A Minimax-Program-Based Approach for Robust Fractional Multi-Objective Optimization

Abstract

In this paper, by making use of some advanced tools from variational analysis and generalized differentiation, we establish necessary optimality conditions for a class of robust fractional minimax programming problems. Sufficient optimality conditions for the considered problem are also obtained by means of generalized convex functions. Additionally, we formulate a dual problem to the primal one and examine duality relations between them. In our results, by using the obtained results, we obtain necessary and sufficient optimality conditions for a class of robust fractional multi-objective optimization problems.

1. Introduction

A fractional multi-objective optimization problem with locally Lipschitzian data in the face of data uncertainty in the constraints is of the form

(1)${\mathrm{Min}}_{{\mathbb{R}}_{+}^l}\left\{f\left(x\right)\right|g_j(x,v_j)\le 0,j\in J\},$

where $x\in {\mathbb{R}}^n$ is the vector of decision variables, $f\left(x\right):=(f_1\left(x\right),\dots ,f_l\left(x\right)),$ $f_k:={\displaystyle \frac{p_k}{q_k}},$ $k\in K:=\{1,\dots ,l\},$ and $v_j\in {\mathcal{V}}_j,$ $j\in J:=\{1,\dots ,m\}$ are uncertain parameters, ${\mathcal{V}}_j\subset {\mathbb{R}}^q,$ $j\in J$ are uncertainty sets; let $p_k:{\mathbb{R}}^n\to \mathbb{R},$ $q_k:{\mathbb{R}}^n\to \mathbb{R},$ $k\in K:=\{1,\dots ,l\}$ be locally Lipschitz functions and let $g_j:{\mathbb{R}}^n\to \mathbb{R},$ $j\in J$ be given functions.

We will treat the problem (1) via a robust optimization approach (see, e.g., [1,2,3,4,5,6,7,8,9]), by following its robust counterpart as shown below,

(2)$\begin{array}{c}\hfill {\mathrm{Min}}_{{\mathbb{R}}_{+}^l}\left\{f\left(x\right)\right|x\in F_r\},\end{array}$

where $F_r$ is the feasible set of the problem (2), defined by

(3)$\begin{array}{c}\hfill F_r=\{x\in {\mathbb{R}}^n|g_j(x,v_j)\le 0,\forall v_j\in {\mathcal{V}}_j,j\in J\}.\end{array}$

In what follows, for the sake of convenience, we assume that $q_k\left(x\right)>0,$ $k\in K$ for all $x\in {\mathbb{R}}^n$ and that $p_k\left(\overline{x}\right)\le 0,$ $k\in K$ for the reference point $\overline{x}\in F_r.$ Furthermore, denote $g:=(g_1,\dots ,g_m)$ for simplicity.

In this paper, we mainly focus on the study of optimality conditions for a weak Pareto solution of the problem (2) by taking the associated minimax optimization problem into account (see, e.g., [10,11,12,13,14,15,16,17,18,19,20,21] for more related works). Let us consider the following robust fractional minimax programming problem,

(4)$\begin{array}{c}\hfill \underset{x\in F_r}{min}\underset{k\in K}{max}{\displaystyle \frac{p_k\left(x\right)}{q_k\left(x\right)}},\end{array}$

where $F_r$ is the feasible set that is the same as defined by the problem (2).

Among others, we first establish the necessary optimality conditions for the problem (4) and its sufficient optimality conditions by means of generalized convex functions. Then, we formulate a dual problem to the problem (4) and examine duality relations between them. Finally, by using the obtained results, we obtain necessary and sufficient optimality conditions for the problem (2), the minimax program-based approach is different from some works in the literature; see, e.g., [22,23,24,25]. The main tools are from variational analysis and generalized differentiation; see, e.g., [26,27].

2. Preliminaries

In this section, we recall some notation and preliminary results that will be used in this paper; see e.g., [26,27]. Let ${\mathbb{R}}^n$ denote the Euclidean space equipped with the usual Euclidean norm $\parallel ·\parallel .$ The nonnegative orthant of ${\mathbb{R}}^n$ is denoted by ${\mathbb{R}}_{+}^n.$ As usual, the polar cone of a set $\mathsf{\Omega }\subset {\mathbb{R}}^n$ is defined by

(5)$\begin{array}{c}\hfill {\mathsf{\Omega }}^{\circ }:=\{y\in {\mathbb{R}}^n|⟨y,x⟩\le 0\mathrm{for}\mathrm{all}x\in \mathsf{\Omega }\}.\end{array}$

A given set-valued mapping $F:\mathsf{\Omega }\subset {\mathbb{R}}^n\rightrightarrows {\mathbb{R}}^m$ is said to be closed at $\overline{x}\in \mathsf{\Omega }$ if for any sequence $\left\{x_k\right\}\subset \mathsf{\Omega },$ $x_k\to \overline{x},$ and for any sequence $\left\{y_k\right\}\subset {\mathbb{R}}^m,$ $y_k\to \overline{y},$ one has $\overline{y}\in F\left(\overline{x}\right).$

Given a multifuction $F:{\mathbb{R}}^n\rightrightarrows {\mathbb{R}}^m$ with values $F\left(x\right)\subset {\mathbb{R}}^m$ in the collection of all the subsets of ${\mathbb{R}}^m$ (and similarly, of course, in infinite dimensions). The limiting construction

$\begin{array}{cc}\hfill \underset{x\to \overline{x}}{Limsup}F\left(x\right):=\{y\in {\mathbb{R}}^m|& \exists x_k\to \overline{x},y_k\to y\hfill \\ \hfill & \mathrm{with}{y}_k\in F\left(x_k\right)\mathrm{for}\mathrm{all}k\in \mathbb{N}:=\{1,2,\dots \}\}\hfill \end{array}$

is known as the Painlevé–Kuratowski upper/outer limit of F at $\overline{x}.$

Given $\mathsf{\Omega }\subset {\mathbb{R}}^n$ and $\overline{x}\in \mathsf{\Omega },$ define the collection of Fréchet/regular normal cone to $\mathsf{\Omega }$ at $\overline{x}$ by

$\begin{array}{c}\hfill \widehat{N}(\overline{x};\mathsf{\Omega })={\widehat{N}}_{\mathsf{\Omega }}\left(\overline{x}\right):=\left\{v\in {\mathbb{R}}^n|\underset{x\stackrel{\mathsf{\Omega }}{\to }\overline{x}}{lim\; sup}{\displaystyle \frac{⟨v,x-\overline{x}⟩}{\parallel x-\overline{x}\parallel }}\le 0\right\}.\end{array}$

If $x\notin \mathsf{\Omega },$ we put $\widehat{N}(\overline{x};\mathsf{\Omega }):=\varnothing .$

The Mordukhovich/limiting normal cone $N(\overline{x};\mathsf{\Omega })$ to $\mathsf{\Omega }$ at $\overline{x}\in \mathsf{\Omega }\subset {\mathbb{R}}^n$ is obtained from regular normal cones by taking the Painlevé–Kurotowski upper limits as

$\begin{array}{c}\hfill N(\overline{x};\mathsf{\Omega }):=\underset{x\stackrel{\mathsf{\Omega }}{\to }\overline{x}}{Limsup}\widehat{N}(x;\mathsf{\Omega }).\end{array}$

If $\overline{x}\notin \mathsf{\Omega },$ we put $N(\overline{x};\mathsf{\Omega }):=\varnothing .$

For an extended real-valued function $\varphi :{\mathbb{R}}^n\to \overline{R}:=[-\infty ,\infty ],$ its domain is defined by

$\mathrm{dom}\varphi :=\left\{x\in {\mathbb{R}}^n|\varphi \left(x\right)<\infty \right\},$

and its epigraph is defined by

$\mathrm{epi}\varphi :=\left\{(x,\mu )\in {\mathbb{R}}^n\times R|\mu \ge \varphi \left(x\right)\right\}.$

Let $\varphi :{\mathbb{R}}^n\to \overline{\mathbb{R}}$ be finite at $\overline{x}\in \mathrm{dom}\varphi .$ Then, the collection of basic subgradients, or the (basic/Mordukhovich/limiting) subdifferential, of $\varphi$ at $\overline{x}$ is defined by

$\begin{array}{c}\hfill \partial \varphi \left(\overline{x}\right):=\left\{v\in {\mathbb{R}}^n|(v,-1)\in N\left((\overline{x},\varphi \left(\overline{x}\right));\mathrm{epi}\varphi \right)\right\}.\end{array}$

We say a function $\phi :{\mathbb{R}}^n\to \overline{\mathbb{R}}$ is locally Lipschitz at $\overline{x}\in {\mathbb{R}}^n$ with rank $L>0$ if there exists $\tau >0$ such that

$\begin{array}{c}\hfill |\phi \left(x_1\right)-\phi \left(x_2\right)| \le L\parallel x_1-x_2\parallel ,\forall x_1,x_2\in \mathbb{B}(\overline{x},\tau ),\end{array}$

where $\mathbb{B}(\overline{x},\tau )$ stands for the closed unit ball centered at $\overline{x}$ of radius $\tau >0.$ Furthermore, it also holds that [27] (Theorem 1.22)

$\begin{array}{c}\hfill \parallel v\parallel \le L,\forall v\in \partial \phi \left(\overline{x}\right).\end{array}$

The generalized Fermat’s rule is formulated as follows [27] (Proposition 1.30): Let $\phi :{\mathbb{R}}^n\to \overline{\mathbb{R}}$ be finite at $\overline{x}.$ If $\overline{x}$ is a local minimizer of $\phi ,$ then

(6)$\begin{array}{c}\hfill 0\in \partial \phi \left(\overline{x}\right).\end{array}$

To establish optimality conditions, the following lemmas, which are related to the Mordukhovich/limiting subdifferential calculus, are very useful.

Lemma 1

([27] (Corollary 2.21 and Theorem 4.10 (ii))).

• (i). Let ${\varphi }_i:{\mathbb{R}}^n\to \overline{\mathbb{R}},$ $i=1,2,\dots ,m,$ $m\ge 2$ be lower semicontinuous around $\overline{x}\in {\mathbb{R}}^n,$ and let all but one of these functions be Lipschitz-continuous around $\overline{x}.$ Then,

  (7)$\begin{array}{c}\hfill \partial ({\varphi }_1+{\varphi }_2+\dots +{\varphi }_m)\left(\overline{x}\right)\subset \partial {\varphi }_1\left(\overline{x}\right)+\partial {\varphi }_2\left(\overline{x}\right)+\dots +\partial {\varphi }_m\left(\overline{x}\right).\end{array}$

• (ii). Let ${\varphi }_i:{\mathbb{R}}^n\to \overline{\mathbb{R}},$ $i=1,2,\dots ,m,$ $m\ge 2$ be lower semicontinuous around $\overline{x}$ for $i\in I_{max}\left(\overline{x}\right)$ and upper semicontinuous at $\overline{x}$ for $i\notin I_{max}\left(\overline{x}\right);$ suppose that each ${\varphi }_i,$ $i=1,\dots ,m,$ is Lipschitz continuous around $\overline{x}.$ Then, we have the inclusion

  $\begin{array}{c}\hfill \partial (max{\varphi }_i)\left(\overline{x}\right)\subset \bigcup \left\{\partial \left(\sum _{i\in I_{max}\left(\overline{x}\right)}{\lambda }_i{\varphi }_i\right)\left(\overline{x}\right)|({\lambda }_1,\dots ,{\lambda }_m)\in \Lambda \left(\overline{x}\right)\right\},\end{array}$

  where the equality holds and the maximum functions are lower regular at $\overline{x}$ if each ${\varphi }_i$ is lower and regular at this point and the sets $I_{max}\left(\overline{x}\right)$ and $\Lambda \left(\overline{x}\right)$ are defined as follows:

  $\begin{array}{cc}\hfill & I_{max}\left(\overline{x}\right):=\left\{i\in \{1,\dots ,m\}|{\varphi }_i\left(\overline{x}\right)=(max{\varphi }_i)\left(\overline{x}\right)\right\},\hfill \\ \hfill & \Lambda \left(\overline{x}\right):=\left\{({\lambda }_1,\dots ,{\lambda }_m)|{\lambda }_i\ge 0,\sum _{i=1}^m{\lambda }_i=1,{\lambda }_i\left({\varphi }_i\left(\overline{x}\right)-(max{\varphi }_i)\left(\overline{x}\right)\right)=0\right\}.\hfill \end{array}$

Lemma 2

([27] (Corollary 4.8)). Let ${\varphi }_i:{\mathbb{R}}^n\to \overline{\mathbb{R}}$ for $i=1,2$ be Lipschitz continuous around $\overline{x}$ with ${\varphi }_2\left(\overline{x}\right)\ne 0.$ Then, we have

(8)$\partial \left({\displaystyle \frac{{\varphi }_1}{{\varphi }_2}}\right)\left(\overline{x}\right)\subset {\displaystyle \frac{\partial \left({\varphi }_2\left(\overline{x}\right){\varphi }_1\right)\left(\overline{x}\right)-\partial \left({\varphi }_1\left(\overline{x}\right){\varphi }_2\right)\left(\overline{x}\right)}{{\left[{\varphi }_2\left(\overline{x}\right)\right]}^2}}.$

Lemma 3

([27] Mean Value Inequality (Corollary 4.14 (ii))). If φ is Lipschitz continuous on an open set containing $[a,b]\subset {\mathbb{R}}^n,$ then

$\begin{array}{c}\hfill ⟨x^{*},b-a⟩\ge \phi \left(b\right)-\phi \left(a\right)\mathrm{for}\mathrm{some}{x}^{*}\in \partial \phi \left(c\right)\mathrm{with}c\in [a,b)\end{array}$

where $[a,b]:=\mathrm{co}\{a,b\},$ and $[a,b):=\mathrm{co}\{a,b\}\backslash \left\{b\right\}.$

For a function $\phi :{\mathbb{R}}^n\to \overline{\mathbb{R}}$ being locally Lipschitz continuous at $\overline{x},$ the generalized (in the sense of Clarke) of $\phi$ at $\overline{x}$ in the direction $v\in {\mathbb{R}}^n$ is defined as follows:

$\begin{array}{c}\hfill {\phi }^{\circ }(\overline{x};v):=\underset{x\to \overline{x},\lambda \downarrow 0}{lim\; sup}{\displaystyle \frac{\phi (x+\lambda v)-\phi \left(x\right)}{\lambda }}.\end{array}$

In this case, the convexified/Clarke subdifferential of $\phi$ at $\overline{x}$ is the set

$\begin{array}{c}\hfill {\partial }^C\phi \left(\overline{x}\right):=\left\{x^{*}\in {\mathbb{R}}^n|⟨x^{*},v⟩\le {\phi }^{\circ }(\overline{x};v),\forall v\in {\mathbb{R}}^n\right\},\end{array}$

which is nonempty, and the Clarke directional derivative is the support function of the Clarke subdifferential, that is,

$\begin{array}{c}\hfill {\phi }^{\circ }(\overline{x},v)=\underset{x^{*}\in {\partial }^C\phi \left(\overline{x}\right)}{max}⟨x^{*},v⟩,\end{array}$

for each $v\in {\mathbb{R}}^n.$

It follows from [27] that the relationship between the above subdifferentials of $\phi$ at $\overline{x}\in {\mathbb{R}}^n$ is as follows:

$\begin{array}{c}\hfill \partial \phi \left(\overline{x}\right)\subset {\partial }^C\phi \left(\overline{x}\right).\end{array}$

3. Optimality Conditions for Robust Fractional Minimax Programming Problems

In this section, we establish optimality conditions for the robust fractional minimax programming problem (4), which will be pretty helpful for Section 5.

Definition 1.

Let $\varphi \left(x\right):={max}_{k\in K}{f}_k\left(x\right),$ $x\in {\mathbb{R}}^n.$ A point $\overline{x}\in F_r$ is called a locally optimal solution to the problem (4) if and only if there is a neighborhood U of $\overline{x}$ such that

$\begin{array}{c}\hfill \varphi \left(\overline{x}\right)\le \varphi \left(x\right),\forall x\in U\cap F_r.\end{array}$

Moreover, we say $\overline{x}\in F_r$ is a globally optimal solution to the problem (4) if the above inequality holds for all $x\in F_r.$

Let us make some assumptions for functions $g_j,$ $j\in J,$ given in (3). We refer the reader to [28] for more details.

• $\left(\mathrm{A}1\right)$

  For a fixed $\overline{x}\in {\mathbb{R}}^n,$ there exists ${\delta }_j^{\overline{x}}>0$ such that the function $v_j\in {\mathcal{V}}_j\mapsto g_j(x,v_j)\in \mathbb{R}$ is upper semicontinuous for each $x\in \mathbb{B}(\overline{x},{\delta }_j^{\overline{x}}),$ and the functions $g_j(·,v_j),v_j\in {\mathcal{V}}_j,$ are Lipschitz of given rank $L_j>0$ on $\mathbb{B}(\overline{x},{\delta }_j^{\overline{x}}),$ i.e.,

  (9)$\begin{array}{c}\hfill |g_j(x_1,v_j)-g_j(x_2,v_j)|\le L_j\parallel x_1-x_2\parallel ,\forall x_1,x_2\in \mathbb{B}(\overline{x},{\delta }_j^{\overline{x}}),\forall v_j\in {\mathcal{V}}_j.\end{array}$

• $\left(\mathrm{A}2\right)$

  The multifunction $(x,v_j)\in \mathbb{B}(\overline{x},{\delta }_j^{\overline{x}})\times {\mathcal{V}}_j\rightrightarrows {\partial }_x{g}_j(x,v_j)\subset {\mathbb{R}}^n$ is closed at $(\overline{x},{\overline{v}}_j)$ for each ${\overline{v}}_j\in {\mathcal{V}}_j\left(\overline{x}\right),$ where the symbol ${\partial }_x$ stands for the limiting subdifferential operation with respect to $x,$ and the notation ${\mathcal{V}}_j\left(\overline{x}\right)$ signifies active indices in ${\mathcal{V}}_j$ at $\overline{x},$ i.e.,

  (10)$\begin{array}{c}\hfill {\mathcal{V}}_j\left(\overline{x}\right):=\left\{v_j\in {\mathcal{V}}_j|g_j(\overline{x},v_j)=G_j\left(\overline{x}\right)\right\}\end{array}$

  with $G_j\left(\overline{x}\right):=\underset{v_j\in {\mathcal{V}}_j}{sup}{g}_j(\overline{x},v_j).$

As usual, in order to obtain the necessary optimality condition of the Karush–Kuhn–Tucker type for a locally optimal solution to the problem (4), the following constraint qualification (see [28] (Definition 3.2)) is needed, which turns into the extended Mangasarian–Fromovitz constraint qualification (see e.g., [29] (p. 72)) in the smooth setting.

Definition 2

([28] (Definition 3.2)). Let $\overline{x}\in F_r.$ Then, the constraint qualification $\left(\mathrm{CQ}\right)$ is fulfilled at $\overline{x}$ if

$\begin{array}{c}\hfill 0\notin \mathrm{co}\left\{\cup {\partial }_x{g}_j(\overline{x},v_j)|v_j\in {\mathcal{V}}_j\left(\overline{x}\right),j\in J\right\}.\end{array}$

Theorem 1.

Let the $\left(\mathrm{CQ}\right)$ be fulfilled at $\overline{x}\in F_r.$ Suppose that $\overline{x}$ is a locally optimal solution to the problem (4); then there are some $\alpha :=({\alpha }_1,\dots ,{\alpha }_l)\in {\mathbb{R}}_{+}^l\backslash \left\{0\right\},$ $\lambda :=({\lambda }_1,\dots ,{\lambda }_m)\in {\mathbb{R}}_{+}^m$ such that

(11) $\begin{array}{cc}\hfill & 0\in \sum _{k\in K}{\alpha }_k\left(\partial p_k\left(\overline{x}\right)-{\displaystyle \frac{p_k\left(\overline{x}\right)}{q_k\left(\overline{x}\right)}}\partial q_k\left(\overline{x}\right)\right)+\sum _{j\in J}{\lambda }_j\mathrm{co}\left\{\cup {\partial }_x{g}_j(\overline{x},v_j)|v_j\in {\mathcal{V}}_j\left(\overline{x}\right)\right\},\hfill \\ \hfill & {\alpha }_k\left(f_k\left(\overline{x}\right)-\underset{k\in K}{max}{f}_k\left(\overline{x}\right)\right)=0,k\in K,\hfill \\ \hfill & {\lambda }_j\underset{v_j\in {\mathcal{V}}_j}{sup}{g}_j(\overline{x},v_j)=0,j\in J.\hfill \end{array}$

Proof. 

Let $\overline{x}$ be a locally optimal solution to the problem (4). For $j\in J,$ we consider ${\delta }_j^{\overline{x}},$ $L_j,$ ${\mathcal{V}}_j\left(\overline{x}\right),$ and $G_j\left(\overline{x}\right)$ as in assumptions $\left(\mathrm{A}1\right)$ and $\left(\mathrm{A}2\right).$ Along with the proof of [28] (Theorem 3.3), we can show that

$\mathrm{co}\left\{\cup {\partial }_x{g}_j(\overline{x},v_j)|v_j\in {\mathcal{V}}_j\left(\overline{x}\right)\right\}$

is closed and compact, and

(12)$\begin{array}{c}\hfill \partial G_j\left(\overline{x}\right)\subset \mathrm{co}\left\{\cup {\partial }_x{g}_j(\overline{x},v_j)|v_j\in {\mathcal{V}}_j\left(\overline{x}\right)\right\}\end{array}$

with the help of Lemma 3 and the relationship between $\partial G_j\left(\overline{x}\right)$ and ${\partial }^C{G}_j\left(\overline{x}\right).$

Set $\varphi \left(x\right):={max}_{k\in K}{f}_k\left(x\right).$ Because $\overline{x}$ is a locally optimal solution to the problem (4), $\overline{x}\in F_r,$ there exists a neighborhood U of $\overline{x}$ such that

(13)$\begin{array}{c}\hfill \varphi \left(\overline{x}\right)\le \varphi \left(x\right),\forall x\in U\cap F_r.\end{array}$

We claim that $\overline{x}$ is also a local minimizer of the following problem

$\begin{array}{c}\hfill min\psi \left(x\right)\mathrm{subject}\mathrm{to}x\in U,\end{array}$

where $\psi \left(x\right):=\underset{j\in J}{max}\left\{\varphi \left(x\right)-\varphi \left(\overline{x}\right),G_j\left(x\right)\right\},$ $x\in {\mathbb{R}}^n.$ Actually, it is easy to see that $\psi \left(\overline{x}\right)=0$ due to $\overline{x}\in F_r.$ Let us justify $\psi \left(\overline{x}\right)\le \psi \left(x\right)$ for $x\in U.$ Suppose that $x\in U\cap F_r$; then, $\psi \left(x\right)\ge 0.$ Otherwise, $\psi \left(x\right)<0$ shows that

$\begin{array}{c}\hfill \varphi \left(x\right)-\varphi \left(\overline{x}\right)<0,\end{array}$

which contradicts (13). If $x\in U\backslash F_r,$ then there exists $j_0\in J$ such that $G_{j_0}\left(x\right)>0,$ which entails that $\psi \left(x\right)>0.$ Therefore, $\overline{x}$ is a local minimizer for $\psi .$

Applying the nonsmooth version of Fermat’s rule (6), we obtain

$\begin{array}{c}\hfill 0\in \partial \psi \left(\overline{x}\right).\end{array}$

Moreover, applying the formula for the limiting subdifferential of maximum functions and the limiting subdifferential sum rule for local Lipschitz functions (Lemma 1), we have

$\begin{array}{c}\hfill 0\in \left\{{\mu }_0\partial \varphi \left(\overline{x}\right)+\sum _{j\in J}{\mu }_j\partial G_j\left(\overline{x}\right)|{\mu }_0\ge 0,{\mu }_j\ge 0,{\mu }_j{G}_j\left(\overline{x}\right)=0,j\in J,{\mu }_0+\sum _{j\in j}{\mu }_j=1\right\}.\end{array}$

From (12) together with (10), we derive

(14)$\begin{array}{cc}\hfill 0\in \{& {\mu }_0\partial \varphi \left(\overline{x}\right)+\sum _{j\in J}{\mu }_j\mathrm{co}\left\{\cup {\partial }_x{g}_j(\overline{x},v_j)|v_j\in {\mathcal{V}}_j\left(\overline{x}\right)\right\}\hfill \\ \hfill & |{\mu }_0\ge 0,{\mu }_j\ge 0,{\mu }_j\underset{v_j\in {\mathcal{V}}_j}{sup}{g}_j(\overline{x},v_j)=0,j\in J,{\mu }_0+\sum _{j\in J}{\mu }_j=1\}.\hfill \end{array}$

Applying the limiting subdifferential of maximum functions and the sum rule to $\varphi \left(\overline{x}\right),$ we obtain

$\begin{array}{cc}\hfill \partial \varphi \left(\overline{x}\right)& =\partial \left(\underset{k\in K}{max}\left\{{\displaystyle \frac{p_k}{q_k}}\right\}\right)\left(\overline{x}\right)\hfill \\ \hfill & \subset \{\sum _{k\in K\left(\overline{x}\right)}{\beta }_k\partial \left({\displaystyle \frac{p_k}{q_k}}\right)\left(\overline{x}\right)|{\beta }_k\ge 0,{\beta }_k\left(f_k\left(\overline{x}\right)-(max{f}_k)\left(\overline{x}\right)\right)=0,\hfill \\ \hfill & k\in K\left(\overline{x}\right),\sum _{k\in K\left(\overline{x}\right)}{\beta }_k=1\},\hfill \end{array}$

where $K\left(\overline{x}\right):=\{k\in K|f_k\left(\overline{x}\right)=\varphi \left(\overline{x}\right)\}\ne \varnothing .$ Taking (8) into account, we arrive at

(15)$\begin{array}{cc}\hfill \sum _{k\in K\left(\overline{x}\right)}{\beta }_k\partial \left({\displaystyle \frac{p_k}{q_k}}\right)\left(\overline{x}\right)& \subset \sum _{k\in K\left(\overline{x}\right)}{\beta }_k{\displaystyle \frac{\partial \left(q_k\left(\overline{x}\right)p_k\right)\left(\overline{x}\right)-\partial \left(p_k\left(\overline{x}\right)q_k\right)\left(\overline{x}\right)}{{\left[q_k\left(\overline{x}\right)\right]}^2}}\hfill \\ \hfill & =\sum _{k\in K\left(\overline{x}\right)}{\beta }_k{\displaystyle \frac{q_k\left(\overline{x}\right)\partial p_k\left(\overline{x}\right)-p_k\left(\overline{x}\right)\partial q_k\left(\overline{x}\right)}{{\left[q_k\left(\overline{x}\right)\right]}^2}}\hfill \\ \hfill & =\sum _{k\in K\left(\overline{x}\right)}{\displaystyle \frac{{\beta }_k}{q_k\left(\overline{x}\right)}}\left(\partial p_k\left(\overline{x}\right)-{\displaystyle \frac{p_k\left(\overline{x}\right)}{q_k\left(\overline{x}\right)}}\partial q_k\left(\overline{x}\right)\right).\hfill \end{array}$

Now, we put ${\beta }_k:=0$ for $k\in K\backslash K\left(\overline{x}\right),$ and let ${\alpha }_k:={\displaystyle \frac{{\beta }_k}{q_k\left(\overline{x}\right)}}$ for $k\in K.$ If the $\left(\mathrm{CQ}\right)$ is satisfied at $\overline{x},$ then ${\mu }_0\ne 0,$ set ${\lambda }_j={\displaystyle \frac{{\mu }_j}{{\mu }_0}},$ $j\in J,$ and (14) with (15) establishes (11), which completes the proof. □

In what follows, we formulate sufficient optimality conditions for the problem (4). For this, we recall the following definition of generalized convexity; see [30,31].

Definition 3.

We say that $(p,q;g)$ is generalized convex on ${\mathbb{R}}^n$ at $\overline{x}\in {\mathbb{R}}^n$ if for any $x\in {\mathbb{R}}^n,$ ${\xi }_k\in \partial p_k\left(\overline{x}\right),$ ${\zeta }_k\in \partial q_k\left(\overline{x}\right),$ $k\in K,$ ${\eta }_j\in {\partial }_x{g}_j(\overline{x},v_j),$ $v_j\in {\mathcal{V}}_j\left(\overline{x}\right),$ $j\in J,$ there exists $h\in {\mathbb{R}}^n$ such that

$\begin{array}{cc}\hfill p_k\left(x\right)-p_k\left(\overline{x}\right)& \ge ⟨{\xi }_k,h⟩,k\in K,\hfill \\ \hfill q_k\left(x\right)-q_k\left(\overline{x}\right)& \ge ⟨{\zeta }_k,h⟩,k\in K,\hfill \\ \hfill g_j(x,v_j)-g_j(\overline{x},v_j)& \ge ⟨{\eta }_j,h⟩,v_j\in {\mathcal{V}}_j\left(\overline{x}\right),j\in J,\hfill \end{array}$

where ${\mathcal{V}}_j\left(\overline{x}\right),$ $j\in J$ are defined as in(10).

The following theorem provides sufficient optimality conditions for a globally optimal solution of (4).

Theorem 2.

Let the condition $\left(\right)$ be satisfied at $\overline{x}\in F_r.$ If $(p,q;g)$ is generalized convex on ${\mathbb{R}}^n$ at $\overline{x},$ then $\overline{x}$ is a globally optimal solution to the problem (4).

Proof. 

Let $\varphi \left(x\right):={max}_{k\in K}{f}_k\left(x\right).$ Since $\overline{x}\in F$ satisfies conditions (11), there exist $\alpha :=({\alpha }_1,\dots ,{\alpha }_l)\in {\mathbb{R}}_{+}^l\backslash \left\{0\right\},$ ${\xi }_k\in \partial p_k\left(\overline{x}\right),$ ${\zeta }_k\in \partial q_k\left(\overline{x}\right),$ $k\in K,$ and ${\lambda }_j\ge 0,$ $j\in J,$ ${\lambda }_{ji}\ge 0,$ ${\eta }_{ji}\in {\partial }_x{g}_j(\overline{x},v_{ji}),$ $v_{ji}\in {\mathcal{V}}_j\left(\overline{x}\right),$ $i\in I_j=\{1,\dots ,i_j\},$ $i_j\in \mathbb{N},$ $\sum _{i\in I_j}{\lambda }_{ji}=1,$ such that

(16)$\begin{array}{cc}\hfill & \sum _{k\in K}{\alpha }_k\left({\xi }_k-{\displaystyle \frac{p_k\left(\overline{x}\right)}{q_k\left(\overline{x}\right)}}{\zeta }_k\right)+\sum _{j\in J}{\lambda }_j\left(\sum _{i\in I_j}{\lambda }_{ji}{\eta }_{ji}\right)=0,\hfill \end{array}$

(17)$\begin{array}{cc}\hfill & {\alpha }_k\left(f_k\left(\overline{x}\right)-\underset{k\in K}{max}{f}_k\left(\overline{x}\right)\right)=0,k\in K,\hfill \end{array}$

(18)$\begin{array}{cc}\hfill & {\lambda }_j\underset{v_j\in {\mathcal{V}}_j}{sup}{g}_j(\overline{x},v_j)=0,j\in J.\hfill \end{array}$

Assume to the contrary that $\overline{x}$ is not a globally optimal solution to the problem (4); then, there is $\widehat{x}\in F_r$ such that

(19)$\begin{array}{c}\hfill \varphi \left(\overline{x}\right)>\varphi \left(\widehat{x}\right).\end{array}$

By the generalized convex on ${\mathbb{R}}^n$ at $\overline{x},$ we deduce from (16) that for such $\widehat{x}$ there is $h\in {\mathbb{R}}^n$ such that

$\begin{array}{cc}\hfill 0=& \sum _{k\in K}{\alpha }_k\left(⟨{\xi }_k,b⟩-{\displaystyle \frac{p_k\left(\overline{x}\right)}{q_k\left(\overline{x}\right)}}⟨{\zeta }_k,b⟩\right)+\sum _{j\in J}{\lambda }_j\left(\sum _{i\in I_j}{\lambda }_{ji}⟨{\eta }_{ji},h⟩\right)\hfill \\ \hfill \le & \sum _{k\in K}{\alpha }_k\left([p_k\left(\widehat{x}\right)-p_k\left(\overline{x}\right)]-{\displaystyle \frac{p_k\left(\overline{x}\right)}{q_k\left(\overline{x}\right)}}[q_k\left(\widehat{x}\right)-q_k\left(\overline{x}\right)]\right)\hfill \\ \hfill & +\sum _{j\in J}{\lambda }_j\left(\sum _{i\in I_j}{\lambda }_{ji}\left[g_j(\widehat{x},v_{ji})-g_j(\overline{x},v_{ji})\right]\right).\hfill \end{array}$

Hence,

(20)$\begin{array}{cc}\hfill & \sum _{j\in J}{\lambda }_j\left(\sum _{i\in I_j}{\lambda }_{ji}{g}_j(\overline{x},v_{ji})\right)\hfill \\ \hfill \le & \sum _{k\in K}{\alpha }_k\left([p_k\left(\widehat{x}\right)-p_k\left(\overline{x}\right)]-{\displaystyle \frac{p_k\left(\overline{x}\right)}{q_k\left(\overline{x}\right)}}[q_k\left(\widehat{x}\right)-q_k\left(\overline{x}\right)]\right)+\sum _{j\in J}{\lambda }_j\left(\sum _{i\in I_j}{\lambda }_{ji}{g}_j(\widehat{x},v_{ji})\right).\hfill \end{array}$

Since $v_{ji}\in {\mathcal{V}}_j\left(\overline{x}\right),$

$\begin{array}{c}\hfill g_j(\overline{x},v_{ji})=\underset{v_j\in {\mathcal{V}}_j}{sup}{g}_j(\overline{x},v_j),\forall j\in J,\forall i\in I_j.\end{array}$

Thus, it follows from (18) that ${\lambda }_j{g}_j(\overline{x},v_{ji})=0$ for $j\in J$ and $i\in I_j.$ In addition, due to $\widehat{x}\in F_r,$ ${\lambda }_j{g}_j(\widehat{x},v_{ji})\le 0$ for $j\in J$ and $i\in I_j.$ So, we obtain by (20) that

$\begin{array}{cc}\hfill 0& \le \sum _{k\in K}{\alpha }_k\left([p_k\left(\widehat{x}\right)-p_k\left(\overline{x}\right)]-{\displaystyle \frac{p_k\left(\overline{x}\right)}{q_k\left(\overline{x}\right)}}[q_k\left(\widehat{x}\right)-q_k\left(\overline{x}\right)]\right)\hfill \\ \hfill & =\sum _{k\in K}{q}_k\left(\widehat{x}\right){\alpha }_k\left({\displaystyle \frac{p_k\left(\widehat{x}\right)}{q_k\left(\widehat{x}\right)}}-{\displaystyle \frac{p_k\left(\overline{x}\right)}{q_k\left(\overline{x}\right)}}\right).\hfill \end{array}$

On the other hand, by (17), it holds that

$\begin{array}{cc}\hfill \sum _{k\in K}{q}_k\left(\widehat{x}\right){\alpha }_k\varphi \left(\overline{x}\right)& =\sum _{k\in K}{q}_k\left(\widehat{x}\right){\alpha }_k{f}_k\left(\overline{x}\right)\hfill \\ \hfill & \le \sum _{k\in K}{q}_k\left(\widehat{x}\right){\alpha }_k{f}_k\left(\widehat{x}\right)\le \sum _{k\in K}{q}_k\left(\widehat{x}\right){\alpha }_k\varphi \left(\widehat{x}\right).\hfill \end{array}$

This implies that

(21)$\begin{array}{c}\hfill \varphi \left(\overline{x}\right)\le \varphi \left(\widehat{x}\right)\end{array}$

due to $\sum _{k\in K}{q}_k\left(\widehat{x}\right){\alpha }_k>0.$ Obviously, (21) contradicts (19), which completes the proof. □

4. Duality for Robust Fractional Minimax Programming Problems

Let $z\in {\mathbb{R}}^n,$ $\alpha :=({\alpha }_1,\dots ,{\alpha }_l)\in {\mathbb{R}}_{+}^l\backslash \left\{0\right\},$

$\begin{array}{c}\hfill \lambda \in {\mathbb{R}}_{+}^{\mathbb{N}}:=\left\{\lambda :=({\lambda }_j,{\lambda }_{ji}),j\in J,i\in I_j=\{1,\dots ,i_j\}|i_j\in \mathbb{N},{\lambda }_j\ge 0,{\lambda }_{ji}\ge 0,\sum _{i\in I_j}{\lambda }_{ji}=1\right\}.\end{array}$

In connection with the robust fractional minimax programming problem (4), we consider a dual problem in the following form:

(22)$\begin{array}{c}\hfill \underset{(z,\alpha ,\lambda )\in F_D}{max}\left\{\tilde{\varphi }(z,\alpha ,\lambda ):=\varphi \left(z\right)\right\}.\end{array}$

Here, we denote $\varphi \left(z\right):=\underset{k\in K}{max}{f}_k\left(z\right)$ and

$\begin{array}{cc}\hfill F_D:=\{& (z,\alpha ,\lambda )\in {\mathbb{R}}^n\times {\mathbb{R}}_{+}^l\times {\mathbb{R}}_{+}^m|0\in \sum _{k\in K}{\alpha }_k\left(\partial p_k\left(z\right)-{\displaystyle \frac{p_k\left(z\right)}{q_k\left(z\right)}}\partial q_k\left(z\right)\right)\hfill \\ \hfill & +\sum _{j\in J}{\lambda }_j\left(\sum _{i\in I_j}{\lambda }_{ji}{\eta }_{ji}\right),\sum _{j\in J}{\lambda }_j\underset{v_j\in {\mathcal{V}}_j}{sup}{g}_j(z,v_j)\ge 0,\hfill \\ \hfill & {\eta }_{ji}\in \{\cup {\partial }_x{g}_j(z,v_{ji})|v_{ji}\in {\mathcal{V}}_j\left(z\right)\},{\alpha }_k\left(f_k\left(z\right)-\varphi \left(z\right)\right)=0,k\in K\},\hfill \end{array}$

where ${\mathcal{V}}_j\left(z\right)$ is defined as in (10) by replacing $\overline{x}$ by $z.$

Definition 4.

We say that a point $(\overline{z},\overline{\alpha },\overline{\lambda })\in F_D$ is called a globally optimal solution to the problem (22) if and only if

$\begin{array}{c}\hfill \tilde{\varphi }(\overline{z},\overline{\alpha },\overline{\lambda })\ge \tilde{\varphi }(z,\alpha ,\lambda ),\forall (z,\alpha ,\lambda )\in F_D.\end{array}$

Theorem 3

(weak duality). Let $x\in F_r$ and let $(z,\alpha ,\lambda )\in F_D.$ If $(p,q;g)$ is generalized convex ${\mathbb{R}}^n$ at $z,$ then

$\begin{array}{c}\hfill \varphi \left(x\right)\ge \tilde{\varphi }(z,\alpha ,\lambda ).\end{array}$

Proof. 

Since $(z,\alpha ,\lambda )\in F_D,$ there exist $\alpha :=({\alpha }_1,\dots ,{\alpha }_l)\in {\mathbb{R}}_{+}^l\backslash \left\{0\right\},$ ${\xi }_k\in \partial p_k\left(z\right),$ ${\zeta }_k\in \partial q_k\left(z\right),$ $k\in K,$ and ${\lambda }_j\ge 0,$ $j\in J,$ ${\lambda }_{ji}\ge 0,$ ${\eta }_{ji}\in {\partial }_x{g}_j(z,v_{ji}),$ $v_{ji}\in {\mathcal{V}}_j\left(z\right),$ $i\in I_j=\{1,\dots ,i_j\},$ $i_j\in \mathbb{N},$ $\sum _{i\in I_j}{\lambda }_{ji}=1,$ such that

(23)$\begin{array}{cc}\hfill & \sum _{k\in K}{\alpha }_k\left({\xi }_k-{\displaystyle \frac{p_k\left(z\right)}{q_k\left(z\right)}}{\zeta }_k\right)+\sum _{j\in J}{\lambda }_j\left(\sum _{i\in I_j}{\lambda }_{ji}{\eta }_{ji}\right)=0,\hfill \end{array}$

(24)$\begin{array}{cc}\hfill & {\alpha }_k\left(f_k\left(z\right)-\varphi \left(z\right)\right)=0,k\in K,\hfill \end{array}$

(25)$\begin{array}{cc}\hfill & \sum _{j\in J}{\lambda }_j\underset{v_j\in {\mathcal{V}}_j}{sup}{g}_j(\overline{x},v_j)\ge 0.\hfill \end{array}$

By the generalized convex on ${\mathbb{R}}^n$ at $z,$ we deduce from (23) that for such x there is $h\in {\mathbb{R}}^n$ such that

$\begin{array}{cc}\hfill 0=& \sum _{k\in K}{\alpha }_k\left(⟨{\xi }_k,h⟩-{\displaystyle \frac{p_k\left(z\right)}{q_k\left(z\right)}}⟨{\zeta }_k,h⟩\right)+\sum _{j\in J}{\lambda }_j\left(\sum _{i\in I_j}{\lambda }_{ji}⟨{\eta }_{ji},h⟩\right)\hfill \\ \hfill \le & \sum _{k\in K}{\alpha }_k\left([p_k\left(x\right)-p_k\left(z\right)]-{\displaystyle \frac{p_k\left(z\right)}{q_k\left(z\right)}}[q_k\left(x\right)-q_k\left(z\right)]\right)+\sum _{j\in J}{\lambda }_j\left(\sum _{i\in I_j}{\lambda }_{ji}[g_j(x,v_{ji})-g_j(z,v_{ji})]\right).\hfill \end{array}$

It stems from $x\in F_r$ that

$\begin{array}{c}\hfill \sum _{j\in J}{\lambda }_j\left(\sum _{i\in I_j}{\lambda }_{ji}{g}_j(x,v_{ji})\right)\le 0.\end{array}$

We obtain

$\begin{array}{cc}\hfill 0\le & \sum _{k\in K}{\alpha }_k\left([p_k\left(x\right)-p_k\left(z\right)]-{\displaystyle \frac{p_k\left(z\right)}{q_k\left(z\right)}}[q_k\left(x\right)-q_k\left(z\right)]\right)-\sum _{j\in J}{\lambda }_j\left(\sum _{i\in I_j}{\lambda }_{ji}{g}_j(z,v_{ji})\right)\hfill \\ \hfill =& \sum _{k\in K}{q}_k\left(x\right){\alpha }_k\left(f_k\left(x\right)-f_k\left(z\right)\right)-\sum _{j\in J}{\lambda }_j\left(\sum _{i\in I_j}{\lambda }_{ji}{g}_j(z,v_{ji})\right).\hfill \end{array}$

Since $v_{ji}\in {\mathcal{V}}_j\left(z\right),$

$g_j(z,v_{ji})=\underset{v_j\in {\mathcal{V}}_j}{sup}{g}_j(z,v_j),\forall j\in J,\forall i\in I_j.$

Thus, we obtain

(26)$\begin{array}{cc}\hfill 0\le & \sum _{k\in K}{q}_k\left(x\right){\alpha }_k\left(f_k\left(x\right)-f_k\left(z\right)\right)-\sum _{j\in J}{\lambda }_j\left(\sum _{i\in I_j}{\lambda }_{ji}{g}_j(z,v_{ji})\right)\hfill \\ \hfill =& \sum _{k\in K}{q}_k\left(x\right){\alpha }_k\left(f_k\left(x\right)-f_k\left(z\right)\right)-\sum _{j\in J}{\lambda }_j\left(\sum _{i\in I_j}{\lambda }_{ji}\underset{v_j\in {\mathcal{V}}_j}{sup}{g}_j(z,v_j)\right)\hfill \\ \hfill =& \sum _{k\in K}{q}_k\left(x\right){\alpha }_k\left(f_k\left(x\right)-f_k\left(z\right)\right)-\sum _{i\in I_j}{\lambda }_{ji}\left(\sum _{j\in J}{\lambda }_j\underset{v_j\in {\mathcal{V}}_j}{sup}{g}_j(z,v_j)\right)\hfill \\ \hfill \le & \sum _{k\in K}{q}_k\left(x\right){\alpha }_k\left(f_k\left(x\right)-f_k\left(z\right)\right),\hfill \end{array}$

where the last inequality holds true due to (25). Combining now (26) and (24), we have

$\begin{array}{c}\hfill \sum _{k\in K}{q}_k\left(x\right){\alpha }_k\varphi \left(z\right)\le \sum _{k\in K}{q}_k\left(x\right){\alpha }_k{f}_k\left(x\right)\le \sum _{k\in K}{q}_k\left(x\right){\alpha }_k\varphi \left(x\right).\end{array}$

This implies that

$\begin{array}{c}\hfill \tilde{\varphi }(z,\alpha ,\mu ,\gamma )=\varphi \left(z\right)\le \varphi \left(x\right)\end{array}$

due to $\sum _{k\in K}{q}_k\left(x\right){\alpha }_k>0.$ The proof is complete. □

Theorem 4

(strong duality). Let $\overline{x}\in F_r$ be a locally optimal solution to the problem (4) such that the $\left(\mathrm{CQ}\right)$ is satisfied at this point. Then, there exists $(\overline{\alpha },\overline{\lambda })\in {\mathbb{R}}_{+}^l\times {\mathbb{R}}_{+}^m$ such that $(\overline{x},\overline{\alpha },\overline{\lambda })\in F_D$ and

$\begin{array}{c}\hfill \varphi \left(\overline{x}\right)=\tilde{\varphi }(\overline{x},\overline{\alpha },\overline{\lambda }).\end{array}$

Furthermore, if $(p,q;g)$ is generalized convex on ${\mathbb{R}}^n$ at any $z\in {\mathbb{R}}^n,$ then $(\overline{x},\overline{\alpha },\overline{\lambda })$ is globally optimal solution to the problem (22).

Proof. 

Thanks to Theorem 1, we can find $\alpha :=({\alpha }_1,\dots ,{\alpha }_l)\in {\mathbb{R}}_{+}^l\backslash \left\{0\right\},$ ${\xi }_k\in \partial p_k\left(\overline{x}\right),$ ${\zeta }_k\in \partial q_k\left(\overline{x}\right),$ $k\in K,$ and ${\lambda }_j\ge 0,$ $j\in J,$ ${\lambda }_{ji}\ge 0,$ ${\eta }_{ji}\in {\partial }_x{g}_j(\overline{x},v_{ji}),$ $v_{ji}\in {\mathcal{V}}_j\left(\overline{x}\right),$ $i\in I_j=\{1,\dots ,i_j\},$ $i_j\in \mathbb{N},$ $\sum _{i\in I_j}{\lambda }_{ji}=1,$ such that

(27)$\begin{array}{cc}\hfill & \sum _{k\in K}{\alpha }_k\left({\xi }_k-{\displaystyle \frac{p_k\left(\overline{x}\right)}{q_k\left(\overline{x}\right)}}{\zeta }_k\right)+\sum _{j\in J}{\lambda }_j\left(\sum _{i\in I_j}{\lambda }_{ji}{\eta }_{ji}\right)=0,\hfill \\ \hfill & {\alpha }_k\left(f_k\left(\overline{x}\right)-\underset{k\in K}{max}{f}_k\left(\overline{x}\right)\right)=0,k\in K,\hfill \end{array}$

(28)$\begin{array}{cc}\hfill & {\lambda }_j\underset{v_j\in {\mathcal{V}}_j}{sup}{g}_j(\overline{x},v_j)=0,j\in J.\hfill \end{array}$

Since $v_{ji}\in {\mathcal{V}}_j\left(\overline{x}\right),$ we have $g_j(\overline{x},v_{ji})=\underset{v_j\in {\mathcal{V}}_j}{sup}{g}_j(\overline{x},v_j)$ for $j\in J,$ and $i\in I_j=\{1,\dots ,i_j\}.$ Thus, it stems from (28) that ${\lambda }_j{g}_j(\overline{x},v_{ji})=0$ for $j\in J,$ and $i\in I_j=\{1,\dots ,i_j\}.$ This entails that

(29)$\begin{array}{c}\hfill \sum _{j\in J}{\lambda }_j\left(\sum _{i\in I_j}{\lambda }_{ji}{g}_j(\overline{x},v_{ji})\right)=\sum _{j\in J}\left(\sum _{i\in I_j}{\lambda }_{ji}{\lambda }_j{g}_j(\overline{x},v_{ji})\right)=0.\end{array}$

Letting $\overline{\alpha }:=({\alpha }_1,\dots ,{\alpha }_l),$ and $\overline{\lambda }:=({\lambda }_j,{\lambda }_{ji}),$ we have

$(\overline{\alpha },\overline{\lambda })\in ({\mathbb{R}}_{+}^l\backslash \left\{0\right\})\times {\mathbb{R}}_{+}^{\mathbb{N}},$

and so $(\overline{x},\overline{\alpha },\overline{\lambda })\in F_{D_5}$ due to (27) and (29). Obviously,

$\varphi \left(\overline{x}\right)=\tilde{\varphi }(\overline{x},\overline{\alpha },\overline{\lambda }).$

Thus, when $(p,q;g)$ is generalized convex on ${\mathbb{R}}^n$ at any $z\in {\mathbb{R}}^n,$ we apply the weak duality result in Theorem 3 to conclude that

$\tilde{\varphi }(\overline{x},\overline{\alpha },\overline{\lambda })=\varphi \left(\overline{x}\right)\ge \tilde{\varphi }(z,\alpha ,\lambda )$

for any $(z,\alpha ,\lambda )\in F_D.$ This means that $(\overline{x},\overline{\alpha },\overline{\lambda })$ is a globally optimal solution to the problem (22). □

Theorem 5

(converse-like duality). Let $(\overline{x},\overline{\alpha },\overline{\lambda })\in F_D$ be such that $\varphi \left(\overline{x}\right)=\tilde{\varphi }(\overline{x},\overline{\alpha },\overline{\lambda }).$ If $\overline{x}\in F_r$ and $(p,q;g)$ is generalized convex on ${\mathbb{R}}^n$ at $\overline{x},$ then $\overline{x}$ is a globally optimal solution to the problem (4).

Proof. 

Since $(\overline{x},\overline{\alpha },\overline{\lambda })\in F_D,$ there exist $\overline{\alpha }:=({\overline{\alpha }}_1,\dots ,{\overline{\alpha }}_l)\in {\mathbb{R}}_{+}^l\backslash \left\{0\right\},$ ${\xi }_k\in \partial p_k\left(\overline{x}\right),$ ${\zeta }_k\in \partial q_k\left(\overline{x}\right),$ $k\in K,$ and ${\lambda }_j\ge 0,$ $j\in J,$ ${\lambda }_{ji}\ge 0,$ ${\eta }_{ji}\in {\partial }_x{g}_j(\overline{x},v_{ji}),$ $v_{ji}\in {\mathcal{V}}_j\left(\overline{x}\right),$ $i\in I_j=\{1,\dots ,i_j\},$ $i_j\in \mathbb{N},$ $\sum _{i\in I_j}{\lambda }_{ji}=1$ such that

(30)$\begin{array}{cc}\hfill & \sum _{k\in K}{\overline{\alpha }}_k\left({\xi }_k-{\displaystyle \frac{p_k\left(\overline{x}\right)}{q_k\left(\overline{x}\right)}}{\zeta }_k\right)+\sum _{j\in J}{\lambda }_j\left(\sum _{i\in I_j}{\lambda }_{ji}{\eta }_{ji}\right)=0,\hfill \end{array}$

(31)$\begin{array}{cc}\hfill & {\overline{\alpha }}_k\left(f_k\left(\overline{x}\right)-\varphi \left(\overline{x}\right)\right)=0,k\in K,\hfill \end{array}$

(32)$\begin{array}{cc}\hfill & \sum _{j\in J}{\lambda }_j\underset{v_j\in {\mathcal{V}}_j}{sup}{g}_j(\overline{x},v_j)\ge 0.\hfill \end{array}$

Let $\overline{x}\in F_r.$ Then, $g_j(\overline{x},v_j)\le 0,$ for all $v_j\in {\mathcal{V}}_j,$ $j\in J,$ and thus,

${\lambda }_j\underset{v_j\in {\mathcal{V}}_j}{sup}{g}_j(\overline{x},v_j)\le 0.$

This, together with (32), yields

$\begin{array}{c}\hfill {\lambda }_j\underset{v_j\in {\mathcal{V}}_j}{sup}{g}_j(\overline{x},v_j)=0,j\in J.\end{array}$

So, we assert by virtue of (30), (31) that $\overline{x}$ satisfies condition (11). To finish the proof, it remains to apply Theorem 2. □

5. Application to Robust Fractional Multi-Objective Optimization Problems

In this section, we will apply the results from Section 3 to study optimality conditions for robust fractional multi-objective optimization problems.

Definition 5.

We say $\overline{x}\in F_r$ is a weak Pareto solution to the problem (2) if and only if

$f\left(x\right)-f\left(\overline{x}\right)\notin -\mathrm{int}{\mathbb{R}}_{+}^l,\forall x\in F_r.$

Theorem 6

([31] (cf. Theorem 5.1)). Suppose that $\overline{x}$ is a weak Pareto solution to the problem (2); then, $\overline{x}$ is an optimal solution of the following problem:

(33)$\begin{array}{c}\hfill \underset{x\in F_r}{min}\underset{k=1,\dots ,l}{max}\left(f_k\left(x\right)-f_k\left(\overline{x}\right)\right).\end{array}$

Proof. 

Let $\overline{x}$ be a weak Pareto solution to the problem (2). We set

${\widehat{f}}_k\left(x\right):=f_k\left(x\right)-f_k\left(\overline{x}\right),k=1,\dots ,l,x\in {\mathbb{R}}^n,$

and $\widehat{\varphi }\left(x\right):={max}_{k=1,\dots ,l}{\widehat{f}}_k\left(x\right).$

On the contrary, assume that $\overline{x}$ is not an optimal solution of the problem (33), and hence there exists $x_0\in F_r$ such that

(34)$\begin{array}{c}\hfill \widehat{\varphi }\left(x_0\right)<\widehat{\varphi }\left(\overline{x}\right).\end{array}$

Since $\widehat{\varphi }\left(\overline{x}\right)=0,$ it holds that $\widehat{\varphi }\left(x_0\right)<0.$ Thus,

$f\left(x_0\right)-f\left(\overline{x}\right)\in -\mathrm{int}{\mathbb{R}}_{+}^l,$

thereby leading to a contradiction that $\overline{x}$ is a weak Pareto solution of problem (2). Hence, $\overline{x}$ is an optimal solution of problem (33). □

The following result is the Karush–Kuhn–Tucker (KKT) necessary conditions for weak Pareto solutions to the problem (2).

Theorem 7

(necessary condition). Let the $\left(\mathrm{CQ}\right)$ be fulfilled at $\overline{x}\in {\mathbb{R}}^n.$ If $\overline{x}$ is a weak Pareto solution of problem $\left(\right),$ then there are some $\alpha :=({\alpha }_1,\dots ,{\alpha }_l)\in {\mathbb{R}}_{+}^l\backslash \left\{0\right\},$ ${\lambda }_j\ge 0,$ $j\in J,$ such that

(35)$\begin{array}{cc}\hfill & 0\in \sum _{k\in K}{\alpha }_k\left(\partial p_k\left(\overline{x}\right)-{\displaystyle \frac{p_k\left(\overline{x}\right)}{q_k\left(\overline{x}\right)}}\partial q_k\left(\overline{x}\right)\right)+\sum _{j\in J}{\lambda }_j\mathrm{co}\left\{\cup {\partial }_x{g}_j(\overline{x},v_j)|v_j\in {\mathcal{V}}_j\left(\overline{x}\right)\right\},\hfill \\ \hfill & {\lambda }_j\underset{v_j\in {\mathcal{V}}_j}{sup}{g}_j(\overline{x},v_j)=0,j\in J.\hfill \end{array}$

Proof. 

Since $\overline{x}$ is a weak Pareto solution to the problem (2), and with the help of Theorem 6, $\overline{x}$ is an optimal solution to the following minimax fractional program problem:

(36)$\begin{array}{c}\hfill \underset{x\in F_r}{min}\underset{k\in K}{max}{\widehat{f}}_k\left(x\right),\end{array}$

where ${\widehat{f}}_k\left(x\right):=f_k\left(x\right)-f_k\left(\overline{x}\right).$ So, we can employ the KKT conditions in Theorem 1 but applying to the problem (36). Then, we find $\alpha :=({\alpha }_1,\dots ,{\alpha }_l)\in {\mathbb{R}}_{+}^l\backslash \left\{0\right\},$ ${\lambda }_j\ge 0,$ $j\in J,$ such that

(37)$\begin{array}{cc}\hfill & 0\in \sum _{k\in K}{\alpha }_k\left(\partial p_k\left(\overline{x}\right)-{\displaystyle \frac{p_k\left(\overline{x}\right)}{q_k\left(\overline{x}\right)}}\partial q_k\left(\overline{x}\right)\right)+\sum _{j\in J}{\lambda }_j\mathrm{co}\left\{\cup {\partial }_x{g}_j(\overline{x},v_j)|v_j\in {\mathcal{V}}_j\left(\overline{x}\right)\right\},\hfill \\ \hfill & {\alpha }_k\left({\widehat{f}}_k\left(\overline{x}\right)-\underset{k\in K}{max}{\widehat{f}}_k\left(\overline{x}\right)\right)=0,k\in K,\hfill \\ \hfill & {\lambda }_j\underset{v_j\in {\mathcal{V}}_j}{sup}{g}_j(\overline{x},v_j)=0,j\in J.\hfill \end{array}$

It is now clear that (37) implies (35) and thus, the proof is complete. □

Theorem 8

(sufficient condition). Let $\overline{x}\in F_r$ satisfy condition (35). Suppose that $(p,q;g)$ is generalized convex on ${\mathbb{R}}^n$ at $\overline{x};$ then $\overline{x}$ is a weak Pareto solution to the problem (2).

Proof. 

Let $\widehat{\varphi }\left(x\right):={max}_{k\in K}{\widehat{f}}_k\left(x\right)$ with ${\widehat{f}}_k\left(x\right):=f_k\left(x\right)-f_k\left(\overline{x}\right),$ $k\in K,$ $x\in {\mathbb{R}}^n.$ Since $\overline{x}\in F_r$ satisfies condition (35), there exist $\alpha :=({\alpha }_1,\dots ,{\alpha }_l)\in {\mathbb{R}}_{+}^l\backslash \left\{0\right\},$ ${\xi }_k\in \partial p_k\left(\overline{x}\right),$ ${\zeta }_k\in \partial q_k\left(\overline{x}\right),$ $k\in K,$ and ${\lambda }_j\ge 0,$ $j\in J,$ ${\lambda }_{ji}\ge 0,$ ${\eta }_{ji}\in {\partial }_x{g}_j(\overline{x},v_{ji}),$ $v_{ji}\in {\mathcal{V}}_j\left(\overline{x}\right),$ $i\in I_j=\{1,\dots ,i_j\},$ $i_j\in \mathbb{N},$ ${\displaystyle \sum _{i\in I_j}}{\lambda }_{ji}=1,$ such that

$\begin{array}{cc}\hfill & \sum _{k\in K}{\alpha }_k\left({\xi }_k-{\displaystyle \frac{p_k\left(\overline{x}\right)}{q_k\left(\overline{x}\right)}}{\zeta }_k\right)+\sum _{j\in J}{\lambda }_j\left(\sum _{i\in I_j}{\lambda }_{ji}{\eta }_{ji}\right)=0,\hfill \\ \hfill & {\lambda }_j\underset{v_j\in {\mathcal{V}}_j}{sup}{g}_j(\overline{x},v_j)=0,j\in J.\hfill \end{array}$

Thanks to the proof of Theorem 2, we obtain

$\begin{array}{c}\hfill 0\le \sum _{k\in K}{q}_k\left(x\right){\alpha }_k\left({\displaystyle \frac{p_k\left(x\right)}{q_k\left(x\right)}}-{\displaystyle \frac{p_k\left(\overline{x}\right)}{q_k\left(\overline{x}\right)}}\right),\forall x\in F_r,\end{array}$

i.e.,

$\begin{array}{c}\hfill \sum _{k\in K}{q}_k\left(x\right){\alpha }_k\left({\displaystyle \frac{p_k\left(\overline{x}\right)}{q_k\left(\overline{x}\right)}}-{\displaystyle \frac{p_k\left(\overline{x}\right)}{q_k\left(\overline{x}\right)}}\right)\le \sum _{k\in K}{q}_k\left(x\right){\alpha }_k\left({\displaystyle \frac{p_k\left(x\right)}{q_k\left(x\right)}}-{\displaystyle \frac{p_k\left(\overline{x}\right)}{q_k\left(\overline{x}\right)}}\right),\forall x\in F_r.\end{array}$

On the other hand, it holds that

$\begin{array}{cc}\hfill \sum _{k\in K}{q}_k\left(x\right){\alpha }_k\widehat{\varphi }\left(\overline{x}\right)& =\sum _{k\in K}{q}_k\left(x\right){\alpha }_k{\widehat{f}}_k\left(\overline{x}\right)\hfill \\ \hfill & \le \sum _{k\in K}{q}_k\left(x\right){\alpha }_k{\widehat{f}}_k\left(x\right)\hfill \\ \hfill & \le \sum _{k\in K}{q}_k\left(x\right){\alpha }_k\widehat{\varphi }\left(x\right),\hfill \end{array}$

where $\widehat{\varphi }\left(x\right):={max}_{k\in K}{\widehat{f}}_k\left(x\right)$ with ${\widehat{f}}_k\left(x\right):=f_k\left(x\right)-f_k\left(\overline{x}\right),$ $k\in K.$

Due to ${\sum }_{k\in K}{q}_k\left(x\right){\alpha }_k>0,$

$\begin{array}{c}\hfill \widehat{\varphi }\left(\overline{x}\right)\le \widehat{\varphi }\left(x\right),\forall x\in F_r.\end{array}$

In other words, we obtain

$0\le \underset{k\in K}{max}\{f_k\left(x\right)-f_k\left(\overline{x}\right)\},\forall x\in F_r,$

which entails that

$f\left(x\right)-f\left(\overline{x}\right)\notin -\mathrm{int}{\mathbb{R}}_{+}^l,\forall x\in F_r,$

Hence, $\overline{x}$ is a weak Pareto solution to the problem (2). □

Author Contributions

Conceptualization, H.L. and Z.H.; methodology, Z.H. and D.S.K.; investigation, H.L. and Z.H.; resources, Z.H.; writing—original draft preparation, Z.H.; writing—review and editing, H.L. and D.S.K. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

No data were used for the research described in the paper.

Conflicts of Interest

The authors declare no conflicts of interest.

Footnotes

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