta },\overline{\ell }){\left({\nabla }_{\phi }{f}_i(\overline{\delta },\overline{\ell })\right)}^T+G_{f_i}^{\prime }\left(f_i(\overline{\delta },\overline{\ell })\right){\nabla }_{\phi \phi }{f}_i(\overline{\delta },\overline{\ell })\}{\overline{q}}_i\right]\notin }$ span $\left\{G_{f_1}^{\prime }\left(f_1(\overline{\delta },\overline{\ell })\right){\nabla }_{\phi }{f}_1(\overline{\delta },\overline{\ell }),G_{f_2}^{\prime }\left(f_2(\overline{\delta },\overline{\ell })\right){\nabla }_{\phi }{f}_2(\overline{\delta },\overline{\ell })\dots ,G_{f_i}^{\prime }\left(f_i(\overline{\delta },\overline{\ell })\right){\nabla }_{\phi }{f}_i(\overline{\delta },\overline{\ell })\right\}.$

(iii) 

the set of vectors $\left\{G_{f_1}^{\prime }\left(f_1(\overline{\delta },\overline{\ell })\right){\nabla }_{\phi }{f}_1(\overline{\delta },\overline{\ell }),G_{f_2}^{\prime }\left(f_2(\overline{\delta },\overline{\ell })\right){\nabla }_{\phi }{f}_2(\overline{\delta },\overline{\ell }),\dots ,G_{f_k}^{\prime }\left(f_k(\overline{\delta },\overline{\ell })\right){\nabla }_{\phi }{f}_k(\overline{\delta },\overline{\ell })\right\}$ are linearly independent,

(iv) 

${\displaystyle \sum _{i=1}^k{\overline{\lambda }}_i{\nabla }_{\vartheta }\left[{\overline{q}}_i\left\{G_{f_i}^{″}\left(f_i(\overline{\delta },\overline{\ell })\right){\nabla }_{\phi }{f}_i(\overline{\delta },\overline{\ell }){\left({\nabla }_{\phi }{f}_i(\overline{\delta },\overline{\ell })\right)}^T+G_{f_i}^{\prime }\left(f_i(\overline{\delta },\overline{\ell })\right){\nabla }_{\phi \phi }{f}_i(\overline{\delta },\overline{\ell })\right\}{\overline{q}}_i\right]=0\Rightarrow {\overline{q}}_i=0,\forall i,}$

(v) 

K is closed convex pointed cone with $R_{+}^k\subseteq K.$

Then, $(\overline{\delta },\overline{\ell },\overline{\lambda },\overline{p}=0)$ is a feasible solution for $\left(GWPP\right)$ and the objective values of $\left(GWDP\right)$ and $\left(GWPP\right)$ are equal. Furthermore, if the hypotheses of Theorem 1 or Theorem 2 are satisfied for all feasible solutions of $\left(GWDP\right)$ and $\left(GWPP\right)$, then $(\overline{\delta },\overline{\ell },\overline{\lambda },\overline{p}=0)$ is an optimal solution of $\left(GWPP\right)$. Also, if the hypotheses of Theorem 1 or Theorem 2 are satisfied for all feasible solutions for $\left(GWDP\right)$ and $\left(GWPP\right)$, then $(\overline{\delta },\overline{\ell },\overline{\lambda },\overline{q})$ and $(\overline{\delta },\overline{\ell },\overline{\lambda },\overline{p})$ is an efficient solution for $\left(GWDP\right)$ and $\left(GWPP\right)$, respectively.

Proof. 

It follows on the lines of Theorem 3. □

4. Conclusions

In this paper, we have presented a novel generalized group of definitions and illustrated various non-trivial numerical examples for existing such type of functions. Numerical examples have also been illustrated to justify the weak duality theorem. Furthermore, we have studied a new class of K-$G_f$-Wolfe type primal-dual model with cone objective as well as constraint and proved duality theorem under K-$G_f$-bonvexity and K-$G_f$-pseudobonvexity. This work can further be extended to higher order symmetric fractional programming problem and variational control problem over cones. This will be feature task for the researchers.