Second-order duality for a nondifferentiable minimax fractional programming under generalized α-univexity

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Research

Abstract

In this paper, we concentrate our study to derive appropriate duality theorems for two types of second-order dual models of a nondifferentiable minimax fractional programming problem involving second-order α-univex functions. Examples to show the existence of α-univex functions have also been illustrated. Several known results including many recent works are obtained as special cases.

MSC:49J35, 90C32, 49N15.

Keywords

minimax programming fractional programming nondifferentiable programming second-order duality α-univexity

1 Introduction

After Schmitendorf , who derived necessary and sufficient optimality conditions for static minimax problems, much attention has been paid to optimality conditions and duality theorems for minimax fractional programming problems [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]. For the theory, algorithms, and applications of some minimax problems, the reader is referred to .

In this paper, we consider the following nondifferentiable minimax fractional programming problem:

where Y is a compact subset of ${R}^{l}$, $f\left(\cdot ,\cdot \right):{R}^{n}×{R}^{l}\to R$, $h\left(\cdot ,\cdot \right):{R}^{n}×{R}^{l}\to R$ are twice continuously differentiable on ${R}^{n}×{R}^{l}$ and $g\left(\cdot \right):{R}^{n}\to {R}^{m}$ is twice continuously differentiable on ${R}^{n}$, B, and D are a $n×n$ positive semidefinite matrix, $f\left(x,y\right)+{\left({x}^{T}Bx\right)}^{1/2}\ge 0$, and $h\left(x,y\right)-{\left({x}^{T}Dx\right)}^{1/2}>0$ for each $\left(x,y\right)\in \mathfrak{J}×Y$, where $\mathfrak{J}=\left\{x\in {R}^{n}:g\left(x\right)\le 0\right\}$.

Motivated by [7, 14, 15], Yang and Hou  formulated a dual model for fractional minimax programming problem and proved duality theorems under generalized convex functions. Ahmad and Husain  extended this model to nondifferentiable and obtained duality relations involving $\left(F,\alpha ,\rho ,d\right)$-pseudoconvex functions. Jayswal  studied duality theorems for another two duals of (P) under α-univex functions. Recently, Ahmad et al. derived the sufficient optimality condition for (P) and established duality relations for its dual problem under $B\text{-}\left(p,r\right)$-invexity assumptions. The papers [2, 4, 5, 6, 7, 11, 12, 13, 14, 15, 17] involved the study of first-order duality for minimax fractional programming problems.

The concept of second-order duality in nonlinear programming problems was first introduced by Mangasarian . One significant practical application of second-order dual over first-order is that it may provide tighter bounds for the value of objective function because there are more parameters involved. Hanson  has shown the other advantage of second-order duality by citing an example, that is, if a feasible point of the primal is given and first-order duality conditions do not apply (infeasible), then we may use second-order duality to provide a lower bound for the value of primal problem.

Recently, several researchers [3, 8, 9, 10, 16] considered second-order dual for minimax fractional programming problems. Husain et al. first formulated second-order dual models for a minimax fractional programming problem and established duality relations involving η-bonvex functions. This work was later on generalized in  by introducing an additional vector r to the dual models, and in Sharma and Gulati  by proving the results under second-order generalized α-type I univex functions. The work cited in [3, 8, 10, 16] involves differentiable minimax fractional programming problems. Recently, Hu et al. proved appropriate duality theorems for a second-order dual model of (P) under η-pseudobonvexity/η-quasibonvexity assumptions. In this paper, we formulate two types of second-order dual models for (P) and then derive weak, strong, and strict converse duality theorems under generalized α-univexity assumptions. Further, examples have been illustrated to show the existence of second-order α-univex functions. Our study extends some of the known results of the literature [5, 6, 11, 12, 14].

2 Notations and preliminaries

For each $\left(x,y\right)\in {R}^{n}×{R}^{l}$ and $M=\left\{1,2,\dots ,m\right\}$, we define
Definition 2.1 Let $\zeta :X\to R$ ($X\subseteq {R}^{n}$) be a twice differentiable function. Then ζ is said to be second-order α-univex at $u\in X$, if there exist $\eta :X×X\to {R}^{n}$, ${b}_{0}:X×X\to {R}_{+}$, ${\varphi }_{0}:R\to R$, and $\alpha :X×X\to {R}_{+}\mathrm{\setminus }\left\{0\right\}$ such that for all $x\in X$ and $p\in {R}^{n}$, we have
$\begin{array}{c}{b}_{0}{\varphi }_{0}\left[\zeta \left(x\right)-\zeta \left(u\right)+\frac{1}{2}{p}^{T}{\mathrm{\nabla }}^{2}\zeta \left(u\right)p\right]\hfill \\ \phantom{\rule{1em}{0ex}}\ge \alpha \left(x,u\right){\eta }^{T}\left(x,u\right)\left[\mathrm{\nabla }\zeta \left(u\right)+{\mathrm{\nabla }}^{2}\zeta \left(u\right)p\right].\hfill \end{array}$
Example 2.1 Let $\zeta :X\to R$ be defined as $\zeta \left(x\right)={e}^{x}+{sin}^{2}x+{x}^{2}$, where $X=\left(-1,\mathrm{\infty }\right)$. Also, let ${\varphi }_{0}\left(t\right)=t+18$, ${b}_{0}\left(x,u\right)=u+1$, $\alpha \left(x,u\right)=\frac{{u}^{2}+2}{x+1}$ and $\eta \left(x,u\right)=x+u$. The function ζ is second-order α-univex at $u=1$, since

But every α-univex function need not be invex. To show this, consider the following example.

Example 2.2 Let $\mathrm{\Omega }:X=\left(0,\mathrm{\infty }\right)\to R$ be defined as $\mathrm{\Omega }\left(x\right)=-{x}^{2}$. Let ${\varphi }_{0}\left(t\right)=-t$, ${b}_{0}\left(x,u\right)=\frac{1}{u}$, $\alpha \left(x,u\right)=2u,$ and $\eta \left(x,u\right)=\frac{1}{2u}$. Then we have
Hence, the function Ω is second-order α-univex but not invex, since for $x=3$, $u=2$, and $p=1$, we obtain
$\mathrm{\Omega }\left(x\right)-\mathrm{\Omega }\left(u\right)+\frac{1}{2}{p}^{T}{\mathrm{\nabla }}^{2}\mathrm{\Omega }\left(u\right)p-{\eta }^{T}\left(x,u\right)\left[\mathrm{\nabla }\mathrm{\Omega }\left(u\right)+{\mathrm{\nabla }}^{2}\mathrm{\Omega }\left(u\right)p\right]=-4.5<0.$

Lemma 2.1 (Generalized Schwartz inequality)

Let B be a positive semidefinite matrix of order n. Then, for all$x,w\in {R}^{n}$,
${x}^{T}Bw\le {\left({x}^{T}Bx\right)}^{1/2}{\left({w}^{T}Bw\right)}^{1/2}.$

The equality holds if$Bx=\lambda Bw$for some$\lambda \ge 0$.

Following Theorem 2.1 (, Theorem 3.1) will be required to prove the strong duality theorem.

Theorem 2.1 (Necessary condition)

If${x}^{\ast }$is an optimal solution of problem (P) satisfying${x}^{\ast T}B{x}^{\ast }>0$, ${x}^{\ast T}D{x}^{\ast }>0$, and$\mathrm{\nabla }{g}_{j}\left({x}^{\ast }\right)$, $j\in J\left({x}^{\ast }\right)$are linearly independent, then there exist$\left({s}^{\ast },{t}^{\ast },\stackrel{˜}{y}\right)\in K\left({x}^{\ast }\right)$, ${k}_{0}\in {R}_{+}$, $w,v\in {R}^{n}$and${\mu }^{\ast }\in {R}_{+}^{m}$such that
In the above theorem, both matrices B and D are positive semidefinite at ${x}^{\ast }$. If either ${x}^{\ast T}B{x}^{\ast }$ or ${x}^{\ast T}D{x}^{\ast }$ is zero, then the functions involved in the objective of problem (P) are not differentiable. To derive necessary conditions under this situation, for $\left({s}^{\ast },{t}^{\ast },\stackrel{˜}{y}\right)\in K\left({x}^{\ast }\right)$, we define
$\begin{array}{rcl}{Z}_{\stackrel{˜}{y}}\left({x}^{\ast }\right)& =& \left\{z\in {R}^{n}:{z}^{T}\mathrm{\nabla }{g}_{j}\left({x}^{\ast }\right)\le 0,j\in J\left({x}^{\ast }\right),\\ \text{with any one of the next conditions (i)-(iii) holds}\right\}.\end{array}$

If in addition, we insert the condition ${Z}_{\stackrel{˜}{y}}\left({x}^{\ast }\right)=\varphi$, then the result of Theorem 2.1 still holds.

For the sake of convenience, let
${\psi }_{1}\left(\cdot \right)={\xi }_{1}\left(\cdot \right)+\sum _{j=1}^{m}{\mu }_{j}\left({g}_{j}\left(\cdot \right)-{g}_{j}\left(z\right)\right)$
(2.6)
and
$\begin{array}{rcl}{\psi }_{2}\left(\cdot \right)& =& \left[\sum _{i=1}^{s}{t}_{i}\left(h\left(z,{\stackrel{˜}{y}}_{i}\right)-{z}^{T}Dv\right)\right]\left[\sum _{i=1}^{s}{t}_{i}\left(f\left(\cdot ,{\stackrel{˜}{y}}_{i}\right)+{\left(\cdot \right)}^{T}Bw\right)+\sum _{j=1}^{m}{\mu }_{j}{g}_{j}\left(\cdot \right)\right]\\ -\left[\sum _{i=1}^{s}{t}_{i}\left(f\left(z,{\stackrel{˜}{y}}_{i}\right)+{z}^{T}Bw\right)+\sum _{j=1}^{m}{\mu }_{j}{g}_{j}\left(z\right)\right]\left[\sum _{i=1}^{s}{t}_{i}\left(h\left(\cdot ,{\stackrel{˜}{y}}_{i}\right)-{\left(\cdot \right)}^{T}Dv\right)\right],\end{array}$
where
${\xi }_{1}\left(\cdot \right)=\sum _{i=1}^{s}{t}_{i}\left[\left(h\left(z,{\stackrel{˜}{y}}_{i}\right)-{z}^{T}Dv\right)\left(f\left(\cdot ,{\stackrel{˜}{y}}_{i}\right)+{\left(\cdot \right)}^{T}Bw\right)-\left(f\left(z,{\stackrel{˜}{y}}_{i}\right)+{z}^{T}Bw\right)\left(h\left(\cdot ,{\stackrel{˜}{y}}_{i}\right)-{\left(\cdot \right)}^{T}Dv\right)\right].$

3 Model I

In this section, we consider the following second-order dual problem for (P):
$\underset{\left(s,t,\stackrel{˜}{y}\right)\in K\left(z\right)}{max}\underset{\left(z,\mu ,w,v,p\right)\in {H}_{1}\left(s,t,\stackrel{˜}{y}\right)}{sup}F\left(z\right),$
(DM1)
where $F\left(z\right)={sup}_{y\in Y}\frac{f\left(z,y\right)+{\left({z}^{T}Bz\right)}^{1/2}}{h\left(z,y\right)-{\left({z}^{T}Dz\right)}^{1/2}}$ and ${H}_{1}\left(s,t,\stackrel{˜}{y}\right)$ denotes the set of all $\left(z,\mu ,w,v,p\right)\in {R}^{n}×{R}_{+}^{m}×{R}^{n}×{R}^{n}×{R}^{n}$ satisfying

If the set ${H}_{1}\left(s,t,\stackrel{˜}{y}\right)=\varphi$, we define the supremum of $F\left(z\right)$ over ${H}_{1}\left(s,t,\stackrel{˜}{y}\right)$ equal to −∞.

Remark 3.1 If $p=0$, then using (3.3), the above dual model reduces to the problems studied in [6, 11, 12]. Further, if B and D are zero matrices of order n, then (DM1) becomes the dual model considered in .

Next, we establish duality relations between primal (P) and dual (DM1).

Theorem 3.1 (Weak duality)

Let x and$\left(z,\mu ,w,v,s,t,\stackrel{˜}{y},p\right)$are feasible solutions of (P) and (DM1), respectively. Assume that
1. (i)

${\psi }_{1}\left(\cdot \right)$ is second-order α-univex at z,

2. (ii)

${\varphi }_{0}\left(a\right)\ge 0⇒a\ge 0$ and ${b}_{0}\left(x,z\right)>0$.

Then
$\underset{\stackrel{˜}{y}\in Y}{sup}\frac{f\left(x,\stackrel{˜}{y}\right)+{\left({x}^{T}Bx\right)}^{1/2}}{h\left(x,\stackrel{˜}{y}\right)-{\left({x}^{T}Dx\right)}^{1/2}}\ge F\left(z\right).$
Proof Assume on contrary to the result that
$\underset{\stackrel{˜}{y}\in Y}{sup}\frac{f\left(x,\stackrel{˜}{y}\right)+{\left({x}^{T}Bx\right)}^{1/2}}{h\left(x,\stackrel{˜}{y}\right)-{\left({x}^{T}Dx\right)}^{1/2}}
(3.4)
Since ${\stackrel{˜}{y}}_{i}\in Y\left(z\right)$, $i=1,2,\dots ,s$, we have
$F\left(z\right)=\frac{f\left(z,{\stackrel{˜}{y}}_{i}\right)+{\left({z}^{T}Bz\right)}^{1/2}}{h\left(z,{\stackrel{˜}{y}}_{i}\right)-{\left({z}^{T}Dz\right)}^{1/2}}.$
(3.5)
From (3.4) and (3.5), for $i=1,2,\dots ,s$, we get
$\frac{f\left(x,{\stackrel{˜}{y}}_{i}\right)+{\left({x}^{T}Bx\right)}^{1/2}}{h\left(x,{\stackrel{˜}{y}}_{i}\right)-{\left({x}^{T}Dx\right)}^{1/2}}\le \underset{\stackrel{˜}{y}\in Y}{sup}\frac{f\left(x,\stackrel{˜}{y}\right)+{\left({x}^{T}Bx\right)}^{1/2}}{h\left(x,\stackrel{˜}{y}\right)-{\left({x}^{T}Dx\right)}^{1/2}}<\frac{f\left(z,{\stackrel{˜}{y}}_{i}\right)+{\left({z}^{T}Bz\right)}^{1/2}}{h\left(z,{\stackrel{˜}{y}}_{i}\right)-{\left({z}^{T}Dz\right)}^{1/2}}.$
This further from ${t}_{i}\ge 0$, $i=1,2,\dots ,s$, $t\ne 0$ and ${\stackrel{˜}{y}}_{i}\in Y\left(z\right)$, we obtain
Now,
Therefore,
${\xi }_{1}\left(x\right)<0={\xi }_{1}\left(z\right).$
(3.7)
By hypothesis (i), we have
${b}_{0}{\varphi }_{0}\left[{\psi }_{1}\left(x\right)-{\psi }_{1}\left(z\right)+\frac{1}{2}{p}^{T}{\mathrm{\nabla }}^{2}{\psi }_{1}\left(z\right)p\right]\ge \alpha \left(x,z\right){\eta }^{T}\left(x,z\right)\left\{\mathrm{\nabla }{\psi }_{1}\left(z\right)+{\mathrm{\nabla }}^{2}{\psi }_{1}\left(z\right)p\right\}.$
This follows from (3.1) that
${b}_{0}{\varphi }_{0}\left[{\psi }_{1}\left(x\right)-{\psi }_{1}\left(z\right)+\frac{1}{2}{p}^{T}{\mathrm{\nabla }}^{2}{\psi }_{1}\left(z\right)p\right]\ge 0$
which using hypothesis (ii) yields
${\psi }_{1}\left(x\right)-{\psi }_{1}\left(z\right)+\frac{1}{2}{p}^{T}{\mathrm{\nabla }}^{2}{\psi }_{1}\left(z\right)p\ge 0.$
This further from (2.6), (3.2), and the feasibility of x implies
${\xi }_{1}\left(x\right)\ge -\sum _{j=1}^{m}{\mu }_{j}{g}_{j}\left(x\right)\ge 0={\xi }_{1}\left(z\right).$

This contradicts (3.7), hence the result. □

Theorem 3.2 (Strong duality)

Let${x}^{\ast }$be an optimal solution for (P) and let$\mathrm{\nabla }{g}_{j}\left({x}^{\ast }\right)$, $j\in J\left({x}^{\ast }\right)$be linearly independent. Then there exist$\left({s}^{\ast },{t}^{\ast },{\stackrel{˜}{y}}^{\ast }\right)\in K\left({x}^{\ast }\right)$and$\left({x}^{\ast },{\mu }^{\ast },{w}^{\ast },{v}^{\ast },{p}^{\ast }=0\right)\in {H}_{1}\left({s}^{\ast },{t}^{\ast },{\stackrel{˜}{y}}^{\ast }\right)$, such that$\left({x}^{\ast },{\mu }^{\ast },{w}^{\ast },{v}^{\ast },{s}^{\ast },{t}^{\ast },{\stackrel{˜}{y}}^{\ast },{p}^{\ast }=0\right)$is feasible solution of (DM1) and the two objectives have same values. If, in addition, the assumptions of Theorem  3.1 hold for all feasible solutions$\left(x,\mu ,w,v,s,t,\stackrel{˜}{y},p\right)$of (DM1), then$\left({x}^{\ast },{\mu }^{\ast },{w}^{\ast },{v}^{\ast },{s}^{\ast },{t}^{\ast },{\stackrel{˜}{y}}^{\ast },{p}^{\ast }=0\right)$is an optimal solution of (DM1).

Proof Since ${x}^{\ast }$ is an optimal solution of (P) and $\mathrm{\nabla }{g}_{j}\left({x}^{\ast }\right)$, $j\in J\left({x}^{\ast }\right)$ are linearly independent, then by Theorem 2.1, there exist $\left({s}^{\ast },{t}^{\ast },{\stackrel{˜}{y}}^{\ast }\right)\in K\left({x}^{\ast }\right)$ and $\left({x}^{\ast },{\mu }^{\ast },{w}^{\ast },{v}^{\ast },{p}^{\ast }=0\right)\in {H}_{1}\left({s}^{\ast },{t}^{\ast },{\stackrel{˜}{y}}^{\ast }\right)$ such that $\left({x}^{\ast },{\mu }^{\ast },{w}^{\ast },{v}^{\ast },{s}^{\ast },{t}^{\ast },{\stackrel{˜}{y}}^{\ast },{p}^{\ast }=0\right)$ is feasible solution of (DM1) and the two objectives have same values. Optimality of $\left({x}^{\ast },{\mu }^{\ast },{w}^{\ast },{v}^{\ast },{s}^{\ast },{t}^{\ast },{\stackrel{˜}{y}}^{\ast },{p}^{\ast }=0\right)$ for (DM1), thus follows from Theorem 3.1. □

Theorem 3.3 (Strict converse duality)

Let${x}^{\ast }$be an optimal solution to (P) and$\left({z}^{\ast },{\mu }^{\ast },{w}^{\ast },{v}^{\ast },{s}^{\ast },{t}^{\ast },{\stackrel{˜}{y}}^{\ast },{p}^{\ast }\right)$be an optimal solution to (DM1). Assume that
1. (i)

${\psi }_{1}\left(\cdot \right)$ is strictly second-order α-univex at ${z}^{\ast }$,

2. (ii)

$\left\{\mathrm{\nabla }{g}_{j}\left({x}^{\ast }\right),j\in J\left({x}^{\ast }\right)\right\}$, are linearly independent,

3. (iii)

${\varphi }_{0}\left(a\right)>0⇒a>0$ and ${b}_{0}\left({x}^{\ast },{z}^{\ast }\right)>0$.

Then${z}^{\ast }={x}^{\ast }$.

Proof By the strict α-univexity of ${\psi }_{1}\left(\cdot \right)$ at ${z}^{\ast }$, we get
which in view of (3.1) and hypothesis (iii) give
${\psi }_{1}\left({x}^{\ast }\right)-{\psi }_{1}\left({z}^{\ast }\right)+\frac{1}{2}{p}^{\ast T}{\mathrm{\nabla }}^{2}{\psi }_{1}\left({z}^{\ast }\right){p}^{\ast }>0.$
Using (2.6), (3.2), and feasibility of ${x}^{\ast }$ in above, we obtain
${\xi }_{1}\left({x}^{\ast }\right)>0={\xi }_{1}\left({z}^{\ast }\right).$
(3.8)
Now, we shall assume that ${z}^{\ast }\ne {x}^{\ast }$ and reach a contradiction. Since ${x}^{\ast }$ and $\left({z}^{\ast },{\mu }^{\ast },{w}^{\ast },{v}^{\ast },{s}^{\ast },{t}^{\ast },{\stackrel{˜}{y}}^{\ast },{p}^{\ast }\right)$ are optimal solutions to (P) and (DM1), respectively, and $\left\{\mathrm{\nabla }{g}_{j}\left({x}^{\ast }\right),j\in J\left({x}^{\ast }\right)\right\}$, are linearly independent, by Theorem 3.2, we get
$\underset{{\stackrel{˜}{y}}^{\ast }\in Y}{sup}\frac{f\left({x}^{\ast },{\stackrel{˜}{y}}^{\ast }\right)+{\left({x}^{\ast T}B{x}^{\ast }\right)}^{1/2}}{h\left({x}^{\ast },{\stackrel{˜}{y}}^{\ast }\right)-{\left({x}^{\ast T}D{x}^{\ast }\right)}^{1/2}}=F\left({z}^{\ast }\right).$
(3.9)
Since ${\stackrel{˜}{y}}_{i}^{\ast }\in Y\left({z}^{\ast }\right)$, $i=1,2,\dots ,{s}^{\ast }$, we have
$F\left({z}^{\ast }\right)=\frac{f\left({z}^{\ast },{\stackrel{˜}{y}}_{i}^{\ast }\right)+{\left({z}^{\ast T}B{z}^{\ast }\right)}^{1/2}}{h\left({z}^{\ast },{\stackrel{˜}{y}}_{i}^{\ast }\right)-{\left({z}^{\ast T}D{z}^{\ast }\right)}^{1/2}}.$
(3.10)
By (3.9) and (3.10), we get
$\begin{array}{c}\left[\left(h\left({z}^{\ast },{\stackrel{˜}{y}}_{i}^{\ast }\right)-{\left({z}^{\ast T}D{z}^{\ast }\right)}^{1/2}\right)\left(f\left({x}^{\ast },{\stackrel{˜}{y}}_{i}^{\ast }\right)+{\left({x}^{\ast T}B{x}^{\ast }\right)}^{1/2}\right)\hfill \\ \phantom{\rule{1em}{0ex}}-\left(f\left({z}^{\ast },{\stackrel{˜}{y}}_{i}^{\ast }\right)+{\left({z}^{\ast T}B{z}^{\ast }\right)}^{1/2}\right)\left(h\left({x}^{\ast },{\stackrel{˜}{y}}_{i}^{\ast }\right)-{\left({x}^{\ast T}D{x}^{\ast }\right)}^{1/2}\right)\right]\le 0,\hfill \end{array}$
for all $i=1,2,\dots ,{s}^{\ast }$ and ${\stackrel{˜}{y}}_{i}^{\ast }\in Y$. From ${\stackrel{˜}{y}}_{i}^{\ast }\in Y\left({z}^{\ast }\right)\subset Y$ and ${t}^{\ast }\in {R}_{+}^{{s}^{\ast }}$, with ${\sum }_{i=1}^{{s}^{\ast }}{t}_{i}^{\ast }=1$, we obtain
From Lemma 2.1, (3.3), and (3.11), we have
$\begin{array}{rcl}{\xi }_{1}\left({x}^{\ast }\right)& =& \sum _{i=1}^{{s}^{\ast }}{t}_{i}^{\ast }\left[\left(h\left({z}^{\ast },{\stackrel{˜}{y}}_{i}^{\ast }\right)-{z}^{\ast T}D{v}^{\ast }\right)\left(f\left({x}^{\ast },{\stackrel{˜}{y}}_{i}^{\ast }\right)+{x}^{\ast T}B{w}^{\ast }\right)\\ -\left(f\left({z}^{\ast },{\stackrel{˜}{y}}_{i}^{\ast }\right)+{z}^{\ast T}B{w}^{\ast }\right)\left(h\left({x}^{\ast },{\stackrel{˜}{y}}_{i}^{\ast }\right)-{x}^{\ast T}D{v}^{\ast }\right)\right]\\ \le & \sum _{i=1}^{{s}^{\ast }}{t}_{i}^{\ast }\left[\left(h\left({z}^{\ast },{\stackrel{˜}{y}}_{i}^{\ast }\right)-{\left({z}^{\ast T}D{z}^{\ast }\right)}^{1/2}\right)\left(f\left({x}^{\ast },{\stackrel{˜}{y}}_{i}^{\ast }\right)+{\left({x}^{\ast T}B{x}^{\ast }\right)}^{1/2}\right)\\ -\left(f\left({z}^{\ast },{\stackrel{˜}{y}}_{i}^{\ast }\right)+{\left({z}^{\ast T}B{z}^{\ast }\right)}^{1/2}\right)\left(h\left({x}^{\ast },{\stackrel{˜}{y}}_{i}^{\ast }\right)-{\left({x}^{\ast T}D{x}^{\ast }\right)}^{1/2}\right)\right]\\ \le & 0={\xi }_{1}\left({z}^{\ast }\right),\end{array}$

which contradicts (3.8), hence the result. □

4 Model II

In this section, we consider another dual problem to (P):
$\underset{\left(s,t,\stackrel{˜}{y}\right)\in K\left(z\right)}{max}\underset{\left(z,\mu ,w,v,p\right)\in {H}_{2}\left(s,t,\stackrel{˜}{y}\right)}{sup}\frac{{\sum }_{i=1}^{s}{t}_{i}\left(f\left(z,{\stackrel{˜}{y}}_{i}\right)+{\left({z}^{T}Bz\right)}^{1/2}\right)+{\sum }_{j=1}^{m}{\mu }_{j}{g}_{j}\left(z\right)}{{\sum }_{i=1}^{s}{t}_{i}\left(h\left(z,{\stackrel{˜}{y}}_{i}\right)-{\left({z}^{T}Dz\right)}^{1/2}\right)},$
(DM2)
where ${H}_{2}\left(s,t,\stackrel{˜}{y}\right)$ denotes the set of all $\left(z,\mu ,w,v,p\right)\in {R}^{n}×{R}_{+}^{m}×{R}^{n}×{R}^{n}×{R}^{n}$ satisfying

If the set ${H}_{2}\left(s,t,\stackrel{˜}{y}\right)$ is empty, we define the supremum in (DM2) over ${H}_{2}\left(s,t,\stackrel{˜}{y}\right)$ equal to −∞.

Remark 4.1 If $p=0$, then using (4.3), the above dual model becomes the dual model considered in [5, 11, 12]. In addition, if B and D are zero matrices of order n, then (DM2) reduces to the problem studied in .

Now, we obtain the following appropriate duality theorems between (P) and (DM2).

Theorem 4.1 (Weak duality)

Let x and$\left(z,\mu ,w,v,s,t,\stackrel{˜}{y},p\right)$are feasible solutions of (P) and (DM2), respectively. Suppose that the following conditions are satisfied:
1. (i)

${\psi }_{2}\left(\cdot \right)$ is second-order α-univex at z,

2. (ii)

${\varphi }_{0}\left(a\right)\ge 0⇒a\ge 0$ and ${b}_{0}\left(x,z\right)>0$.

Then
$\underset{\stackrel{˜}{y}\in Y}{sup}\frac{f\left(x,\stackrel{˜}{y}\right)+{\left({x}^{T}Bx\right)}^{1/2}}{h\left(x,\stackrel{˜}{y}\right)-{\left({x}^{T}Dx\right)}^{1/2}}\ge \frac{{\sum }_{i=1}^{s}{t}_{i}\left(f\left(z,{\stackrel{˜}{y}}_{i}\right)+{\left({z}^{T}Bz\right)}^{1/2}\right)+{\sum }_{j=1}^{m}{\mu }_{j}{g}_{j}\left(z\right)}{{\sum }_{i=1}^{s}{t}_{i}\left(h\left(z,{\stackrel{˜}{y}}_{i}\right)-{\left({z}^{T}Dz\right)}^{1/2}\right)}.$
Proof Assume on contrary to the result that
$\underset{\stackrel{˜}{y}\in Y}{sup}\frac{f\left(x,\stackrel{˜}{y}\right)+{\left({x}^{T}Bx\right)}^{1/2}}{h\left(x,\stackrel{˜}{y}\right)-{\left({x}^{T}Dx\right)}^{1/2}}<\frac{{\sum }_{i=1}^{s}{t}_{i}\left(f\left(z,{\stackrel{˜}{y}}_{i}\right)+{\left({z}^{T}Bz\right)}^{1/2}\right)+{\sum }_{j=1}^{m}{\mu }_{j}{g}_{j}\left(z\right)}{{\sum }_{i=1}^{s}{t}_{i}\left(h\left(z,{\stackrel{˜}{y}}_{i}\right)-{\left({z}^{T}Dz\right)}^{1/2}\right)}$
Using ${t}_{i}\ge 0$, $i=1,2,\dots ,s$ and (4.3) in above, we have
Now,
Hence,
${\psi }_{2}\left(x\right)<0={\psi }_{2}\left(z\right).$
(4.5)
Now, by the second-order α-univexity of ${\psi }_{2}\left(\cdot \right)$ at z, we get
${b}_{0}{\varphi }_{0}\left[{\psi }_{2}\left(x\right)-{\psi }_{2}\left(z\right)+\frac{1}{2}{p}^{T}{\mathrm{\nabla }}^{2}{\psi }_{2}\left(z\right)p\right]\ge {\eta }^{T}\left(x,z\right)\alpha \left(x,z\right)\left\{\mathrm{\nabla }{\psi }_{2}\left(z\right)+{\mathrm{\nabla }}^{2}{\psi }_{2}\left(z\right)p\right\}$
which using (4.1) and hypothesis (ii) give
${\psi }_{2}\left(x\right)-{\psi }_{2}\left(z\right)+\frac{1}{2}{p}^{T}{\mathrm{\nabla }}^{2}{\psi }_{2}\left(z\right)p\ge 0.$
This from (4.2) follows that
${\psi }_{2}\left(x\right)\ge {\psi }_{2}\left(z\right)$

which contradicts (4.5). This proves the theorem. □

By a similar way, we can prove the following theorems between (P) and (DM2).

Theorem 4.2 (Strong duality)

Let${x}^{\ast }$be an optimal solution for (P) and let$\mathrm{\nabla }{g}_{j}\left({x}^{\ast }\right)$, $j\in J\left({x}^{\ast }\right)$be linearly independent. Then there exist$\left({s}^{\ast },{t}^{\ast },{\stackrel{˜}{y}}^{\ast }\right)\in K\left({x}^{\ast }\right)$and$\left({x}^{\ast },{\mu }^{\ast },{w}^{\ast },{v}^{\ast },{p}^{\ast }=0\right)\in {H}_{2}\left({s}^{\ast },{t}^{\ast },{\stackrel{˜}{y}}^{\ast }\right)$, such that$\left({x}^{\ast },{\mu }^{\ast },{w}^{\ast },{v}^{\ast },{s}^{\ast },{t}^{\ast },{\stackrel{˜}{y}}^{\ast },{p}^{\ast }=0\right)$is feasible solution of (DM2) and the two objectives have same values. If, in addition, the assumptions of weak duality hold for all feasible solutions$\left(x,\mu ,w,v,s,t,\stackrel{˜}{y},p\right)$of (DM2), then$\left({x}^{\ast },{\mu }^{\ast },{w}^{\ast },{v}^{\ast },{s}^{\ast },{t}^{\ast },{\stackrel{˜}{y}}^{\ast },{p}^{\ast }=0\right)$is an optimal solution of (DM2).

Theorem 4.3 (Strict converse duality)

Let${x}^{\ast }$and$\left({z}^{\ast },{\mu }^{\ast },{w}^{\ast },{v}^{\ast },{s}^{\ast },{t}^{\ast },{\stackrel{˜}{y}}^{\ast },{p}^{\ast }\right)$are optimal solutions of (P) and (DM2), respectively. Assume that
1. (i)

${\psi }_{2}\left(\cdot \right)$ is strictly second-order α-univex at z,

2. (ii)

$\left\{\mathrm{\nabla }{g}_{j}\left({x}^{\ast }\right),j\in J\left({x}^{\ast }\right)\right\}$ are linearly independent,

3. (iii)

${\varphi }_{0}\left(a\right)>0⇒a>0$ and ${b}_{0}\left({x}^{\ast },{z}^{\ast }\right)>0$.

Then${z}^{\ast }={x}^{\ast }$.

5 Concluding remarks

In the present work, we have formulated two types of second-order dual models for a nondifferentiable minimax fractional programming problems and proved appropriate duality relations involving second-order α-univex functions. Further, examples have been illustrated to show the existence of such type of functions. Now, the question arises whether or not the results can be further extended to a higher-order nondifferentiable minimax fractional programming problem.

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