Mond-Weir and Wolfe Duality of Set-Valued Fractional Minimax Problems in Terms of Contingent Epi-Derivative of Second-Order

Abstract

This paper is devoted to provide sufficient Karush Kuhn Tucker (in short, KKT) conditions of optimality of second-order for a set-valued fractional minimax problem. In addition, we define duals of the types Mond-Weir and Wolfe of second-order for the problem. Further we obtain the theorems of duality under contingent epi-derivative together with generalized cone convexity suppositions of second-order.

1. Introduction

A fractional minimax optimization programming is a particular type of optimization problem. It has applications in various fields of engineering sciences, mathematics, and economics. In the year of 1985, Bector and Bhatia [1], and in 1992, Weir [2] developed the conditions of optimality and developed the theorems of duality of differentiable fractional minimax problems. In 1987, Zamlai [3] demonstrated the sufficient and necessary conditions of optimality and constituted theorems of duality for fractional minimax problems using invexity suppositions in its broader sense. Liu and Wu [4] determined the sufficient conditions of optimality and developed the theorems of duality for fractional minimax problems with the assistance of $(\mathcal{F},\rho )$ convexity suppositions in 1998. In the year of 2003, Ahmad [5] demonstrated the sufficient conditions of optimality and provided the theorems of duality for fractional minimax problems via $\rho$ invexity suppositions. In 1999, Lai et al. [6] determined the sufficient and necessary conditions of optimality and developed the theorems of duality for parametric type in terms of convexity in its broader sense. In 2002, Lai and Lee [7] constituted the theorems of duality for fractional minimax problems involving convexity in its larger sense. In 2003, Liang and Shi [8] studied the sufficient conditions of optimality and provided the theorems of duality for fractional minimax problems involving the notion of $(F,\alpha ,\rho ,d)$ convexity. In the year of 2006, Ahmad and Husain [9] studied the sufficient conditions of optimality and developed the theorems of duality for fractional minimax problems in terms of $(F,\alpha ,\rho ,d)$ convexity.

A set-valued function is one for which image of elements of domain set of the map is a set. A set-valued optimization problem (in short, SOP) is an optimization problem with a set-valued objective function and constraints involving set-valued functions. Since last two decades, many authors have been expressing their interests in extending many results of vector optimization problems (in short, VOP) to SOP. In 2008, Li et al. [10,11] constituted the sufficient and necessary conditions of optimality via contingent derivative of higher-order. They also developed the kind of Mond-Weir theorems of duality of higher-order for SOP under convexity suppositions. In the year of 2015, Das and Nahak [12] demonstrated the sufficient optimality conditions of SOP in terms of generalized convexity and contingent derivative of higher-order. They also developed theorems of duality of Wolfe, Mond-Weir, and mixed kinds. They [12] also introduced the notion $\rho$-cone convexity of second-order using contingent epi-derivative of second-order. Das and Nahak [13] developed the KKT sufficient conditions of optimality of set-valued minimax problems in terms of contingent epi-derivative along with cone convexity in generalized sense. They also provided the theorems of duality of Wolfe, Mond-Weir, and mixed kinds. In 2020, Das and Nahak [14] constituted the sufficient conditions of optimality of KKT type for set-valued fractional minimax problems. They also defined the duals of Wolfe, Mond-Weir, and mixed kinds and provided the theorems of duality. For other different but connected ideas on this subject, the reader is directed to the following books [15,16].

In this study, we will look upon the sufficient conditions of KKT type of second-order for a set-valued fractional minimax problem (FMP) via cone convexity in its generalized sense and contingent epi-derivative of second-order. We provide duality relationships of Mond-Weir and Wolfe kinds between the problem (FMP) and the dual problems of second-order.

The paper is organized in the following manner. We give some introductory ideas and definitions of set-valued functions in Section 2. In the Section 3, we look at the sufficient conditions of KKT type of second-order and prove the theorems of duality of Mond-Weir and Wolfe kinds for (FMP) in terms of cone convexity in its generalized sense of second-order.

2. Definitions and Preliminaries

Let us assume that ${\mathbb{R}}^k$ denotes the k-dimensional Euclidean space and $\Delta$ is non-empty subset of ${\mathbb{R}}^k$. Then we call $\Delta$ a cone if $t\Delta \subseteq \Delta$, for all $t\ge 0$. In addition, the cone $\Delta$ is known as solid if the interior of $\Delta$, denoted by int$(\Delta )$, is non-empty. Moreover, the cone $\Delta$ is referred to as pointed if $\Delta \cap (-\Delta )=\left\{0_{{\mathbb{R}}^k}\right\}$, $0_{{\mathbb{R}}^k}$ being the zero element of ${\mathbb{R}}^k$ and closed if closure of $\Delta$, represented by $\overline{\Delta }$, becomes $\Delta$ itself. Additionally, the cone $\Delta$ is called convex if $t\Delta +(1-t)\Delta \subseteq \Delta$, for all $t\in [0,1]$.

The non-negative orthant of ${\mathbb{R}}^k$, denoted by ${\mathbb{R}}_{+}^k$, is defined by

$${\mathbb{R}}_{+}^k=\{r=(r_1,\dots ,r_j,\dots ,r_k)\in {\mathbb{R}}^k:r_j\ge 0,\mathrm{for}\mathrm{all}j=1,2,\dots ,k\}.$$

It is clear that int$\left({\mathbb{R}}_{+}^k\right)\cup \left\{0_{{\mathbb{R}}^k}\right\}$ is a pointed solid convex cone and ${\mathbb{R}}_{+}^k$ is a closed pointed solid convex cone of ${\mathbb{R}}^k$.

With respect to (in short, w.r.t.) the pointed solid convex cone ${\mathbb{R}}_{+}^k$ of ${\mathbb{R}}^k$, there are basically two types of cone-orders in ${\mathbb{R}}^k$. For two elements $r_1,r_2\in {\mathbb{R}}^k$, we have

$$r_1\le r_2\mathrm{if}{r}_2-r_1\in {\mathbb{R}}_{+}^k$$

and

$$r_1<r_2\mathrm{if}{r}_2-r_1\in \mathrm{int}\left({\mathbb{R}}_{+}^k\right).$$

In finite dimensional Euclidean space, the concepts of minimality and weak minimality are defined as follows.

Definition 1.

Let Θ be a non-empty subset of ${\mathbb{R}}^k$. Then

(i)

$r^{\prime }\in \Theta$ is stated to be a minimal point of Θ if there is not any $r\in \Theta \backslash \left\{r^{\prime }\right\}$, such that

$$r\le r^{\prime }.$$

(ii)

$r^{\prime }\in \Theta$ is stated to be a weakly minimal point of Θ if there is not any $r\in \Theta$, such that

$$r<r^{\prime }.$$

Let us consider that min$(\Theta )$ and w-min$(\Theta )$ denote the sets of minimal points and weakly minimal points of $\Theta$. These two sets can be expressed in the following ways.

$$\min (\Theta )=\{r^{\prime }\in \Theta :(r^{\prime }-{\mathbb{R}}_{+}^k)\cap \Theta =\left\{r^{\prime }\right\}\}$$

and

$$\mathrm{w}-\min (\Theta )=\{r^{\prime }\in \Theta :(r^{\prime }-\mathrm{int}\left({\mathbb{R}}_{+}^k\right))\cap \Theta =\varnothing \}.$$

In real normed space, we recall the concepts of contingent cone and second-order contingent set.

Definition 2

([17,18]). Suppose that W is a real normed space, Θ is a non-empty subset of W, and $w^{\prime }$ is an element of $\overline{\Theta }$. We denote the contingent cone to the set Θ at the point $w^{\prime }$ by $\mathcal{C}(\Theta ,w^{\prime })$.

$w\in W$ is an element of $\mathcal{C}(\Theta ,w^{\prime })$ if there are sequences $\left\{t_n\right\}$ in $\mathbb{R}$ and $\left\{w_n\right\}$ in W, with $t_n\to 0^{+}$ and $w_n\to w$, as $n\to +\infty$, such that

$$w^{\prime }+t_n{w}_n\in \Theta ,\mathrm{for}\mathrm{all}n\in \mathbb{N},$$

or, there are sequences $\left\{s_n\right\}$ in $\mathbb{R}$ and $\left\{w_n^{\prime }\right\}$ in Θ, with $s_n>0$ and $w_n^{\prime }\to w^{\prime }$, as $n\to +\infty$, such that

$$s_n(w_n^{\prime }-w^{\prime })\to w,\mathrm{as}n\to +\infty .$$

Definition 3

([17,18,19]). Suppose that W is a real normed space, Θ is a non-empty subset of W, $w^{\prime }\in \overline{\Theta }$, and $\widehat{w}\in W$. We denote the second-order contingent set to the set Θ at the point $w^{\prime }$ in the direction $\widehat{w}$ by ${\mathcal{C}}^2(\Theta ,w^{\prime },\widehat{w})$.

$w\in W$ is an element of ${\mathcal{C}}^2(\Theta ,w^{\prime },\widehat{w})$ if there are sequences $\left\{t_n\right\}$ in $\mathbb{R}$ and $\left\{w_n\right\}$ in W, with $t_n\to 0^{+}$ and $w_n\to w$, as $n\to +\infty$, such that

$$w^{\prime }+t_n\widehat{w}+\frac{1}{2}{t_n}^2{w}_n\in \Theta ,\mathrm{for}\mathrm{all}n\in \mathbb{N},$$

or, there are sequences $\left\{s_n\right\}$, $\left\{s_n^{\prime }\right\}$ in $\mathbb{R}$ and $\left\{w_n^{\prime }\right\}$ in Θ, with $s_n,s_n^{\prime }>0$, $s_n\to +\infty$, $s_n^{\prime }\to +\infty$, $\frac{s_n^{\prime }}{s_n}\to 2$, and $w_n^{\prime }\to w^{\prime }$, as $n\to +\infty$, such that

$$s_n(w_n^{\prime }-w^{\prime })\to \widehat{w}\mathrm{and}{s}_n^{\prime }(s_n(w_n^{\prime }-w^{\prime })-\widehat{w})\to w,\mathrm{as}n\to +\infty .$$

Suppose that Z and W are real normed spaces, $2^W$ denotes the set of all subsets of W, and $\Delta$ is a pointed solid convex cone in W. Assume that $\mathcal{F}:Z\to 2^W$ is a set-valued function from Z to W. So, $\mathcal{F}\left(z\right)\subseteq W$, $\mathrm{for}\mathrm{all}z\in Z$. The domain, graph, and epi-graph of $\mathcal{F}$ can be defined like this:

$$\mathrm{dom}\left(\mathcal{F}\right)=\{z\in Z:\mathcal{F}(z)\ne \varnothing \},$$

$$\mathrm{gr}\left(\mathcal{F}\right)=\left\{\right(z,w)\in Z\times W:w\in \mathcal{F}(z\left)\right\},$$

and

$$\mathrm{epi}\left(\mathcal{F}\right)=\left\{\right(z,w)\in Z\times W:w\in \mathcal{F}(z)+\Delta \}.$$

Suppose that $Z^{\prime }$ is a non-empty subset of Z, $z^{\prime }\in Z^{\prime }$, $z\in Z$, $w\in W$, $\mathcal{F}:Z\to 2^W$ be a set-valued function, along with $Z^{\prime }\subseteq \mathrm{dom}\left(\mathcal{F}\right)$ and $w^{\prime }\in \mathcal{F}\left(z^{\prime }\right)$. Jahn and Rauh [20] introduced the idea of contingent epi-derivative. It is quite important in the theory of SOP.

Definition 4

([20]). A single-valued map $\overrightarrow{d}\mathcal{F}(z^{\prime },w^{\prime }):Z\to W$ is stated to be contingent epi-derivative of $\mathcal{F}$ at $(z^{\prime },w^{\prime })$ if

$$\mathrm{epi}(\overrightarrow{d}\mathcal{F}(z^{\prime },w^{\prime }))=\mathcal{C}(\mathrm{epi}\left(\mathcal{F}\right),(z^{\prime },w^{\prime })).$$

Jahn et al. [21] developed the concept of contingent epi-derivative of second-order. It has very crucial role in the theory of SOP.

Definition 5

([21]). A single-valued map ${\overrightarrow{d}}^2\mathcal{F}(z^{\prime },w^{\prime },z,w):Z\to W$ is stated to be contingent epi-derivative of second-order of $\mathcal{F}$ at $(z^{\prime },w^{\prime })$ in a direction $(z,w)$ if

$$\mathrm{epi}({\overrightarrow{d}}^2\mathcal{F}(z^{\prime },w^{\prime },z,w))={\mathcal{C}}^2(\mathrm{epi}\left(\mathcal{F}\right),(z^{\prime },w^{\prime }),(z,w)).$$

The concept of cone convexity was introduced by Borwein [22] in the theory of SOP.

Definition 6

([22]). Suppose that $Z^{\prime }$ is a convex subset of Z. A set-valued function $\mathcal{F}:Z\to 2^W$, along with $Z^{\prime }\subseteq \mathrm{dom}\left(\mathcal{F}\right)$, is said to be Δ-convex on $Z^{\prime }$ if $forall{z}_1^{\prime },z_2^{\prime }\in Z^{\prime }$ and $t\in [0,1]$,

$$t\mathcal{F}\left(z_1^{\prime }\right)+(1-t)\mathcal{F}\left(z_2^{\prime }\right)\subseteq \mathcal{F}(t{z}_1^{\prime }+(1-t)z_2^{\prime })+\Delta .$$

The cone convexity can be expressed with the assistance of contingent epi-derivative.

Lemma 1

([20]). Assume that $Z^{\prime }$ is a convex subset of Z and $\mathcal{F}:Z\to 2^W$ is Δ-convex on $Z^{\prime }$. Then

$$\mathcal{F}\left(z\right)-w^{\prime }\subseteq \overrightarrow{d}\mathcal{F}(z^{\prime },w^{\prime })(z-z^{\prime })+\Delta ,\mathrm{for}\mathrm{all}z,z^{\prime }\in Z^{\prime }\mathrm{and}{w}^{\prime }\in \mathcal{F}\left(z^{\prime }\right).$$

Zhu et al. [23] presented the concept of cone convexity of second-order in the theory of SOP.

Definition 7

([23]). Suppose that $Z^{\prime }$ is a non-empty subset of Z and $\mathcal{F}:Z\to 2^W$ is a set-valued function, along with $Z^{\prime }\subseteq \mathrm{dom}\left(\mathcal{F}\right)$. Let $z^{\prime },\widehat{z}\in Z^{\prime }$, $w^{\prime }\in \mathcal{F}\left(z^{\prime }\right)$, and $\widehat{w}\in \mathcal{F}\left(\widehat{z}\right)+\Delta$. We suppose that $\mathcal{F}$ is contingent epiderivable of second-order at $(z^{\prime },w^{\prime })$ in the direction $(\widehat{z}-z^{\prime },\widehat{w}-w^{\prime })$. Then F is stated to be second-order Δ-convex at $(z^{\prime },w^{\prime })$ in the direction $(\widehat{z}-z^{\prime },\widehat{w}-w^{\prime })$ on $Z^{\prime }$ if

$$\mathcal{F}\left(z\right)-w^{\prime }\subseteq {\overrightarrow{d}}^2\mathcal{F}(z^{\prime },w^{\prime },\widehat{z}-z^{\prime },\widehat{w}-w^{\prime })(z-z^{\prime })+\Delta ,\mathrm{for}\mathrm{all}z\in Z^{\prime }.$$

Definition 8.

A set-valued function $\mathcal{F}:{\mathbb{R}}^k\to 2^{{\mathbb{R}}^l}$ is stated to be upper semicontinuous on ${\mathbb{R}}^k$ if for any open subset W of ${\mathbb{R}}^l$, ${\mathcal{F}}^{+}\left(W\right)=\{z\in {\mathbb{R}}^k:\mathcal{F}\left(z\right)\subseteq W\}$ is an open set in ${\mathbb{R}}^k$.

Definition 9.

Suppose that W is a non-empty subset of ${\mathbb{R}}^l$. Then W is stated to be ${\mathbb{R}}_{+}^l$-semicompact in ${\mathbb{R}}^l$ if every open cover being of the form $\{{(w_j+{\mathbb{R}}_{+}^l)}^c:w_j\in W,j\in J\}$ has a finite subcover, where J is an index set.

Definition 10.

A set-valued function $\mathcal{F}:{\mathbb{R}}^k\to 2^{{\mathbb{R}}^l}$ is stated to be ${\mathbb{R}}_{+}^l$-semicompact-valued if for all $z\in \mathrm{dom}\left(\mathcal{F}\right)$, the set $\mathcal{F}\left(z\right)$ is ${\mathbb{R}}_{+}^l$-semicompact in ${\mathbb{R}}^l$.

Corley [24] provided the existence theorems for optimization problems having cone semicompact-valued and upper semicontinuous set-valued functions.

Theorem 1

([24]). Suppose that $Z^{\prime }$ is an compact subset of Z, and Δ is a convex cone in W such that $\overline{\Delta }$ is pointed. Let $\mathcal{F}:Z\to 2^W$ be an upper semicontinuous and Δ-semicompact-valued. Then there is a solution of the optimization problem defined as $\max {\displaystyle \bigcup _{z\in Z^{\prime }}}\mathcal{F}\left(z\right)$.

Suppose that C is a non-empty subset of ${\mathbb{R}}^k$ and D is a non-empty compact subset of ${\mathbb{R}}^l$. Suppose that $M_1$ and $M_2$ is two positive semidefinite matrices of order $k\times k$. Let $\mathcal{F},\mathcal{H}:{\mathbb{R}}^k\times {\mathbb{R}}^l\to 2^{\mathbb{R}}$ and $\mathcal{G}:{\mathbb{R}}^k\to 2^{{\mathbb{R}}^m}$ be set-valued functions, along with

$$C\times D\subseteq \mathrm{dom}\left(\mathcal{F}\right)\cap \mathrm{dom}\left(\mathcal{H}\right)\mathrm{and}C\subseteq \mathrm{dom}\left(\mathcal{G}\right).$$

Let us consider the following set-valued fractional minimax problem.

$$\begin{array}{cccc}\hfill & \underset{z\in C}{\mathrm{minimize}}\hfill & \hfill & \max \bigcup _{w\in D}\frac{\mathcal{F}(z,w)+{\left(z^T{M}_{1}z\right)}^{\frac{1}{2}}}{\mathcal{H}(z,w)-{\left(z^T{M}_{2}z\right)}^{\frac{1}{2}}}\hfill \\ \hfill & \mathrm{subject}\mathrm{to}\hfill & \hfill & \mathcal{G}\left(z\right)\cap (-{\mathbb{R}}_{+}^m)\ne \varnothing .\hfill \end{array}$$

(FMP)

Let us define a set-valued function $\Psi :C\times D\subseteq {\mathbb{R}}^k\times {\mathbb{R}}^l\to 2^{\mathbb{R}}$ by

$$\Psi (z,w)=\frac{\mathcal{F}(z,w)+{\left(z^T{M}_{1}z\right)}^{\frac{1}{2}}}{\mathcal{H}(z,w)-{\left(z^T{M}_{2}z\right)}^{\frac{1}{2}}},forall(z,w)\in C\times D.$$

assuming that

$$\mathcal{F}(z,w)+{\left(z^T{M}_{1}z\right)}^{\frac{1}{2}}\ge 0,\mathrm{for}\mathrm{all}(z,w)\in C\times D$$

and

$$\mathcal{H}(z,w)-{\left(z^T{M}_{2}z\right)}^{\frac{1}{2}}>0,\mathrm{for}\mathrm{all}(z,w)\in C\times D.$$

We suppose that for all $z\in C$, $\Psi (z,.):{\mathbb{R}}^l\to 2^{\mathbb{R}}$ is upper semicontinuous and ${\mathbb{R}}_{+}$-semicompact-valued on D. In Theorem 1, we suppose that $Z={\mathbb{R}}^l$, $W=\mathbb{R}$, and $\Delta ={\mathbb{R}}_{+}$. Therefore, $\max {\displaystyle \bigcup _{w\in D}}\Psi (z,w)$ always exists, for all $z\in C$. Since $\Psi (z,w)\subseteq \mathbb{R}$, there is one and only one solution of the problem $\max {\displaystyle \bigcup _{w\in D}}\Psi (z,w)$, for each $z\in C$. The feasible set of (FMP), denoted by $S^{\prime }$, can be defined as

$$S^{\prime }=\{z\in C:\mathcal{G}\left(z\right)\cap (-{\mathbb{R}}_{+}^m)\ne \varnothing \}.$$

The minimizer of (FMP) is defined in following manner.

Definition 11.

We suppose that $z^{\prime }\in S^{\prime }$ and $y^{\prime }=\max {\displaystyle \bigcup _{w\in D}}\Psi (z^{\prime },w)$. Then the point $(z^{\prime },y^{\prime })$ is stated to minimize (FMP) if there does not exist any point $(z,y)$, with $z\in S^{\prime }$, $y=\max {\displaystyle \bigcup _{w\in D}}\Psi (z,w)$, and $z\ne z^{\prime }$, such that $y<y^{\prime }$.

For $z^{\prime }\in C$, let us define

$$I\left(z^{\prime }\right)=\{j\in \{1,\dots ,m\}:0\in {\mathcal{G}}_j\left(z^{\prime }\right)\},$$

$$J\left(z^{\prime }\right)=\{1,\dots ,m\}\backslash I\left(z^{\prime }\right),$$

$$B\left(z^{\prime }\right)=\left\{w^{\prime }\in D:\max \bigcup _{w\in D}\Psi (z^{\prime },w)\in \Psi (z^{\prime },w^{\prime })\right\},$$

and

$$\begin{array}{c}\hfill K\left(z^{\prime }\right)=\{(n,z^{\ast },\tilde{w})\in \mathbb{N}\times {\mathbb{R}}_{+}^n\times {\mathbb{R}}^{ln}:1\le n\le k+1,z^{\ast }=(z_1^{\ast },\dots ,z_n^{\ast })\in {\mathbb{R}}_{+}^n,\\ \hfill \mathrm{with}\sum _{i=1}^n{z}_i^{\ast }=1,\tilde{w}=(\overline{w_1},\dots ,\overline{w_n}),\mathrm{with}\overline{w_i}\in B\left(z^{\prime }\right),i=1,\dots ,n\}.\end{array}$$

Since $\Psi (z^{\prime },.)$ is upper semicontinuous and ${\mathbb{R}}_{+}$-semicompact-valued on $D,forall{z}^{\prime }\in C$, we have

$$B\left(z^{\prime }\right)\ne \varnothing ,forall{z}^{\prime }\in S^{\prime }.$$

Let M be an $k\times k$ positive semidefinite matrix. So,

$$z^{T}Mw\le {\left(z^{T}Mz\right)}^{\frac{1}{2}}{\left(w^{T}Mw\right)}^{\frac{1}{2}},\mathrm{for}\mathrm{all}z,w\in {\mathbb{R}}^k.$$

Moreover, if ${\left(w^{T}Mw\right)}^{\frac{1}{2}}\le 1$, we have

$$\begin{array}{c}\hfill z^{T}Mw\le {\left(z^{T}Mz\right)}^{\frac{1}{2}}.\end{array}$$

(1)

3. Main Results

Das and Nahak [12,13,14,25,26] and Das and Treanţă [27] introduced the concept of $\rho$-cone convexity in set-valued optimization theory. They obtained the sufficient conditions of KKT type and constituted the theorems of duality of different types of SOP in terms of $\rho$-cone convexity and contingent epi-derivative. For $\rho =0$, $\rho$-cone convexity becomes the usual concept of cone convexity (Borwein [22]).

Definition 12

([25,26]). Suppose that C is a convex subset of ${\mathbb{R}}^k$, $\rho \in \mathbb{R}$, $\mathbf{1}=(1,\dots ,1)\in \mathrm{int}\left({\mathbb{R}}_{+}^l\right)$, and $\mathcal{F}:{\mathbb{R}}^k\to 2^{{\mathbb{R}}^l}$ is a set-valued function, along with $C\subseteq \mathrm{dom}\left(\mathcal{F}\right)$. Then $\mathcal{F}$ is stated to be ρ-${\mathbb{R}}_{+}^l$-convex w.r.t. 1 on C if

$$\begin{array}{c}\hfill t\mathcal{F}\left(z_1\right)+(1-t)\mathcal{F}\left(z_2\right)\subseteq \mathcal{F}(t{z}_1+(1-t)z_2)+\rho t(1-t){\parallel z_1-z_2\parallel }^2\mathbf{1}+{\mathbb{R}}_{+}^l,\\ \hfill \mathrm{for}\mathrm{all}{z}_1,z_2\in C\mathrm{and}\mathrm{for}\mathrm{all}t\in [0,1].\end{array}$$

Das and Nahak [26] composed an example of set-valued function which is $\rho$-cone convex for some $\rho \in \mathbb{R}$, but not cone convex. They represented $\rho$-cone convexity with the assistance of contingent epi-derivative.

Theorem 2

([26]). Suppose that C be is convex subset of ${\mathbb{R}}^k$, $z^{\prime }\in C$, $\rho \in \mathbb{R}$, $\mathbf{1}\in \mathrm{int}\left({\mathbb{R}}_{+}^l\right)$, $\mathcal{F}:{\mathbb{R}}^k\to 2^{{\mathbb{R}}^l}$ is ρ-${\mathbb{R}}_{+}^l$-convex w.r.t. 1 on C, and $w^{\prime }\in \mathcal{F}\left(z^{\prime }\right)$. Then,

$$\mathcal{F}\left(z\right)-w^{\prime }\subseteq \overrightarrow{d}\mathcal{F}(z^{\prime },w^{\prime })(z-z^{\prime })+\rho {\parallel z-z^{\prime }\parallel }^2\mathit{1}+{\mathbb{R}}_{+}^l,\mathrm{for}\mathrm{all}z\in C.$$

Das and Nahak [12] introduced $\rho$-cone convexity of second-order with the assistance of contingent epi-derivative of second-order.

Definition 13

([12]). Suppose that C is a non-empty subset of ${\mathbb{R}}^k$, $z^{\prime },\widehat{z}\in C$, $\rho \in \mathbb{R}$, 1 $\in \mathrm{int}\left({\mathbb{R}}_{+}^l\right)$, and $\mathcal{F}:{\mathbb{R}}^k\to 2^{{\mathbb{R}}^l}$ is a set-valued function, along with $C\subseteq \mathrm{dom}\left(\mathcal{F}\right)$, $w^{\prime }\in \mathcal{F}\left(z^{\prime }\right)$, and $\widehat{w}\in \mathcal{F}\left(\widehat{z}\right)+{\mathbb{R}}_{+}^l$. We suppose that $\mathcal{F}$ is contingent epiderivable of second-order at $(z^{\prime },w^{\prime })$ in the direction $(\widehat{z}-z^{\prime },\widehat{w}-w^{\prime })$. Then $\mathcal{F}$ is stated to be ρ-${\mathbb{R}}_{+}^l$-convex of second-order w.r.t. 1 at $(z^{\prime },w^{\prime })$ in the direction $(\widehat{z}-z^{\prime },\widehat{w}-w^{\prime })$ on C if

$$\mathcal{F}\left(z\right)-w^{\prime }\subseteq {\overrightarrow{d}}^2\mathcal{F}(z^{\prime },w^{\prime },\widehat{z}-z^{\prime },\widehat{w}-w^{\prime })(z-z^{\prime })+\rho {\parallel z-z^{\prime }\parallel }^2\mathit{1}+{\mathbb{R}}_{+}^l,\mathrm{for}\mathrm{all}z\in C.$$

Remark 1.

For $\widehat{z}=z^{\prime }$ and $\widehat{w}=w^{\prime }$,

$$\mathcal{F}\left(z\right)-w^{\prime }\subseteq \overrightarrow{d}\mathcal{F}(z^{\prime },w^{\prime })(z-z^{\prime })+\rho {\parallel z-z^{\prime }\parallel }^2\mathbf{1}+{\mathbb{R}}_{+}^l,\mathrm{for}\mathrm{all}z\in C.$$

This case reduces to the concept of ρ-${\mathbb{R}}_{+}^l$-convexity. If $\rho >0$, then $\mathcal{F}$ is stated to be strongly ρ-${\mathbb{R}}_{+}^l$-convex of second-order, for $\rho =0$, it reduces to the usual concept of ${\mathbb{R}}_{+}^l$-convexity of second-order, and if $\rho <0$, then $\mathcal{F}$ is stated to be weakly ρ-${\mathbb{R}}_{+}^l$-convex of second-order. It is straightforward that strongly ρ-${\mathbb{R}}_{+}^l$-convexity of second-order ⇒ ${\mathbb{R}}_{+}^l$-convexity of second-order ⇒ weakly ρ-${\mathbb{R}}_{+}^l$-convexity of second-order.

Das and Nahak [12] composed an example of set-valued function $\mathcal{F}:\mathbb{R}\to 2^{{\mathbb{R}}^2}$, which is $\rho$-${\mathbb{R}}_{+}^2$-convex of second-order for some $\rho \in \mathbb{R}$, but is not ${\mathbb{R}}_{+}^2$-convex of second-order.

3.1. Sufficient Conditions of Optimality of Second-Order

We provide the sufficient conditions of second-order of KKT type of the set-valued fractional minimax problem (FMP). We first prove the following lemma which will be utilized to find out our results.

Lemma 2.

Let $z^{\prime }\in C$, $y^{\prime }=\max {\displaystyle \bigcup _{w\in D}}\Psi (z^{\prime },w)$, and $\overline{w_i}\in B\left(z^{\prime }\right)$, $1\le i\le n$. Then there are $\overline{x_i^{\prime }}\in \mathcal{F}(z^{\prime },\overline{w_i})$ and $\overline{x_i^{″}}\in \mathcal{H}(z^{\prime },\overline{w_i})$ such that

$$y^{\prime }=\frac{\overline{x_i^{\prime }}+{\left({z^{\prime }}^T{M}_1{z}^{\prime }\right)}^{\frac{1}{2}}}{\overline{x_i^{″}}-{\left({z^{\prime }}^T{M}_2{z}^{\prime }\right)}^{\frac{1}{2}}}.$$

Proof. 

As $\overline{w_i}\in B\left(z^{\prime }\right)$, $i=1,\dots ,n$,

$$\max \bigcup _{w\in D}\Psi (z^{\prime },w)\in \Psi (z^{\prime },\overline{w_i}).$$

Since $y^{\prime }=\max {\displaystyle \bigcup _{w\in D}}\Psi (z^{\prime },w)$, we have

$$y^{\prime }\in \Psi (z^{\prime },\overline{w_i}),i=1,\dots ,n.$$

As $y^{\prime }\in \Psi (z^{\prime },\overline{w_i})$, there are $\overline{x_i^{\prime }}\in \mathcal{F}(z^{\prime },\overline{w_i})$ and $\overline{x_i^{″}}\in \mathcal{H}(z^{\prime },\overline{w_i})$ such that

$$y^{\prime }=\frac{\overline{x_i^{\prime }}+{\left({z^{\prime }}^T{M}_1{z}^{\prime }\right)}^{\frac{1}{2}}}{\overline{x_i^{″}}-{\left({z^{\prime }}^T{M}_2{z}^{\prime }\right)}^{\frac{1}{2}}},$$

which completes the proof. □

Theorem 3.

Suppose that C is a convex subset of ${\mathbb{R}}^k$, $z^{\prime }$ is a feasible solution of (FMP), and $y^{\prime }=\max {\displaystyle \bigcup _{w\in D}}\Psi (z^{\prime },w)$. Let $n\in \mathbb{N}$, (where $1\le n\le k+1$), $w,v\in {\mathbb{R}}^k$, $\overline{w_i}\in B\left(z^{\prime }\right)$, $(i=1,\dots ,n)$, $\overline{x_i^{\prime }}\in \mathcal{F}(z^{\prime },\overline{w_i})$, and $\overline{x_i^{″}}\in \mathcal{H}(z^{\prime },\overline{w_i})$ such that

$$\begin{array}{c}\hfill y^{\prime }=\frac{\overline{x_i^{\prime }}+{\left({z^{\prime }}^T{M}_1{z}^{\prime }\right)}^{\frac{1}{2}}}{\overline{x_i^{″}}-{\left({z^{\prime }}^T{M}_2{z}^{\prime }\right)}^{\frac{1}{2}}}.\end{array}$$

(2)

Let $u\in C$, $r_i\in \mathcal{F}(u,\overline{w_i})+{\mathbb{R}}_{+}$, $s_i\in (-\mathcal{H})(u,\overline{w_i})+{\mathbb{R}}_{+}$, $(i=1,\dots ,n)$ and $t_j\in {\mathcal{G}}_j\left(u\right)$, $(j=1,\dots ,m)$. We suppose that $\mathcal{F}(.,\overline{w_i})$ is ${\rho }_i$-${\mathbb{R}}_{+}$-convex of second-order at $(z^{\prime },\overline{x_i^{\prime }})$ in the direction $(u-z^{\prime },r_i-\overline{x_i^{\prime }})$, ${(.)}^T{M}_{1}w$ is ${\overline{\rho }}_i$-${\mathbb{R}}_{+}$-convex, $-\mathcal{H}(.,\overline{w_i})$ is ${\rho }_i^{\prime }$-${\mathbb{R}}_{+}$-convex of second-order at $(z^{\prime },-\overline{x_i^{″}})$ in the direction $(u-z^{\prime },s_i+\overline{x_i^{″}})$, ${(.)}^T{M}_{2}v$ is ${\overline{{\rho }^{\prime }}}_i$-${\mathbb{R}}_{+}$-convex and ${\mathcal{G}}_j$ is ${\nu }_j$-${\mathbb{R}}_{+}$-convex of second-order at $(z^{\prime },w_j^{\prime })$ in the direction $(u-z^{\prime },t_j-w_j^{\prime })$, respectively w.r.t. 1, on C. Suppose that there exist $z^{\ast }=(z_1^{\ast },\dots ,z_n^{\ast })\in {\mathbb{R}}_{+}^n$, with ${\displaystyle \sum _{i=1}^n{z}_i^{\ast }=1}$, $w^{\ast }=(w_1^{\ast },\dots ,w_m^{\ast })\in {\mathbb{R}}_{+}^m$, and $w_j^{\prime }\in {\mathcal{G}}_j\left(x^{\prime }\right)\cap (-{\mathbb{R}}_{+})$, $(j=1,\dots ,m)$, such that

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \sum _{i=1}^n{z}_i^{\ast }({\overrightarrow{d}}^2\mathcal{F}(.,\overline{w_i})(z^{\prime },\overline{x_i^{\prime }},u-z^{\prime },r_i-\overline{x_i^{\prime }})+M_{1}w\\ \hfill +y^{\prime }({\overrightarrow{d}}^2(-\mathcal{H})(.,\overline{w_i})(z^{\prime },-\overline{x_i^{″}},u-z^{\prime },s_i+\overline{x_i^{″}})+M_{2}v))(z-z^{\prime })\\ \hfill +\sum _{j=1}^m{w}_j^{\ast }{\overrightarrow{d}}^2{\mathcal{G}}_j(z^{\prime },w_j^{\prime },u-z^{\prime },t_j-w_j^{\prime })(z-z^{\prime })\ge 0,\mathrm{for}\mathrm{all}z\in C,\end{array}\end{array}$$

(3)

$$\begin{array}{c}\hfill \sum _{j=1}^m{w}_j^{\ast }{w}_j^{\prime }=0,\end{array}$$

(4)

$$\begin{array}{c}\hfill w^T{M}_{1}w\le 1,v^T{M}_{2}v\le 1,\end{array}$$

(5)

$$\begin{array}{c}\hfill {\left({z^{\prime }}^T{M}_1{z}^{\prime }\right)}^{\frac{1}{2}}={z^{\prime }}^T{M}_{1}w,\end{array}$$

(6)

$$\begin{array}{c}\hfill {\left({z^{\prime }}^T{M}_2{z}^{\prime }\right)}^{\frac{1}{2}}={z^{\prime }}^T{M}_{2}v,\end{array}$$

(7)

and

$$\begin{array}{c}\hfill \sum _{i=1}^n{z}_i^{\ast }\left({\rho }_i+\overline{{\rho }_i}+y^{\prime }({\rho }_i^{\prime }+\overline{{\rho }_i^{\prime }})\right)+\sum _{j=1}^m{w}_j^{\ast }{\nu }_j\ge 0.\end{array}$$

(8)

Then the point $(z^{\prime },y^{\prime })$ minimizes the problem (FMP).

Proof. 

Suppose that the point $(z^{\prime },y^{\prime })$ does not minimize the problem (FMP). Then there are $z\in S^{\prime }$ and $y=\max {\displaystyle \bigcup _{w\in D}}\Psi (z,w)$, with $z\ne z^{\prime }$, such that

$$y<y^{\prime }.$$

Let $y_i\in \Psi (z,\overline{w_i})$. Since $y=\max {\displaystyle \bigcup _{w\in D}}\Psi (z,w)$ and $\overline{w_i}\in B\left(z^{\prime }\right)\subseteq D$, we have

$$y_i\le y.$$

Hence,

$$y_i<y^{\prime }.$$

As $y^{\prime }=\frac{\overline{x_i^{\prime }}+{\left({z^{\prime }}^T{M}_1{z}^{\prime }\right)}^{\frac{1}{2}}}{\overline{x_i^{″}}-{\left({z^{\prime }}^T{M}_2{z}^{\prime }\right)}^{\frac{1}{2}}}$, we have

$$\begin{array}{c}\hfill \overline{x_i^{\prime }}+{\left({z^{\prime }}^T{M}_1{z}^{\prime }\right)}^{\frac{1}{2}}-y^{\prime }(\overline{x_i^{″}}-{\left({z^{\prime }}^T{M}_2{z}^{\prime }\right)}^{\frac{1}{2}})=0,\mathrm{for}\mathrm{all}i=1,\dots ,n.\end{array}$$

(9)

As $y_i\in \Psi (z,\overline{w_i})$, there are $x_i^{\prime }\in \mathcal{F}(z,\overline{w_i})$ and $x_i^{″}\in \mathcal{H}(z,\overline{w_i})$ such that

$$\begin{array}{c}\hfill y_i=\frac{x_i^{\prime }+{\left(z^T{M}_{1}z\right)}^{\frac{1}{2}}}{x_i^{″}-{\left(z^T{M}_{2}z\right)}^{\frac{1}{2}}}.\end{array}$$

Therefore,

$$\frac{x_i^{\prime }+{\left(z^T{M}_{1}z\right)}^{\frac{1}{2}}}{x_i^{″}-{\left(z^T{M}_{2}z\right)}^{\frac{1}{2}}}<y^{\prime }.$$

So,

$$\begin{array}{c}\hfill x_i^{\prime }+{\left(z^T{M}_{1}z\right)}^{\frac{1}{2}}-y^{\prime }(x_i^{″}-{\left(z^T{M}_{2}z\right)}^{\frac{1}{2}})<0,\mathrm{for}\mathrm{all}i=1,\dots ,n.\end{array}$$

(10)

Hence,

$$\begin{array}{cc}& \sum _{i=1}^n{z}_i^{\ast }\left(x_i^{\prime }+\left(z^T{M}_{1}w\right)-y^{\prime }(x_i^{″}-\left(z^T{M}_{2}v\right))\right)\hfill \\ & \le \sum _{i=1}^n{z}_i^{\ast }\left(x_i^{\prime }+{\left(z^T{M}_{1}z\right)}^{\frac{1}{2}}-y^{\prime }(x_i^{″}-{\left(z^T{M}_{2}z\right)}^{\frac{1}{2}})\right)\hfill \\ & <0\hfill \\ & =\sum _{i=1}^n{z}_i^{\ast }\left(\overline{x_i^{\prime }}+{\left({z^{\prime }}^T{M}_1{z}^{\prime }\right)}^{\frac{1}{2}}-y^{\prime }(\overline{x_i^{″}}-{\left({z^{\prime }}^T{M}_2{z}^{\prime }\right)}^{\frac{1}{2}})\right)\hfill \\ & =\sum _{i=1}^n{z}_i^{\ast }\left(\overline{x_i^{\prime }}+\left({z^{\prime }}^T{M}_{1}w\right)-y^{\prime }(\overline{x_i^{″}}-\left({z^{\prime }}^T{M}_{2}v\right))\right).\hfill \end{array}$$

Since $z\in S^{\prime }$, there is

$$w_j\in {\mathcal{G}}_j\left(z\right)\cap (-{\mathbb{R}}_{+}).$$

As $w_j^{\ast }\ge 0$$(j=1,\dots ,m)$,

$$w_j^{\ast }{w}_j\le 0,\mathrm{for}\mathrm{all}j,j=1,\dots ,m.$$

So,

$$\sum _{j=1}^m{w}_j^{\ast }{w}_j\le 0.$$

Again, from (4), we have

$$\sum _{j=1}^m{w}_j^{\ast }{w}_j^{\prime }=0.$$

Therefore,

$$\sum _{j=1}^m{w}_j^{\ast }{w}_j\le \sum _{j=1}^m{w}_j^{\ast }{w}_j^{\prime }.$$

Hence,

$$\begin{array}{c}\hfill \begin{array}{cc}& \sum _{i=1}^n{z}_i^{\ast }\left(x_i^{\prime }+\left(z^T{M}_{1}w\right)-y^{\prime }(x_i^{″}-\left(z^T{M}_{2}v\right))\right)+\sum _{j=1}^m{w}_j^{\ast }{w}_j\hfill \\ & <\sum _{i=1}^n{z}_i^{\ast }\left(\overline{x_i^{\prime }}+\left({z^{\prime }}^T{M}_{1}w\right)-y^{\prime }(\overline{x_i^{″}}-\left({z^{\prime }}^T{M}_{2}v\right))\right)+\sum _{j=1}^m{w}_j^{\ast }{w}_j^{\prime }.\hfill \end{array}\end{array}$$

(11)

As $\mathcal{F}(.,\overline{w_i})$ is ${\rho }_i$-${\mathbb{R}}_{+}$-convex of second-order at $(z^{\prime },\overline{x_i^{\prime }})$ in the direction $(u-z^{\prime },r_i-\overline{w_i^{\prime }})$ w.r.t. 1, on C and $\overline{x_i^{\prime }}\in \mathcal{F}(z^{\prime },\overline{w_i})$, we have

$$\mathcal{F}(z,\overline{w_i})-\overline{x_i^{\prime }}\subseteq {\overrightarrow{d}}^2\mathcal{F}(.,\overline{w_i})(z^{\prime },\overline{x_i^{\prime }},u-z^{\prime },r_i-\overline{x_i^{\prime }})(z-z^{\prime })+{\rho }_i{\parallel z-z^{\prime }\parallel }^2+{\mathbb{R}}_{+}.$$

Since $x_i^{\prime }\in \mathcal{F}(z,\overline{w_i})$, we have

$$\begin{array}{c}\hfill x_i^{\prime }-\overline{x_i^{\prime }}\in {\overrightarrow{d}}^2\mathcal{F}(.,\overline{w_i})(z^{\prime },\overline{x_i^{\prime }},u-z^{\prime },r_i-\overline{x_i^{\prime }})(z-z^{\prime })+{\rho }_i{\parallel z-z^{\prime }\parallel }^2+{\mathbb{R}}_{+}.\end{array}$$

(12)

Since ${(.)}^T{M}_{1}w$ is ${\overline{\rho }}_i$-${\mathbb{R}}_{+}$-convex w.r.t. 1, on C, we have

$$\begin{array}{c}\hfill z^T{M}_{1}w-{z^{\prime }}^T{M}_{1}w\ge M_{1}w(z-z^{\prime })+\overline{{\rho }_i}{\parallel z-z^{\prime }\parallel }^2+{\mathbb{R}}_{+}.\end{array}$$

(13)

As $-\mathcal{H}(.,\overline{w_i})$ is ${\rho }_i^{\prime }$-${\mathbb{R}}_{+}$-convex of second-order at $(z^{\prime },-\overline{x_i^{″}})$ in the direction $(u-z^{\prime },s_i+\overline{x_i^{″}})$ w.r.t. 1, on C and $\overline{x_i^{″}}\in \mathcal{H}(z^{\prime },\overline{w_i})$, we have

$$-\mathcal{H}(z,\overline{w_i})+\overline{x_i^{″}}\subseteq {\overrightarrow{d}}^2(-\mathcal{H})(.,\overline{w_i})(z^{\prime },-\overline{x_i^{″}},u-z^{\prime },s_i+\overline{x_i^{″}})(z-z^{\prime })+{\rho }_i^{\prime }{\parallel z-z^{\prime }\parallel }^2+{\mathbb{R}}_{+}.$$

Since $x_i^{″}\in \mathcal{H}(z,\overline{w_i})$, we have

$$\begin{array}{c}\hfill \begin{array}{c}\hfill -x_i^{″}+\overline{x_i^{″}}\in {\overrightarrow{d}}^2(-\mathcal{H})(.,\overline{w_i})(z^{\prime },-\overline{x_i^{″}},u-z^{\prime },s_i+\overline{x_i^{″}})(z-z^{\prime })\\ \hfill +{\rho }_i^{\prime }{\parallel z-z^{\prime }\parallel }^2+{\mathbb{R}}_{+}.\end{array}\end{array}$$

(14)

As ${(.)}^T{M}_{2}v$ is ${\overline{{\rho }^{\prime }}}_i$-${\mathbb{R}}_{+}$-convex w.r.t. 1, on C, we have

$$\begin{array}{c}\hfill z^T{M}_{2}v-{z^{\prime }}^T{M}_{2}v\ge M_{2}v(z-z^{\prime })+\overline{{\rho }_i^{\prime }}{\parallel z-z^{\prime }\parallel }^2+{\mathbb{R}}_{+}.\end{array}$$

(15)

As ${\mathcal{G}}_j$ is ${\nu }_j$-${\mathbb{R}}_{+}$-convex of second-order at $(z^{\prime },w_j^{\prime })$ in the direction $(u-z^{\prime },t_j-w_j^{\prime })$, respectively w.r.t. 1, on C and $w_j^{\prime }\in {\mathcal{G}}_j\left(z^{\prime }\right)\cap (-{\mathbb{R}}_{+})$, we have

$${\mathcal{G}}_j\left(z\right)-w_j^{\prime }\subseteq {\overrightarrow{d}}^2{\mathcal{G}}_j(z^{\prime },w_j^{\prime },u-z^{\prime },t_j-w_j^{\prime })(z-z^{\prime })+{\nu }_j{\parallel z-z^{\prime }\parallel }^2+{\mathbb{R}}_{+}.$$

Since $w_j\in {\mathcal{G}}_j\left(z\right)\cap (-{\mathbb{R}}_{+})$, we have

$$\begin{array}{c}\hfill w_j-w_j^{\prime }\in {\overrightarrow{d}}^2{\mathcal{G}}_j(z^{\prime },w_j^{\prime },u-z^{\prime },t_j-w_j^{\prime })(z-z^{\prime })+{\nu }_j{\parallel z-z^{\prime }\parallel }^2+{\mathbb{R}}_{+}.\end{array}$$

(16)

From (3), (8), and (12)–(16), we have

$$\begin{array}{c}\hfill \begin{array}{cc}& \sum _{i=1}^n{z}_i^{\ast }\left(x_i^{\prime }+\left(z^T{M}_{1}w\right)-y^{\prime }(x_i^{″}-\left(z^T{M}_{2}v\right))\right)+\sum _{j=1}^m{w}_j^{\ast }{w}_j\hfill \\ & \ge \sum _{i=1}^n{z}_i^{\ast }\left(\overline{x_i^{\prime }}+\left({z^{\prime }}^T{M}_{1}w\right)-y^{\prime }(\overline{x_i^{″}}-\left({z^{\prime }}^T{M}_{2}v\right))\right)+\sum _{j=1}^m{w}_j^{\ast }{w}_j^{\prime }.\hfill \end{array}\end{array}$$

It contradicts Equation (11). Hence, the point $(z^{\prime },y^{\prime })$ minimizes the problem (FMP). □

The succeeding example can be adopted to verify Theorem 3.

Example 1.

Suppose that for all $(z,w)\in {\mathbb{R}}^2\times {\mathbb{R}}^2$, the set-valued maps $\mathcal{F}:{\mathbb{R}}^2\times {\mathbb{R}}^2\to 2^{\mathbb{R}}$, $\mathcal{H}:{\mathbb{R}}^2\times {\mathbb{R}}^2\to 2^{\mathbb{R}}$, and $\mathcal{G}:{\mathbb{R}}^2\to 2^{{\mathbb{R}}^2}$ are defined as

$$\mathcal{F}(z,w)=\left\{\begin{array}{cc}\{\sqrt{x}-|z_1-z_2|+\parallel w\parallel :\sqrt{x}\ge \parallel z{\parallel }^2\},\hfill & if z\ne (0,0),z\ne (2,2),z=(z_1,z_2),\hfill \\ \{\sqrt{-x}+\parallel w\parallel :x\in [-4,0]\},\hfill & if z=(2,2),\hfill \\ \left\{0\right\},\hfill & if z=(0,0),\hfill \\ \{-|z_1-z_2\left|\right\},\hfill & if w=(0,0),\hfill \end{array}\right.$$

$$\mathcal{H}(z,w)=1,$$

and

$$\mathcal{G}\left(z\right)=\left\{\begin{array}{cc}\left\{\right(x^2+{\parallel z\parallel }^2,x^2+{\parallel z\parallel }^2):x\in \mathbb{R}\},\hfill & if z\ne (0,0),z\ne (2,2),\hfill \\ \left\{\right(x+4,x+4):x\ge 0\},\hfill & if z=(2,2),\hfill \\ \left\{\right(0,0\left)\right\},\hfill & if z=(0,0).\hfill \end{array}\right.$$

Consider two positive semi-definite (symmetric) real matrices

$$M_1=\left(\begin{array}{cc}1& -1\\ -1& 1\end{array}\right)$$

and

$$M_2=\left(\begin{array}{cc}0& 0\\ 0& 0\end{array}\right).$$

Let $C=[0,1]\times [0,1]$ and $D=[-1,0]\times [-1,0]$. Then C is a convex subset of ${\mathbb{R}}^2$ and D is a non-empty compact subset of ${\mathbb{R}}^2$. Clearly,

$$\mathcal{F}(z,w)+{\left(z^T{M}_{1}z\right)}^{\frac{1}{2}}\ge 0,\mathrm{for}\mathrm{all}(z,w)\in C\times D$$

and

$$\mathcal{H}(z,w)-{\left(z^T{M}_{2}z\right)}^{\frac{1}{2}}=1>0,\mathrm{for}\mathrm{all}(z,w)\in C\times D.$$

The set-valued map $\Psi :{\mathbb{R}}^2\times {\mathbb{R}}^2\to 2^{\mathbb{R}}$, defined as

$$\Psi (z,w)=\frac{\mathcal{F}(z,w)+{\left(z^T{M}_{1}z\right)}^{\frac{1}{2}}}{\mathcal{H}(z,w)-{\left(z^T{M}_{2}z\right)}^{\frac{1}{2}}},\mathrm{for}\mathrm{all}(z,w)\in C\times D,$$

can be considered as

$$\Psi (z,w)=\left\{\begin{array}{cc}\{\sqrt{x}+\parallel w\parallel :\sqrt{x}\ge \parallel z{\parallel }^2\},\hfill & if z\ne (0,0),z\ne (2,2),\hfill \\ \{\sqrt{-x}+\parallel w\parallel :x\in [-4,0]\},\hfill & if z=(2,2),\hfill \\ \left\{0\right\},\hfill & if z=(0,0),\hfill \\ \left\{0\right\},\hfill & if w=(0,0),\hfill \end{array}\right.$$

It is clear that for all $z\in C$, $\Psi (z,.):{\mathbb{R}}^2\to 2^{\mathbb{R}}$ is upper semicontinuous and ${\mathbb{R}}_{+}$-semicompact-valued on D. Let $z^{\prime }=(0,0)$, $y^{\prime }=\max {\displaystyle \bigcup _{w\in D}}\Psi (z^{\prime },w)=0$, and $w_j^{\prime }=0\in {\mathcal{G}}_j\left(z^{\prime }\right)\cap (-{\mathbb{R}}_{+})$, $(j=1,2)$. Obviously, $B\left(z^{\prime }\right)=D$. Let $n=2$, $w=(1,0)\in {\mathbb{R}}^2,v=(0,1)\in {\mathbb{R}}^2$, $\overline{w_1}=(-1,0)\in B\left(z^{\prime }\right)$, and $\overline{w_2}=(0,-1)\in B\left(z^{\prime }\right)$. Suppose that there exist $\overline{x_i^{\prime }}=0\in \mathcal{F}(z^{\prime },\overline{w_i})$ and $\overline{x_i^{″}}=1\in \mathcal{H}(z^{\prime },\overline{w_i})$ such that

$$\begin{array}{c}\hfill 0=y^{\prime }=\frac{\overline{x_i^{\prime }}+{\left({z^{\prime }}^T{M}_1{z}^{\prime }\right)}^{\frac{1}{2}}}{\overline{x_i^{″}}-{\left({z^{\prime }}^T{M}_2{z}^{\prime }\right)}^{\frac{1}{2}}}.\end{array}$$

Let $u=(1,1)\in C$, $r_i=3\in \mathcal{F}(u,\overline{w_i})+{\mathbb{R}}_{+}$, $s_i=-1\in (-\mathcal{H})(u,\overline{w_i})+{\mathbb{R}}_{+}$, $(i=1,2)$, and $t_j=2\in {\mathcal{G}}_j\left(u\right)$, $(j=1,2)$. Choose ${\rho }_i=-1$, ${\overline{\rho }}_i=0$, ${\rho }_i^{\prime }=0$, ${\overline{{\rho }^{\prime }}}_i=0$, and ${\nu }_j=1$. We can prove that $\mathcal{F}(.,\overline{w_i})$ is ${\rho }_i$-${\mathbb{R}}_{+}$-convex of second-order at $(z^{\prime },\overline{x_i^{\prime }})=((0,0),0)$ in the direction $(u-z^{\prime },r_i-\overline{x_i^{\prime }})=((1,1),3)$, ${(.)}^T{M}_{1}w$ is ${\overline{\rho }}_i$-${\mathbb{R}}_{+}$-convex, $-\mathcal{H}(.,\overline{w_i})$ is ${\rho }_i^{\prime }$-${\mathbb{R}}_{+}$-convex of second-order at $(z^{\prime },-\overline{x_i^{″}})=((0,0),-1)$ in the direction $(u-z^{\prime },s_i+\overline{x_i^{″}})=((1,1),0)$, ${(.)}^T{M}_{2}v$ is ${\overline{{\rho }^{\prime }}}_i$-${\mathbb{R}}_{+}$-convex and ${\mathcal{G}}_j$ is ${\nu }_j$-${\mathbb{R}}_{+}$-convex of second-order at $(z^{\prime },w_j^{\prime })=((0,0),0)$ in the direction $(u-z^{\prime },t_j-w_j^{\prime })=((1,1),2)$, respectively w.r.t. 1, on C. So, there exist $z^{\ast }=(z_1^{\ast },z_2^{\ast })=(\frac{1}{2},\frac{1}{2})\in {\mathbb{R}}_{+}^2$, with ${\displaystyle \sum _{i=1}^2{z}_i^{\ast }=1}$ and $w^{\ast }=(w_1^{\ast },w_2^{\ast })=(1,1)\in {\mathbb{R}}_{+}^2$ such that Equations (3)–(7) along with (8) are satisfied. Then the point $(z^{\prime },y^{\prime })=((0,0),0)$ minimizes the problem (FMP).

3.2. Mond-Weir Type Dual of Second-Order

Let us consider a Mond-Weir dual of second-order (MWD) associated with (FMP), assuming that $\mathcal{F}(.,\overline{w_i})$, $-\mathcal{H}(.,\overline{w_i})$ and ${\mathcal{G}}_j$ are contingent epiderivable of second-order, where $\overline{w_i}\in B\left(z^{\prime }\right)$ and $z^{\prime }\in C$.

$$\begin{array}{cccc}\hfill & \mathrm{maximize}\hfill & \hfill & y^{\prime }\hfill \\ \hfill & \mathrm{subject}\mathrm{to}\hfill \\ \hfill & \hfill & & \sum _{i=1}^n{z}_i^{\ast }({\overrightarrow{d}}^2\mathcal{F}(.,\overline{w_i})(z^{\prime },\overline{x_i^{\prime }},u-z^{\prime },r_i-\overline{x_i^{\prime }})+M_{1}w\hfill \\ \hfill & \hfill & & +y^{\prime }({\overrightarrow{d}}^2(-\mathcal{H})(.,\overline{w_i})(x^{\prime },-\overline{x_i^{″}},u-z^{\prime },s_i+\overline{x_i^{″}})+M_{2}v))(z-z^{\prime })\hfill \\ \hfill & \hfill & & +\sum _{j=1}^m{w}_j^{\ast }{\overrightarrow{d}}^2{\mathcal{G}}_j(z^{\prime },w_j^{\prime },u-z^{\prime },t_j-w_j^{\prime })(z-z^{\prime })\ge 0,\mathrm{for}\mathrm{all}z\in C,\hfill \\ \hfill & \hfill & & \mathrm{for}\mathrm{some}n\in \mathbb{N},(1\le n\le k+1),\overline{w_i}\in B\left(z^{\prime }\right),\overline{x_i^{\prime }}\in \mathcal{F}(z^{\prime },\overline{w_i}),\hfill \\ \hfill & \hfill & & \overline{x_i^{″}}\in \mathcal{H}(z^{\prime },\overline{w_i}),\mathrm{and}{y}^{\prime }=\frac{\overline{x_i^{\prime }}+{\left({z^{\prime }}^T{M}_1{z}^{\prime }\right)}^{\frac{1}{2}}}{\overline{x_i^{″}}-{\left({z^{\prime }}^T{M}_2{z}^{\prime }\right)}^{\frac{1}{2}}},\hfill \\ \hfill & \hfill & & \sum _{j=1}^m{w}_j^{\ast }{w}_j^{\prime }\ge 0,\hfill \\ \hfill & \hfill & & w^T{M}_{1}w\le 1,v^T{M}_{2}v\le 1,{\left({z^{\prime }}^T{M}_1{z}^{\prime }\right)}^{\frac{1}{2}}={z^{\prime }}^T{M}_{1}w,\hfill \\ \hfill & \hfill & & {\left({z^{\prime }}^T{M}_2{z}^{\prime }\right)}^{\frac{1}{2}}={z^{\prime }}^T{M}_{2}v,\mathrm{for}\mathrm{some}w,v\in {\mathbb{R}}^k,\hfill \\ \hfill & \hfill & & z^{\prime }\in C,y^{\prime }=\max {\displaystyle \bigcup _{w\in D}}\Psi (z^{\prime },w),w^{\prime }=(w_1^{\prime },\dots ,w_m^{\prime }),w_j^{\prime }\in {\mathcal{G}}_j\left(z^{\prime }\right),\hfill \\ \hfill & \hfill & & z^{\ast }=(z_1^{\ast },\dots ,z_n^{\ast }),w^{\ast }=(w_1^{\ast },\dots ,w_m^{\ast }),z_i^{\ast }\ge 0,w_j^{\ast }\ge 0,\mathrm{and}\sum _{i=1}^n{z}_i^{\ast }=1,\hfill \\ \hfill & \hfill & & \mathrm{where}i=1,\dots ,n\mathrm{and}j=1,\dots ,m.\hfill \end{array}$$

(MWD)

Definition 14.

A feasible solution $(z^{\prime },y^{\prime },w^{\prime },z^{\ast },w^{\ast })$ of (MWD) is stated to maximize the problem (MWD) if there is not any feasible solution $(z,y,w,z_1^{\ast },w_1^{\ast })$ of the problem (MWD) such that

$$y^{\prime }<y.$$

Theorem 4.

Suppose that C is a convex subset of ${\mathbb{R}}^k$, $z_0$ is a feasible solution of (FMP) and $(z^{\prime },y^{\prime },w^{\prime },z^{\ast },w^{\ast })$ is a feasible solution of the dual problem (MWD). Let $u\in C$, $r_i\in \mathcal{F}(u,\overline{w_i})+{\mathbb{R}}_{+}$, $s_i\in (-\mathcal{H})(u,\overline{w_i})+{\mathbb{R}}_{+}$, $(i=1,\dots ,n)$ and $t_j\in {\mathcal{G}}_j\left(u\right)$, $(j=1,\dots ,m)$. We suppose that $\mathcal{F}(.,\overline{w_i})$ is ${\rho }_i$-${\mathbb{R}}_{+}$-convex of second-order at $(z^{\prime },\overline{x_i^{\prime }})$ in the direction $(u-z^{\prime },r_i-\overline{x_i^{\prime }})$, ${(.)}^T{M}_{1}w$ is ${\overline{\rho }}_i$-${\mathbb{R}}_{+}$-convex, $-\mathcal{H}(.,\overline{w_i})$ is ${\rho }_i^{\prime }$-${\mathbb{R}}_{+}$-convex of second-order at $(z^{\prime },-\overline{x_i^{″}})$ in the direction $(u-z^{\prime },s_i+\overline{x_i^{″}})$, ${(.)}^T{M}_{2}v$ is ${\overline{{\rho }^{\prime }}}_i$-${\mathbb{R}}_{+}$-convex and ${\mathcal{G}}_j$ is ${\nu }_j$-${\mathbb{R}}_{+}$-convex of second-order at $(z^{\prime },w_j^{\prime })$ in the direction $(u-z^{\prime },t_j-w_j^{\prime })$, respectively w.r.t. 1, on C, satisfying

$$\begin{array}{c}\hfill \sum _{i=1}^n{z}_i^{\ast }\left({\rho }_i+\overline{{\rho }_i}+y^{\prime }({\rho }_i^{\prime }+\overline{{\rho }_i^{\prime }})\right)+\sum _{j=1}^m{w}_j^{\ast }{\nu }_j\ge 0.\end{array}$$

(17)

Then,

$$\max \bigcup _{w\in D}\Psi (z_0,w)\nless y^{\prime }.$$

Proof. 

Let us assume that for $y_0=\max {\displaystyle \bigcup _{w\in D}}\Psi (z_0,w)$,

$$y_0<y^{\prime }.$$

Let $y_i\in \Psi (z_0,\overline{w_i})$. Since $y_0=\max {\displaystyle \bigcup _{w\in D}}\Psi (z_0,w)$ and $\overline{w_i}\in B\left(z^{\prime }\right)\subseteq D$, we have

$$y_i\le y_0.$$

Hence,

$$y_i<y^{\prime }.$$

As $y^{\prime }=\frac{\overline{x_i^{\prime }}+{\left({z^{\prime }}^T{M}_1{z}^{\prime }\right)}^{\frac{1}{2}}}{\overline{x_i^{″}}-{\left({z^{\prime }}^T{M}_2{x}^{\prime }\right)}^{\frac{1}{2}}}$, we have

$$\begin{array}{c}\hfill \overline{x_i^{\prime }}+{\left({z^{\prime }}^T{M}_1{z}^{\prime }\right)}^{\frac{1}{2}}-y^{\prime }(\overline{x_i^{″}}-{\left({z^{\prime }}^T{M}_2{z}^{\prime }\right)}^{\frac{1}{2}})=0,\mathrm{for}\mathrm{all}i=1,\dots ,n.\end{array}$$

(18)

Since $y_i\in \Psi (z_0,\overline{w_i})$, there exist $x_i^{\prime }\in \mathcal{F}(z_0,\overline{w_i})$ and $x_i^{″}\in \mathcal{H}(z_0,\overline{w_i})$ such that

$$\begin{array}{c}\hfill y_i=\frac{x_i^{\prime }+{\left(z_0^T{M}_1{z}_0\right)}^{\frac{1}{2}}}{x_i^{″}-{\left(z_0^T{M}_2{z}_0\right)}^{\frac{1}{2}}}.\end{array}$$

Therefore,

$$\frac{x_i^{\prime }+{\left(z_0^T{M}_1{z}_0\right)}^{\frac{1}{2}}}{x_i^{″}-{\left(z_0^T{M}_2{z}_0\right)}^{\frac{1}{2}}}<y^{\prime }.$$

So,

$$\begin{array}{c}\hfill x_i^{\prime }+{\left(z_0^T{M}_1{z}_0\right)}^{\frac{1}{2}}-y^{\prime }(x_i^{″}-{\left(z_0^T{M}_2{z}_0\right)}^{\frac{1}{2}})<0,\mathrm{for}\mathrm{all}i=1,\dots ,n.\end{array}$$

(19)

We have

$$\begin{array}{cc}& \sum _{i=1}^n{z}_i^{\ast }\left(x_i^{\prime }+\left(z_0^T{M}_{1}w\right)-y^{\prime }(x_i^{″}-\left(z_0^T{M}_{2}v\right))\right)\hfill \\ & \le \sum _{i=1}^n{z}_i^{\ast }\left(x_i^{\prime }+{\left(z_0^T{M}_1{z}_0\right)}^{\frac{1}{2}}-y^{\prime }(x_i^{″}-{\left(z_0^T{M}_2{z}_0\right)}^{\frac{1}{2}})\right),\left[\mathrm{by}\mathrm{applying}\left(\mathrm{MWD}\right)\right]\hfill \\ & <0\hfill \\ & =\sum _{i=1}^n{z}_i^{\ast }\left(\overline{x_i^{\prime }}+{\left({z^{\prime }}^T{M}_1{z}^{\prime }\right)}^{\frac{1}{2}}-y^{\prime }(\overline{x_i^{″}}-{\left({z^{\prime }}^T{M}_2{z}^{\prime }\right)}^{\frac{1}{2}})\right)\hfill \\ & =\sum _{i=1}^n{z}_i^{\ast }\left(\overline{x_i^{\prime }}+\left({z^{\prime }}^T{M}_{1}w\right)-y^{\prime }(\overline{x_i^{″}}-\left({z^{\prime }}^T{M}_{2}v\right))\right),\left[\mathrm{by}\mathrm{applying}\mathrm{the}\mathrm{constraints}\mathrm{of}\left(\mathrm{MWD}\right)\right].\hfill \end{array}$$

As $z_0\in S^{\prime }$, there exists

$$w_j\in {\mathcal{G}}_j\left(z_0\right)\cap (-{\mathbb{R}}_{+}).$$

As $w_j^{\ast }\ge 0$$(j=1,\dots ,m)$, we have

$$w_j^{\ast }{w}_j\le 0,\mathrm{for}\mathrm{all}j,(j=1,\dots ,m).$$

So,

$$\sum _{j=1}^m{w}_j^{\ast }{w}_j\le 0.$$

By applying constraints of (MWD),

$$\sum _{j=1}^m{w}_j^{\ast }{w}_j^{\prime }\ge 0.$$

Therefore,

$$\sum _{j=1}^m{w}_j^{\ast }{w}_j\le \sum _{j=1}^m{w}_j^{\ast }{w}_j^{\prime }.$$

Hence,

$$\begin{array}{c}\hfill \begin{array}{cc}& \sum _{i=1}^n{z}_i^{\ast }\left(x_i^{\prime }+\left(z_0^T{M}_{1}w\right)-y^{\prime }(x_i^{″}-\left(z_0^T{M}_{2}v\right))\right)+\sum _{j=1}^m{w}_j^{\ast }{w}_j\hfill \\ & <\sum _{i=1}^n{z}_i^{\ast }\left(\overline{x_i^{\prime }}+\left({z^{\prime }}^T{M}_{1}w\right)-y^{\prime }(\overline{x_i^{″}}-\left({z^{\prime }}^T{M}_{2}v\right))\right)+\sum _{j=1}^m{w}_j^{\ast }{w}_j^{\prime }.\hfill \end{array}\end{array}$$

(20)

As $\mathcal{F}(.,\overline{w_i})$ is ${\rho }_i$-${\mathbb{R}}_{+}$-convex of second-order at $(z^{\prime },\overline{x_i^{\prime }})$ in the direction $(u-z^{\prime },r_i-\overline{x_i^{\prime }})$ w.r.t. 1, on C and $\overline{x_i^{\prime }}\in \mathcal{F}(z^{\prime },\overline{w_i})$, we have

$$\mathcal{F}(z_0,\overline{w_i})-\overline{x_i^{\prime }}\subseteq {\overrightarrow{d}}^2\mathcal{F}(.,\overline{w_i})(z^{\prime },\overline{x_i^{\prime }},u-z^{\prime },r_i-\overline{x_i^{\prime }})(z_0-z^{\prime })+{\rho }_i{\parallel z_0-z^{\prime }\parallel }^2+{\mathbb{R}}_{+}.$$

As $x_i^{\prime }\in \mathcal{F}(z_0,\overline{w_i})$, we have

$$\begin{array}{c}\hfill x_i^{\prime }-\overline{x_i^{\prime }}\in {\overrightarrow{d}}^2\mathcal{F}(.,\overline{w_i})(z^{\prime },\overline{x_i^{\prime }},u-z^{\prime },r_i-\overline{x_i^{\prime }})(z_0-z^{\prime })+{\rho }_i{\parallel z_0-z^{\prime }\parallel }^2+{\mathbb{R}}_{+}.\end{array}$$

(21)

Since ${(.)}^T{M}_{1}w$ is ${\overline{\rho }}_i$-${\mathbb{R}}_{+}$-convex w.r.t. 1, on C, we have

$$\begin{array}{c}\hfill z_0^T{M}_{1}w-{z^{\prime }}^T{M}_{1}w\ge M_{1}w(z_0-z^{\prime })+\overline{{\rho }_i}{\parallel z_0-z^{\prime }\parallel }^2+{\mathbb{R}}_{+}.\end{array}$$

(22)

As $-\mathcal{H}(.,\overline{w_i})$ is ${\rho }_i^{\prime }$-${\mathbb{R}}_{+}$-convex of second-order at $(z^{\prime },-\overline{x_i^{″}})$ in the direction $(u-z^{\prime },s_i+\overline{x_i^{″}})$ w.r.t. 1, on C and $\overline{x_i^{″}}\in \mathcal{H}(z^{\prime },\overline{w_i})$, we have

$$-\mathcal{H}(z_0,\overline{w_i})+\overline{x_i^{″}}\subseteq {\overrightarrow{d}}^2(-\mathcal{H})(.,\overline{w_i})(z^{\prime },-\overline{x_i^{″}},u-z^{\prime },s_i+\overline{x_i^{″}})(z_0-x^{\prime })+{\rho }_i^{\prime }{\parallel z_0-z^{\prime }\parallel }^2+{\mathbb{R}}_{+}.$$

As $x_i^{″}\in \mathcal{H}(z_0,\overline{w_i})$, we have

$$\begin{array}{c}\hfill \begin{array}{c}\hfill -x_i^{″}+\overline{x_i^{″}}\in {\overrightarrow{d}}^2(-\mathcal{H})(.,\overline{w_i})(z^{\prime },-\overline{x_i^{″}},u-z^{\prime },s_i+\overline{x_i^{″}})(z_0-z^{\prime })+\\ \hfill {\rho }_i^{\prime }{\parallel z_0-z^{\prime }\parallel }^2+{\mathbb{R}}_{+}.\end{array}\end{array}$$

(23)

Since ${(.)}^T{M}_{2}v$ is ${\overline{{\rho }^{\prime }}}_i$-${\mathbb{R}}_{+}$-convex w.r.t. 1, on C, we have

$$\begin{array}{c}\hfill z_0^T{M}_{2}v-{z^{\prime }}^T{M}_{1}v\ge M_{2}v(z_0-z^{\prime })+\overline{{\rho }_i^{\prime }}{\parallel z_0-z^{\prime }\parallel }^2+{\mathbb{R}}_{+}.\end{array}$$

(24)

As ${\mathcal{G}}_j$ is ${\nu }_j$-${\mathbb{R}}_{+}$-convex of second-order at $(z^{\prime },w_j^{\prime })$ in the direction $(u-z^{\prime },t_j-w_j^{\prime })$, respectively w.r.t. 1, on C and $w_j^{\prime }\in {\mathcal{G}}_j\left(z^{\prime }\right)\cap (-{\mathbb{R}}_{+})$, we have

$${\mathcal{G}}_j\left(z_0\right)-w_j^{\prime }\subseteq {\overrightarrow{d}}^2{\mathcal{G}}_j(z^{\prime },w_j^{\prime },u-z^{\prime },t_j-w_j^{\prime })(z_0-z^{\prime })+{\nu }_j{\parallel z_0-x^{\prime }\parallel }^2+{\mathbb{R}}_{+}.$$

Since $w_j\in {\mathcal{G}}_j\left(z_0\right)\cap (-{\mathbb{R}}_{+})$, we have

$$\begin{array}{c}\hfill w_j-w_j^{\prime }\in {\overrightarrow{d}}^2{\mathcal{G}}_j(z^{\prime },w_j^{\prime },u-z^{\prime },t_j-w_j^{\prime })(z_0-x^{\prime })+{\nu }_j{\parallel z_0-z^{\prime }\parallel }^2+{\mathbb{R}}_{+}.\end{array}$$

(25)

By applying the constraints of the dual problem (MWD) along with Equations (17), (21)–(25), we have

$$\begin{array}{c}\hfill \begin{array}{cc}& \sum _{i=1}^n{z}_i^{\ast }\left(x_i^{\prime }+\left(z_0^T{M}_{1}w\right)-y^{\prime }(x_i^{″}-\left(z_0^T{M}_{2}v\right))\right)+\sum _{j=1}^m{w}_j^{\ast }{w}_j\hfill \\ & \ge \sum _{i=1}^n{z}_i^{\ast }\left(\overline{x_i^{\prime }}+\left({z^{\prime }}^T{M}_{1}w\right)-y^{\prime }(\overline{x_i^{″}}-\left({z^{\prime }}^T{M}_{2}v\right))\right)+\sum _{j=1}^m{w}_j^{\ast }{w}_j^{\prime }.\hfill \end{array}\end{array}$$

It contradicts Equation (20). Therefore,

$$\max \bigcup _{w\in D}\Psi (z_0,w)\nless y^{\prime }.$$

□

Theorem 5.

Suppose that the point $(z^{\prime },y^{\prime })$ minimizes the problem (FMP) and $w_j^{\prime }\in {\mathcal{G}}_j\left(z^{\prime }\right)\cap (-{\mathbb{R}}_{+})$, $(j=1,\dots ,m)$. We suppose that for some $n\in \mathbb{N}$, $(1\le n\le k+1)$, $z_i^{\ast }\ge 0$, $\overline{w_i}\in B\left(z^{\prime }\right)$, $(i=1,\dots ,n)$ with ${\displaystyle \sum _{i=1}^n{z}_i^{\ast }=1}$ and $w_j^{\ast }\ge 0$, $(j=1,\dots ,m)$, the point $(z^{\prime },y^{\prime },w^{\prime },z^{\ast },w^{\ast })$ satisfies Equations (3)–(7). Then $(z^{\prime },y^{\prime },w^{\prime },z^{\ast },w^{\ast })$ is a feasible solution of the dual problem (MWD). If Theorem 4 is satisfied, then the point $(z^{\prime },y^{\prime },w^{\prime },z^{\ast },w^{\ast })$ maximizes the problem (MWD).

Proof. 

Since the point $(z^{\prime },y^{\prime },w^{\prime },z^{\ast },w^{\ast })$ satisfies Equations (3)–(7),

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \sum _{i=1}^n{z}_i^{\ast }({\overrightarrow{d}}^2\mathcal{F}(.,\overline{w_i})(z^{\prime },\overline{x_i^{\prime }},u-z^{\prime },r_i-\overline{x_i^{\prime }})+M_{1}w\\ \hfill +y^{\prime }({\overrightarrow{d}}^2(-H)(.,\overline{w_i})(z^{\prime },-\overline{x_i^{″}},u-z^{\prime },s_i+\overline{x_i^{″}})+M_{2}v))(z-z^{\prime })\\ \hfill +\sum _{j=1}^m{w}_j^{\ast }{\overrightarrow{d}}^2{\mathcal{G}}_j(z^{\prime },w_j^{\prime },u-z^{\prime },t_j-w_j^{\prime })(z-z^{\prime })\ge 0,\mathrm{for}\mathrm{all}z\in C,\end{array}\end{array}$$

$$\begin{array}{c}\hfill \sum _{j=1}^m{w}_j^{\ast }{w}_j^{\prime }=0,\end{array}$$

$$\begin{array}{c}\hfill w^T{M}_{1}w\le 1,v^T{M}_{2}v\le 1,\end{array}$$

$$\begin{array}{c}\hfill {\left({z^{\prime }}^T{M}_1{z}^{\prime }\right)}^{\frac{1}{2}}={z^{\prime }}^T{M}_{1}w,\end{array}$$

and

$$\begin{array}{c}\hfill {\left({z^{\prime }}^T{M}_2{z}^{\prime }\right)}^{\frac{1}{2}}={z^{\prime }}^T{M}_{2}v.\end{array}$$

So, $(z^{\prime },y^{\prime },w^{\prime },z^{\ast },w^{\ast })$ is a feasible solution of the dual problem (MWD). Suppose that Theorem 4 is satisfied and the point $(z^{\prime },y^{\prime },w^{\prime },z^{\ast },w^{\ast })$ does not maximize the problem (MWD). Assume that $(z,y,w,z_1^{\ast },w_1^{\ast })$ is a feasible solution for the dual problem (MWD) such that

$$y^{\prime }<y.$$

It contradicts Theorem 4. Hence, the point $(z^{\prime },y^{\prime },w^{\prime },z^{\ast },w^{\ast })$ maximizes the problem (MWD). □

Theorem 6.

Suppose that C is a convex subset of ${\mathbb{R}}^k$ and $(z^{\prime },y^{\prime },w^{\prime },z^{\ast },w^{\ast })$ is a feasible solution of (MWD). Let $u\in C$, $r_i\in \mathcal{F}(u,\overline{w_i})+{\mathbb{R}}_{+}$, $s_i\in (-\mathcal{H})(u,\overline{w_i})+{\mathbb{R}}_{+}$, $(i=1,\dots ,n)$ and $t_j\in {\mathcal{G}}_j\left(u\right)$, $(j=1,\dots ,m)$. We suppose that $\mathcal{F}(.,\overline{w_i})$ is ${\rho }_i$-${\mathbb{R}}_{+}$-convex of second-order at $(z^{\prime },\overline{x_i^{\prime }})$ in the direction $(u-z^{\prime },r_i-\overline{x_i^{\prime }})$, ${(.)}^T{M}_{1}w$ is ${\overline{\rho }}_i$-${\mathbb{R}}_{+}$-convex, $-\mathcal{H}(.,\overline{w_i})$ is ${\rho }_i^{\prime }$-${\mathbb{R}}_{+}$-convex of second-order at $(z^{\prime },-\overline{x_i^{″}})$ in the direction $(u-z^{\prime },s_i+\overline{x_i^{″}})$, ${(.)}^T{M}_{2}v$ is ${\overline{{\rho }^{\prime }}}_i$-${\mathbb{R}}_{+}$-convex and ${\mathcal{G}}_j$ is ${\nu }_j$-${\mathbb{R}}_{+}$-convex of second-order at $(z^{\prime },w_j^{\prime })$ in the direction $(u-z^{\prime },t_j-w_j^{\prime })$, respectively w.r.t. 1, on C, satisfying Equation (17). If $z^{\prime }$ is a feasible solution of (FMP), then the point $(z^{\prime },y^{\prime })$ minimizes the problem (FMP).

Proof. 

Let us assume that the point $(z^{\prime },y^{\prime })$ does not minimize the problem (FMP). Then there is $z\in S^{\prime }$ and $y=\max {\displaystyle \bigcup _{w\in D}}\Psi (z,w)$, with $y\ne y^{\prime }$, such that

$$y<y^{\prime }.$$

Let $y_i\in \Psi (z,\overline{w_i})$. Since $y=\max {\displaystyle \bigcup _{w\in D}}\Psi (z,w)$ and $\overline{w_i}\in B\left(z^{\prime }\right)\subseteq D$,

$$y_i\le y.$$

Hence,

$$y_i<y^{\prime }.$$

As $y^{\prime }=\frac{\overline{x_i^{\prime }}+{\left({z^{\prime }}^T{M}_1{z}^{\prime }\right)}^{\frac{1}{2}}}{\overline{x_i^{″}}-{\left({z^{\prime }}^T{M}_2{z}^{\prime }\right)}^{\frac{1}{2}}}$, we have

$$\begin{array}{c}\hfill \overline{x_i^{\prime }}+{\left({z^{\prime }}^T{M}_1{z}^{\prime }\right)}^{\frac{1}{2}}-y^{\prime }(\overline{x_i^{″}}-{\left({z^{\prime }}^T{M}_2{z}^{\prime }\right)}^{\frac{1}{2}})=0,\mathrm{for}\mathrm{all}i=1,\dots ,n.\end{array}$$

(26)

Since $y_i\in \Psi (z,\overline{w_i})$, there are $x_i^{\prime }\in \mathcal{F}(z,\overline{w_i})$ and $x_i^{″}\in \mathcal{H}(z,\overline{w_i})$ such that

$$\begin{array}{c}\hfill y_i=\frac{x_i^{\prime }+{\left(z^T{M}_{1}z\right)}^{\frac{1}{2}}}{x_i^{″}-{\left(z^T{M}_{2}z\right)}^{\frac{1}{2}}}.\end{array}$$

Therefore,

$$\frac{x_i^{\prime }+{\left(z^T{M}_{1}z\right)}^{\frac{1}{2}}}{x_i^{″}-{\left(z^T{M}_{2}z\right)}^{\frac{1}{2}}}<y^{\prime }.$$

So,

$$\begin{array}{c}\hfill x_i^{\prime }+{\left(z^T{M}_{1}z\right)}^{\frac{1}{2}}-y^{\prime }(x_i^{″}-{\left(z^T{M}_{2}z\right)}^{\frac{1}{2}})<0,\mathrm{for}\mathrm{all}i=1,\dots ,n.\end{array}$$

(27)

Hence,

$$\begin{array}{cc}& \sum _{i=1}^n{z}_i^{\ast }\left(x_i^{\prime }+\left(z^T{M}_{1}w\right)-y^{\prime }(x_i^{″}-\left(z^T{M}_{2}v\right))\right)\hfill \\ & \le \sum _{i=1}^n{z}_i^{\ast }\left(x_i^{\prime }+{\left(z^T{M}_{1}z\right)}^{\frac{1}{2}}-y^{\prime }(x_i^{″}-{\left(z^T{M}_{2}z\right)}^{\frac{1}{2}})\right),\left[\mathrm{by}\mathrm{applying}\left(\mathrm{MWD}\right)\right]\hfill \\ & <0\hfill \\ & =\sum _{i=1}^n{z}_i^{\ast }\left(\overline{x_i^{\prime }}+{\left({z^{\prime }}^T{M}_1{z}^{\prime }\right)}^{\frac{1}{2}}-y^{\prime }(\overline{x_i^{″}}-{\left({z^{\prime }}^T{M}_2{z}^{\prime }\right)}^{\frac{1}{2}})\right)\hfill \\ & =\sum _{i=1}^n{z}_i^{\ast }\left(\overline{x_i^{\prime }}+\left({z^{\prime }}^T{M}_{1}w\right)-y^{\prime }(\overline{x_i^{″}}-\left({z^{\prime }}^T{M}_{2}v\right))\right),\left[\mathrm{by}\mathrm{applying}\left(\mathrm{MWD}\right)\right].\hfill \end{array}$$

As $z\in S^{\prime }$, there is

$$w_j\in {\mathcal{G}}_j\left(z\right)\cap (-{\mathbb{R}}_{+}).$$

Since $w_j^{\ast }\ge 0$$(j=1,\dots ,m)$,

$$w_j^{\ast }{w}_j\le 0,\mathrm{for}\mathrm{all}j,j=1,\dots ,m.$$

So,

$$\sum _{j=1}^m{w}_j^{\ast }{w}_j\le 0.$$

By applying the constraints of the dual problem (MWD), we have

$$\sum _{j=1}^m{w}_j^{\ast }{w}_j^{\prime }\ge 0.$$

Therefore,

$$\sum _{j=1}^m{w}_j^{\ast }{w}_j\le \sum _{j=1}^m{w}_j^{\ast }{w}_j^{\prime }.$$

Hence,

$$\begin{array}{c}\hfill \begin{array}{cc}& \sum _{i=1}^n{z}_i^{\ast }\left(x_i^{\prime }+\left(z^T{M}_{1}w\right)-y^{\prime }(x_i^{″}-\left(z^T{M}_{2}v\right))\right)+\sum _{j=1}^m{w}_j^{\ast }{w}_j\hfill \\ & <\sum _{i=1}^n{z}_i^{\ast }\left(\overline{x_i^{\prime }}+\left({z^{\prime }}^T{M}_{1}w\right)-y^{\prime }(\overline{x_i^{″}}-\left({z^{\prime }}^T{M}_{2}v\right))\right)+\sum _{j=1}^m{w}_j^{\ast }{w}_j^{\prime }.\hfill \end{array}\end{array}$$

(28)

As $\mathcal{F}(.,\overline{w_i})$ is ${\rho }_i$-${\mathbb{R}}_{+}$-convex of second-order at $(z^{\prime },\overline{x_i^{\prime }})$ in the direction $(u-z^{\prime },r_i-\overline{x_i^{\prime }})$ w.r.t. 1, on C and $\overline{x_i^{\prime }}\in \mathcal{F}(z^{\prime },\overline{w_i})$, we have

$$\mathcal{F}(z,\overline{w_i})-\overline{x_i^{\prime }}\subseteq {\overrightarrow{d}}^2\mathcal{F}(.,\overline{w_i})(z^{\prime },\overline{x_i^{\prime }},u-z^{\prime },r_i-\overline{x_i^{\prime }})(z-z^{\prime })+{\rho }_i{\parallel z-z^{\prime }\parallel }^2+{\mathbb{R}}_{+}.$$

As $x_i^{\prime }\in \mathcal{F}(z,\overline{w_i})$, we have

$$\begin{array}{c}\hfill x_i^{\prime }-\overline{x_i^{\prime }}\in {\overrightarrow{d}}^2\mathcal{F}(.,\overline{w_i})(z^{\prime },\overline{x_i^{\prime }},u-z^{\prime },r_i-\overline{x_i^{\prime }})(z-z^{\prime })+{\rho }_i{\parallel z-z^{\prime }\parallel }^2+{\mathbb{R}}_{+}.\end{array}$$

(29)

Since ${(.)}^T{M}_{1}w$ is ${\overline{\rho }}_i$-${\mathbb{R}}_{+}$-convex w.r.t. 1, on C, we have

$$\begin{array}{c}\hfill z^T{M}_{1}w-{z^{\prime }}^T{M}_{1}w\ge M_{1}w(z-z^{\prime })+\overline{{\rho }_i}{\parallel z-z^{\prime }\parallel }^2+{\mathbb{R}}_{+}.\end{array}$$

(30)

As $-\mathcal{H}(.,\overline{w_i})$ is ${\rho }_i^{\prime }$-${\mathbb{R}}_{+}$-convex of second-order at $(z^{\prime },-\overline{x_i^{″}})$ in the direction $(u-z^{\prime },s_i+\overline{x_i^{″}})$ w.r.t. 1, on C and $\overline{x_i^{″}}\in \mathcal{H}(z^{\prime },\overline{w_i})$, we have

$$-\mathcal{H}(z,\overline{w_i})+\overline{x_i^{″}}\subseteq {\overrightarrow{d}}^2(-\mathcal{H})(.,\overline{w_i})(z^{\prime },-\overline{x_i^{″}},u-z^{\prime },s_i+\overline{x_i^{″}})(z-z^{\prime })+{\rho }_i^{\prime }{\parallel z-z^{\prime }\parallel }^2+{\mathbb{R}}_{+}.$$

As $x_i^{″}\in \mathcal{H}(z,\overline{w_i})$, we have

$$\begin{array}{c}\hfill \begin{array}{c}\hfill -x_i^{″}+\overline{x_i^{″}}\in {\overrightarrow{d}}^2(-\mathcal{H})(.,\overline{w_i})(z^{\prime },-\overline{x_i^{″}},u-z^{\prime },s_i+\overline{x_i^{″}})(z-z^{\prime })\\ \hfill +{\rho }_i^{\prime }{\parallel z-z^{\prime }\parallel }^2+{\mathbb{R}}_{+}.\end{array}\end{array}$$

(31)

As ${(.)}^T{M}_{2}v$ is ${\overline{{\rho }^{\prime }}}_i$-${\mathbb{R}}_{+}$-convex w.r.t. 1, on C, we have

$$\begin{array}{c}\hfill z^T{M}_{2}v-{z^{\prime }}^T{M}_{1}v\ge M_{2}v(z-z^{\prime })+\overline{{\rho }_i^{\prime }}{\parallel z-z^{\prime }\parallel }^2+{\mathbb{R}}_{+}.\end{array}$$

(32)

As ${\mathcal{G}}_j$ is ${\nu }_j$-${\mathbb{R}}_{+}$-convex of second-order at $(z^{\prime },w_j^{\prime })$ in the direction $(u-z^{\prime },t_j-w_j^{\prime })$, respectively w.r.t. 1, on C and $w_j^{\prime }\in {\mathcal{G}}_j\left(z^{\prime }\right)\cap (-{\mathbb{R}}_{+})$, we have

$${\mathcal{G}}_j\left(z\right)-w_j^{\prime }\subseteq {\overrightarrow{d}}^2{\mathcal{G}}_j(z^{\prime },w_j^{\prime },u-z^{\prime },t_j-w_j^{\prime })(z-z^{\prime })+{\nu }_j{\parallel z-z^{\prime }\parallel }^2+{\mathbb{R}}_{+}.$$

Since $w_j\in {\mathcal{G}}_j\left(z\right)\cap (-{\mathbb{R}}_{+})$, we have

$$\begin{array}{c}\hfill w_j-w_j^{\prime }\in {\overrightarrow{d}}^2{\mathcal{G}}_j(z^{\prime },w_j^{\prime },u-z^{\prime },t_j-w_j^{\prime })(z-z^{\prime })+{\nu }_j{\parallel z-z^{\prime }\parallel }^2+{\mathbb{R}}_{+}.\end{array}$$

(33)

By applying the constraints of the dual problem (MWD) along with Equations (17), (29)–(33), we have

$$\begin{array}{c}\hfill \begin{array}{cc}& \sum _{i=1}^n{z}_i^{\ast }\left(x_i^{\prime }+\left(z^T{M}_{1}w\right)-y^{\prime }(x_i^{″}-\left(z^T{M}_{2}v\right))\right)+\sum _{j=1}^m{w}_j^{\ast }{w}_j\hfill \\ & \ge \sum _{i=1}^n{z}_i^{\ast }\left(\overline{x_i^{\prime }}+\left({z^{\prime }}^T{M}_{1}w\right)-y^{\prime }(\overline{x_i^{″}}-\left({z^{\prime }}^T{M}_{2}v\right))\right)+\sum _{j=1}^m{w}_j^{\ast }{w}_j^{\prime },\hfill \end{array}\end{array}$$

which contradicts (28). Hence, the point $(z^{\prime },y^{\prime })$ minimizes the problem (FMP). □

3.3. Wolfe Type Dual of Second-Order

Let us consider a Wolfe type dual of second-order (WD) associated with (FMP), assuming that $\mathcal{F}(.,\overline{w_i})$, $-\mathcal{H}(.,\overline{w_i})$ and ${\mathcal{G}}_j$ are contingent epiderivable of second-order, where $\overline{w_i}\in B\left(z^{\prime }\right)$ and $z^{\prime }\in C$.

$$\begin{array}{cccc}\hfill & \mathrm{maximize}\hfill & \hfill & y^{\prime }+\sum _{j=1}^m{w}_j^{\ast }{w}_j^{\prime }\hfill \\ \hfill & \mathrm{subject}\mathrm{to}\hfill \\ \hfill & \hfill & & \sum _{i=1}^n{z}_i^{\ast }({\overrightarrow{d}}^2\mathcal{F}(.,\overline{w_i})(z^{\prime },\overline{x_i^{\prime }},u-z^{\prime },r_i-\overline{x_i^{\prime }})+M_{1}w\hfill \\ \hfill & \hfill & & +y^{\prime }({\overrightarrow{d}}^2(-\mathcal{H})(.,\overline{w_i})(z^{\prime },-\overline{x_i^{″}},u-z^{\prime },s_i+\overline{x_i^{″}})+M_{2}v))(z-z^{\prime })\hfill \\ \hfill & \hfill & & +\sum _{j=1}^m{w}_j^{\ast }{\overrightarrow{d}}^2{\mathcal{G}}_j(z^{\prime },w_j^{\prime },u-z^{\prime },t_j-w_j^{\prime })(z-z^{\prime })\ge 0,\mathrm{for}\mathrm{all}z\in C,\hfill \\ \hfill & \hfill & & \mathrm{for}\mathrm{some}n\in \mathbb{N},(1\le n\le k+1),\overline{w_i}\in B\left(z^{\prime }\right),\overline{x_i^{\prime }}\in \mathcal{F}(z^{\prime },\overline{w_i}),\hfill \\ \hfill & \hfill & & \overline{x_i^{″}}\in \mathcal{H}(z^{\prime },\overline{w_i}),\mathrm{and}{y}^{\prime }=\frac{\overline{x_i^{\prime }}+{\left({z^{\prime }}^T{M}_1{z}^{\prime }\right)}^{\frac{1}{2}}}{\overline{x_i^{″}}-{\left(z^{\prime T}{M}_2{z}^{\prime }\right)}^{\frac{1}{2}}},\hfill \\ \hfill & \hfill & & w^T{M}_{1}w\le 1,v^T{M}_{2}v\le 1,{\left({z^{\prime }}^T{M}_1{z}^{\prime }\right)}^{\frac{1}{2}}={z^{\prime }}^T{M}_{1}w,\hfill \\ \hfill & \hfill & & {\left({z^{\prime }}^T{M}_2{z}^{\prime }\right)}^{\frac{1}{2}}={z^{\prime }}^T{M}_{2}v,\mathrm{for}\mathrm{some}w,v\in {\mathbb{R}}^k,\hfill \\ \hfill & \hfill & & z^{\prime }\in C,y^{\prime }=\max \bigcup _{w\in D}\Psi (z^{\prime },w),w^{\prime }=(w_1^{\prime },\dots ,w_m^{\prime }),w_j^{\prime }\in {\mathcal{G}}_j\left(z^{\prime }\right),\hfill \\ \hfill & \hfill & & z^{\ast }=(z_1^{\ast },\dots ,z_n^{\ast }),w^{\ast }=(w_1^{\ast },\dots ,w_m^{\ast }),z_i^{\ast }\ge 0,w_j^{\ast }\ge 0,\mathrm{and}\sum _{i=1}^n{z}_i^{\ast }=1,\hfill \\ \hfill & \hfill & & \mathrm{where}i=1,\dots ,n\mathrm{and}j=1,\dots ,m.\hfill \end{array}$$

(WD)

Definition 15.

A feasible solution $(z^{\prime },y^{\prime },w^{\prime },z^{\ast },w^{\ast })$ of the dual problem (WD) is stated to maximize the problem (WD) if there is not any feasible solution $(z,y,w,{\overline{z}}^{\ast },{\overline{w}}^{\ast })$ of the dual problem (WD) such that

$$y^{\prime }+\sum _{j=1}^m{w}_j^{\ast }{w}_j^{\prime }<y+\sum _{j=1}^m{\overline{w}}_j^{\ast }{w}_j,$$

where $w=(w_1,\dots ,w_m)$, $w^{\prime }=(w_1^{\prime },\dots ,w_m^{\prime })$, $w^{\ast }=(w_1^{\ast },\dots ,w_m^{\ast })$, and ${\overline{w}}^{\ast }=({\overline{w}}_1^{\ast },\dots ,{\overline{w}}_m^{\ast })$.

We can prove the theorems of duality of Wolfe kind. The proofs resemble those of Theorems 4–6. As a result, we do not include it.

Theorem 7.

Suppose that C is a convex subset of ${\mathbb{R}}^k$, $z_0$ is a feasible solution of (FMP) and $(z^{\prime },y^{\prime },w^{\prime },z^{\ast },w^{\ast })$ is a feasible solution of (WD). Let $u\in C$, $r_i\in \mathcal{F}(u,\overline{w_i})+{\mathbb{R}}_{+}$, $s_i\in (-\mathcal{H})(u,\overline{w_i})+{\mathbb{R}}_{+}$, $(i=1,\dots ,n)$ and $t_j\in {\mathcal{G}}_j\left(u\right)$, $(j=1,\dots ,m)$. We suppose that $\mathcal{F}(.,\overline{w_i})$ is ${\rho }_i$-${\mathbb{R}}_{+}$-convex of second-order at $(z^{\prime },\overline{x_i^{\prime }})$ in the direction $(u-z^{\prime },r_i-\overline{x_i^{\prime }})$, ${(.)}^T{M}_{1}w$ is ${\overline{\rho }}_i$-${\mathbb{R}}_{+}$-convex, $-\mathcal{H}(.,\overline{w_i})$ is ${\rho }_i^{\prime }$-${\mathbb{R}}_{+}$-convex of second-order at $(z^{\prime },-\overline{x_i^{″}})$ in the direction $(u-z^{\prime },s_i+\overline{x_i^{″}})$, ${(.)}^T{M}_{2}v$ is ${\overline{{\rho }^{\prime }}}_i$-${\mathbb{R}}_{+}$-convex and ${\mathcal{G}}_j$ is ${\nu }_j$-${\mathbb{R}}_{+}$-convex of second-order at $(z^{\prime },w_j^{\prime })$ in the direction $(u-z^{\prime },t_j-w_j^{\prime })$, respectively w.r.t. 1, on C, satisfying Equation (17). Then,

$$\max \bigcup _{w\in D}\Psi (z_0,w)\nless y^{\prime }+\sum _{j=1}^m{w}_j^{\ast }{w}_j^{\prime }.$$

Theorem 8.

Suppose that the point $(z^{\prime },y^{\prime })$ minimizes the problem (FMP) and $w_j^{\prime }\in {\mathcal{G}}_j\left(z^{\prime }\right)\cap (-{\mathbb{R}}_{+})$, $(j=1,\dots ,m)$. We suppose that for some $n\in \mathbb{N}$, $(1\le n\le k+1)$, $z_i^{\ast }\ge 0$, $\overline{w_i}\in B\left(z^{\prime }\right)$, $(i=1,\dots ,n)$ with ${\displaystyle \sum _{i=1}^n{z}_i^{\ast }=1}$ and $w_j^{\ast }\ge 0$, $(j=1,\dots ,m)$, the point $(z^{\prime },y^{\prime },w^{\prime },z^{\ast },w^{\ast })$ satisfies Equations (3)–(7). Then the point $(z^{\prime },y^{\prime },w^{\prime },z^{\ast },w^{\ast })$ is a feasible solution of the dual problem (WD). If Theorem 7 is satisfied, then the point $(z^{\prime },y^{\prime },w^{\prime },z^{\ast },w^{\ast })$ maximizes the problem (WD).

Theorem 9.

Suppose that C is a convex subset of ${\mathbb{R}}^k$ and $(z^{\prime },y^{\prime },w^{\prime },z^{\ast },w^{\ast })$ is a feasible solution of (WD), along with ${\displaystyle \sum _{j=1}^m{w}_j^{\ast }{w}_j^{\prime }\ge 0}$. Let $u\in C$, $r_i\in \mathcal{F}(u,\overline{w_i})+{\mathbb{R}}_{+}$, $s_i\in (-\mathcal{H})(u,\overline{w_i})+{\mathbb{R}}_{+}$, $(i=1,\dots ,n)$ and $t_j\in {\mathcal{G}}_j\left(u\right)$, $(j=1,\dots ,m)$. We suppose that $\mathcal{F}(.,\overline{w_i})$ is ${\rho }_i$-${\mathbb{R}}_{+}$-convex of second-order at $(z^{\prime },\overline{x_i^{\prime }})$ in the direction $(u-z^{\prime },r_i-\overline{x_i^{\prime }})$, ${(.)}^T{M}_{1}w$ is ${\overline{\rho }}_i$-${\mathbb{R}}_{+}$-convex, $-\mathcal{H}(.,\overline{w_i})$ is ${\rho }_i^{\prime }$-${\mathbb{R}}_{+}$-convex of second-order at $(z^{\prime },-\overline{x_i^{″}})$ in the direction $(u-z^{\prime },s_i+\overline{x_i^{″}})$, ${(.)}^T{M}_{2}v$ is ${\overline{{\rho }^{\prime }}}_i$-${\mathbb{R}}_{+}$-convex and ${\mathcal{G}}_j$ is ${\nu }_j$-${\mathbb{R}}_{+}$-convex of second-order at $(z^{\prime },w_j^{\prime })$ in the direction $(u-z^{\prime },t_j-w_j^{\prime })$, respectively w.r.t. 1, on C, satisfying Equation (17). If $z^{\prime }$ is a feasible solution of (FMP), then the point $(z^{\prime },y^{\prime })$ minimizes the problem (FMP).

4. Conclusions

We deal with a set-valued fractional minimax problem where set-valued maps are used for the constraint and objective functions. We use the idea of second-order $\rho$-cone convexity of set-valued maps as a generalization of cone convex set-valued maps. We investigate the required conditions of the KKT type of second-order for the set-valued fractional minimax problem and prove the theorems of duality of second-order Mond-Weir and Wolfe types in terms of contingent epi-derivative, as well as cone convexity in its broad sense. As a special case, our results reduce to the existing ones of scalar-valued fractional minimax problems. One can study conditions of optimality for approximate solutions and symmetric duals of the set-valued fractional minimax problem under second-order $\rho$-cone convexity suppositions. Under second-order $\rho$-cone convexity suppositions, one can study variational inequality, complementarity problems, and control problems in the setting of set-valued maps. These types of problems can also be studied for approximate quasi efficient solutions.