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

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.

1 Introduction

After Schmitendorf [1], 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 [2–17]. For the theory, algorithms, and applications of some minimax problems, the reader is referred to [18].

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

(P)

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

Motivated by [7, 14, 15], Yang and Hou [17] formulated a dual model for fractional minimax programming problem and proved duality theorems under generalized convex functions. Ahmad and Husain [5] extended this model to nondifferentiable and obtained duality relations involving $(F,\alpha ,\rho ,d)$-pseudoconvex functions. Jayswal [11] studied duality theorems for another two duals of (P) under α-univex functions. Recently, Ahmad et al.[4] derived the sufficient optimality condition for (P) and established duality relations for its dual problem under $B\text{-}(p,r)$-invexity assumptions. The papers [2, 4–7, 11–15, 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 [19]. 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 [20] 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, 8–10, 16] considered second-order dual for minimax fractional programming problems. Husain et al.[8] 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 [10] by introducing an additional vector r to the dual models, and in Sharma and Gulati [16] 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.[9] 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 $(x,y)\in R^n\times R^l$ and $M=\{1,2,\dots ,m\}$, 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 $\eta :X\times X\to R^n$, $b_0:X\times X\to R_{+}$, ${\varphi }_0:R\to R$, and $\alpha :X\times X\to R_{+}\mathrm{\setminus }\{0\}$ such that for all $x\in X$ and $p\in R^n$, we have

$$\begin{array}{c}{b}_0{\varphi }_0[\zeta (x)-\zeta (u)+\frac{1}{2}{p}^T{\mathrm{\nabla }}^2\zeta (u)p]\hfill \\ \ge \alpha (x,u){\eta }^T(x,u)[\mathrm{\nabla }\zeta (u)+{\mathrm{\nabla }}^2\zeta (u)p].\hfill \end{array}$$

Example 2.1 Let $\zeta :X\to R$ be defined as $\zeta (x)=e^x+{sin}^{2}x+x^2$, where $X=(-1,\mathrm{\infty })$. Also, let ${\varphi }_0(t)=t+18$, $b_0(x,u)=u+1$, $\alpha (x,u)=\frac{u^2+2}{x+1}$ and $\eta (x,u)=x+u$. The function ζ is second-order α-univex at $u=1$, since

$$\begin{array}{c}{b}_0{\varphi }_0[\zeta (x)-\zeta (u)+\frac{1}{2}{p}^T{\mathrm{\nabla }}^2\zeta (u)p]-\alpha (x,u){\eta }^T(x,u)[\mathrm{\nabla }\zeta (u)+{\mathrm{\nabla }}^2\zeta (u)p]\hfill \\ =2(e^x+{sin}^{2}x+x^2)+1.521+3.886{(p-1.5)}^2\hfill \\ \ge 0\text{for all }x\in X\text{ and }p\in R.\hfill \end{array}$$

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

Example 2.2 Let $\mathrm{\Omega }:X=(0,\mathrm{\infty })\to R$ be defined as $\mathrm{\Omega }(x)=-x^2$. Let ${\varphi }_0(t)=-t$, $b_0(x,u)=\frac{1}{u}$, $\alpha (x,u)=2u,$ and $\eta (x,u)=\frac{1}{2u}$. Then we have

$$\begin{array}{c}{b}_0{\varphi }_0[\mathrm{\Omega }(x)-\mathrm{\Omega }(u)+\frac{1}{2}{p}^T{\mathrm{\nabla }}^2\mathrm{\Omega }(u)p]-\alpha (x,u){\eta }^T(x,u)[\mathrm{\nabla }\mathrm{\Omega }(u)+{\mathrm{\nabla }}^2\mathrm{\Omega }(u)p]\hfill \\ =\frac{1}{u}[x^2+{(p+u)}^2]\ge 0\text{for all }x,u\in X\text{ and }p\in R.\hfill \end{array}$$

Hence, the function Ω is second-order α-univex but not invex, since for $x=3$, $u=2$, and $p=1$, we obtain

$$\mathrm{\Omega }(x)-\mathrm{\Omega }(u)+\frac{1}{2}{p}^T{\mathrm{\nabla }}^2\mathrm{\Omega }(u)p-{\eta }^T(x,u)[\mathrm{\nabla }\mathrm{\Omega }(u)+{\mathrm{\nabla }}^2\mathrm{\Omega }(u)p]=-4.5<0.$$

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 ([13], 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$\mathrm{\nabla }{g}_j(x^{\ast })$, $j\in J(x^{\ast })$are linearly independent, then there exist$(s^{\ast },t^{\ast },\tilde{y})\in K(x^{\ast })$, $k_0\in R_{+}$, $w,v\in R^n$and${\mu }^{\ast }\in R_{+}^m$such that

(2.1)

(2.2)

(2.3)

(2.4)

(2.5)

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 $(s^{\ast },t^{\ast },\tilde{y})\in K(x^{\ast })$, we define

$$\begin{array}{rcl}{Z}_{\tilde{y}}\left(x^{\ast }\right)& =& \{z\in R^n:z^T\mathrm{\nabla }{g}_j\left(x^{\ast }\right)\le 0,j\in J\left(x^{\ast }\right),\\ \text{with any one of the next conditions (i)-(iii) holds}\}.\end{array}$$

If in addition, we insert the condition $Z_{\tilde{y}}(x^{\ast })=\varphi$, then the result of Theorem 2.1 still holds.

For the sake of convenience, let

$${\psi }_1(\cdot )={\xi }_1(\cdot )+\sum _{j=1}^m{\mu }_j(g_j(\cdot )-g_j(z))$$

(2.6)

and

$$\begin{array}{rcl}{\psi }_2(\cdot )& =& \left[\sum _{i=1}^s{t}_i(h(z,{\tilde{y}}_i)-z^{T}Dv)\right][\sum _{i=1}^s{t}_i(f(\cdot ,{\tilde{y}}_i)+{(\cdot )}^{T}Bw)+\sum _{j=1}^m{\mu }_j{g}_j(\cdot )]\\ -[\sum _{i=1}^s{t}_i(f(z,{\tilde{y}}_i)+z^{T}Bw)+\sum _{j=1}^m{\mu }_j{g}_j(z)]\left[\sum _{i=1}^s{t}_i(h(\cdot ,{\tilde{y}}_i)-{(\cdot )}^{T}Dv)\right],\end{array}$$

where

$${\xi }_1(\cdot )=\sum _{i=1}^s{t}_i[(h(z,{\tilde{y}}_i)-z^{T}Dv)(f(\cdot ,{\tilde{y}}_i)+{(\cdot )}^{T}Bw)-(f(z,{\tilde{y}}_i)+z^{T}Bw)(h(\cdot ,{\tilde{y}}_i)-{(\cdot )}^{T}Dv)].$$

3 Model I

In this section, we consider the following second-order dual problem for (P):

$$\underset{(s,t,\tilde{y})\in K(z)}{max}\underset{(z,\mu ,w,v,p)\in H_1(s,t,\tilde{y})}{sup}F(z),$$

(DM1)

where $F(z)={sup}_{y\in Y}\frac{f(z,y)+{(z^{T}Bz)}^{1/2}}{h(z,y)-{(z^{T}Dz)}^{1/2}}$ and $H_1(s,t,\tilde{y})$ denotes the set of all $(z,\mu ,w,v,p)\in R^n\times R_{+}^m\times R^n\times R^n\times R^n$ satisfying

(3.1)

(3.2)

(3.3)

If the set $H_1(s,t,\tilde{y})=\varphi$, we define the supremum of $F(z)$ over $H_1(s,t,\tilde{y})$ 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 [14].

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

Theorem 3.1 (Weak duality)

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

• (i) ${\psi }_1(\cdot )$ is second-order α-univex at z,

• (ii) ${\varphi }_0(a)\ge 0\Rightarrow a\ge 0$ and $b_0(x,z)>0$.

Then

$$\underset{\tilde{y}\in Y}{sup}\frac{f(x,\tilde{y})+{(x^{T}Bx)}^{1/2}}{h(x,\tilde{y})-{(x^{T}Dx)}^{1/2}}\ge F(z).$$

Proof Assume on contrary to the result that

$$\underset{\tilde{y}\in Y}{sup}\frac{f(x,\tilde{y})+{(x^{T}Bx)}^{1/2}}{h(x,\tilde{y})-{(x^{T}Dx)}^{1/2}}<F(z).$$

(3.4)

Since ${\tilde{y}}_i\in Y(z)$, $i=1,2,\dots ,s$, we have

$$F(z)=\frac{f(z,{\tilde{y}}_i)+{(z^{T}Bz)}^{1/2}}{h(z,{\tilde{y}}_i)-{(z^{T}Dz)}^{1/2}}.$$

(3.5)

From (3.4) and (3.5), for $i=1,2,\dots ,s$, we get

$$\frac{f(x,{\tilde{y}}_i)+{(x^{T}Bx)}^{1/2}}{h(x,{\tilde{y}}_i)-{(x^{T}Dx)}^{1/2}}\le \underset{\tilde{y}\in Y}{sup}\frac{f(x,\tilde{y})+{(x^{T}Bx)}^{1/2}}{h(x,\tilde{y})-{(x^{T}Dx)}^{1/2}}<\frac{f(z,{\tilde{y}}_i)+{(z^{T}Bz)}^{1/2}}{h(z,{\tilde{y}}_i)-{(z^{T}Dz)}^{1/2}}.$$

This further from $t_i\ge 0$, $i=1,2,\dots ,s$, $t\ne 0$ and ${\tilde{y}}_i\in Y(z)$, we obtain

(3.6)

Now,

$$\begin{array}{rcl}{\xi }_1(x)& =& \sum _{i=1}^s{t}_i[(h(z,{\tilde{y}}_i)-z^{T}Dv)(f(x,{\tilde{y}}_i)+x^{T}Bw)\\ -(f(z,{\tilde{y}}_i)+z^{T}Bw)(h(x,{\tilde{y}}_i)-x^{T}Dv)]\\ \le & \sum _{i=1}^s{t}_i[(h(z,{\tilde{y}}_i)-{\left(z^{T}Dz\right)}^{1/2})(f(x,{\tilde{y}}_i)+{\left(x^{T}Bx\right)}^{1/2})\\ -(f(z,{\tilde{y}}_i)+{\left(z^{T}Bz\right)}^{1/2})(h(x,{\tilde{y}}_i)-{\left(x^{T}Dx\right)}^{1/2})]\left(\text{using Lemma 2.1 and (3.3)}\right)\\ <& 0\left(\text{from (3.6)}\right).\end{array}$$

Therefore,

$${\xi }_1(x)<0={\xi }_1(z).$$

(3.7)

By hypothesis (i), we have

$$b_0{\varphi }_0[{\psi }_1(x)-{\psi }_1(z)+\frac{1}{2}{p}^T{\mathrm{\nabla }}^2{\psi }_1(z)p]\ge \alpha (x,z){\eta }^T(x,z)\{\mathrm{\nabla }{\psi }_1(z)+{\mathrm{\nabla }}^2{\psi }_1(z)p\}.$$

This follows from (3.1) that

$$b_0{\varphi }_0[{\psi }_1(x)-{\psi }_1(z)+\frac{1}{2}{p}^T{\mathrm{\nabla }}^2{\psi }_1(z)p]\ge 0$$

which using hypothesis (ii) yields

$${\psi }_1(x)-{\psi }_1(z)+\frac{1}{2}{p}^T{\mathrm{\nabla }}^2{\psi }_1(z)p\ge 0.$$

This further from (2.6), (3.2), and the feasibility of x implies

$${\xi }_1(x)\ge -\sum _{j=1}^m{\mu }_j{g}_j(x)\ge 0={\xi }_1(z).$$

This contradicts (3.7), hence the result. □

Theorem 3.2 (Strong duality)

Let$x^{\ast }$be an optimal solution for (P) and let$\mathrm{\nabla }{g}_j(x^{\ast })$, $j\in J(x^{\ast })$be linearly independent. Then there exist$(s^{\ast },t^{\ast },{\tilde{y}}^{\ast })\in K(x^{\ast })$and$(x^{\ast },{\mu }^{\ast },w^{\ast },v^{\ast },p^{\ast }=0)\in H_1(s^{\ast },t^{\ast },{\tilde{y}}^{\ast })$, such that$(x^{\ast },{\mu }^{\ast },w^{\ast },v^{\ast },s^{\ast },t^{\ast },{\tilde{y}}^{\ast },p^{\ast }=0)$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$(x,\mu ,w,v,s,t,\tilde{y},p)$of (DM1), then$(x^{\ast },{\mu }^{\ast },w^{\ast },v^{\ast },s^{\ast },t^{\ast },{\tilde{y}}^{\ast },p^{\ast }=0)$is an optimal solution of (DM1).

Proof Since $x^{\ast }$ is an optimal solution of (P) and $\mathrm{\nabla }{g}_j(x^{\ast })$, $j\in J(x^{\ast })$ are linearly independent, then by Theorem 2.1, there exist $(s^{\ast },t^{\ast },{\tilde{y}}^{\ast })\in K(x^{\ast })$ and $(x^{\ast },{\mu }^{\ast },w^{\ast },v^{\ast },p^{\ast }=0)\in H_1(s^{\ast },t^{\ast },{\tilde{y}}^{\ast })$ such that $(x^{\ast },{\mu }^{\ast },w^{\ast },v^{\ast },s^{\ast },t^{\ast },{\tilde{y}}^{\ast },p^{\ast }=0)$ is feasible solution of (DM1) and the two objectives have same values. Optimality of $(x^{\ast },{\mu }^{\ast },w^{\ast },v^{\ast },s^{\ast },t^{\ast },{\tilde{y}}^{\ast },p^{\ast }=0)$ for (DM1), thus follows from Theorem 3.1. □

Theorem 3.3 (Strict converse duality)

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

• (i) ${\psi }_1(\cdot )$ is strictly second-order α-univex at $z^{\ast }$,

• (ii) $\{\mathrm{\nabla }{g}_j(x^{\ast }),j\in J(x^{\ast })\}$, are linearly independent,

• (iii) ${\varphi }_0(a)>0\Rightarrow a>0$ and $b_0(x^{\ast },z^{\ast })>0$.

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

Proof By the strict α-univexity of ${\psi }_1(\cdot )$ 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 $(z^{\ast },{\mu }^{\ast },w^{\ast },v^{\ast },s^{\ast },t^{\ast },{\tilde{y}}^{\ast },p^{\ast })$ are optimal solutions to (P) and (DM1), respectively, and $\{\mathrm{\nabla }{g}_j(x^{\ast }),j\in J(x^{\ast })\}$, are linearly independent, by Theorem 3.2, we get

$$\underset{{\tilde{y}}^{\ast }\in Y}{sup}\frac{f(x^{\ast },{\tilde{y}}^{\ast })+{(x^{\ast T}B{x}^{\ast })}^{1/2}}{h(x^{\ast },{\tilde{y}}^{\ast })-{(x^{\ast T}D{x}^{\ast })}^{1/2}}=F\left(z^{\ast }\right).$$

(3.9)

Since ${\tilde{y}}_i^{\ast }\in Y(z^{\ast })$, $i=1,2,\dots ,s^{\ast }$, we have

$$F\left(z^{\ast }\right)=\frac{f(z^{\ast },{\tilde{y}}_i^{\ast })+{(z^{\ast T}B{z}^{\ast })}^{1/2}}{h(z^{\ast },{\tilde{y}}_i^{\ast })-{(z^{\ast T}D{z}^{\ast })}^{1/2}}.$$

(3.10)

By (3.9) and (3.10), we get

$$\begin{array}{c}[(h(z^{\ast },{\tilde{y}}_i^{\ast })-{\left(z^{\ast T}D{z}^{\ast }\right)}^{1/2})(f(x^{\ast },{\tilde{y}}_i^{\ast })+{\left(x^{\ast T}B{x}^{\ast }\right)}^{1/2})\hfill \\ -(f(z^{\ast },{\tilde{y}}_i^{\ast })+{\left(z^{\ast T}B{z}^{\ast }\right)}^{1/2})(h(x^{\ast },{\tilde{y}}_i^{\ast })-{\left(x^{\ast T}D{x}^{\ast }\right)}^{1/2})]\le 0,\hfill \end{array}$$

for all $i=1,2,\dots ,s^{\ast }$ and ${\tilde{y}}_i^{\ast }\in Y$. From ${\tilde{y}}_i^{\ast }\in Y(z^{\ast })\subset Y$ and $t^{\ast }\in R_{+}^{s^{\ast }}$, with ${\sum }_{i=1}^{s^{\ast }}{t}_i^{\ast }=1$, we obtain

(3.11)

From Lemma 2.1, (3.3), and (3.11), we have

$$\begin{array}{rcl}{\xi }_1\left(x^{\ast }\right)& =& \sum _{i=1}^{s^{\ast }}{t}_i^{\ast }[(h(z^{\ast },{\tilde{y}}_i^{\ast })-z^{\ast T}D{v}^{\ast })(f(x^{\ast },{\tilde{y}}_i^{\ast })+x^{\ast T}B{w}^{\ast })\\ -(f(z^{\ast },{\tilde{y}}_i^{\ast })+z^{\ast T}B{w}^{\ast })(h(x^{\ast },{\tilde{y}}_i^{\ast })-x^{\ast T}D{v}^{\ast })]\\ \le & \sum _{i=1}^{s^{\ast }}{t}_i^{\ast }[(h(z^{\ast },{\tilde{y}}_i^{\ast })-{\left(z^{\ast T}D{z}^{\ast }\right)}^{1/2})(f(x^{\ast },{\tilde{y}}_i^{\ast })+{\left(x^{\ast T}B{x}^{\ast }\right)}^{1/2})\\ -(f(z^{\ast },{\tilde{y}}_i^{\ast })+{\left(z^{\ast T}B{z}^{\ast }\right)}^{1/2})(h(x^{\ast },{\tilde{y}}_i^{\ast })-{\left(x^{\ast T}D{x}^{\ast }\right)}^{1/2})]\\ \le & 0={\xi }_1\left(z^{\ast }\right),\end{array}$$

which contradicts (3.8), hence the result. □

4 Model II

In this section, we consider another dual problem to (P):

$$\underset{(s,t,\tilde{y})\in K(z)}{max}\underset{(z,\mu ,w,v,p)\in H_2(s,t,\tilde{y})}{sup}\frac{{\sum }_{i=1}^s{t}_i(f(z,{\tilde{y}}_i)+{(z^{T}Bz)}^{1/2})+{\sum }_{j=1}^m{\mu }_j{g}_j(z)}{{\sum }_{i=1}^s{t}_i(h(z,{\tilde{y}}_i)-{(z^{T}Dz)}^{1/2})},$$

(DM2)

where $H_2(s,t,\tilde{y})$ denotes the set of all $(z,\mu ,w,v,p)\in R^n\times R_{+}^m\times R^n\times R^n\times R^n$ satisfying

(4.1)

(4.2)

(4.3)

If the set $H_2(s,t,\tilde{y})$ is empty, we define the supremum in (DM2) over $H_2(s,t,\tilde{y})$ 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 [14].

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

Theorem 4.1 (Weak duality)

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

• (i) ${\psi }_2(\cdot )$ is second-order α-univex at z,

• (ii) ${\varphi }_0(a)\ge 0\Rightarrow a\ge 0$ and $b_0(x,z)>0$.

Then

$$\underset{\tilde{y}\in Y}{sup}\frac{f(x,\tilde{y})+{(x^{T}Bx)}^{1/2}}{h(x,\tilde{y})-{(x^{T}Dx)}^{1/2}}\ge \frac{{\sum }_{i=1}^s{t}_i(f(z,{\tilde{y}}_i)+{(z^{T}Bz)}^{1/2})+{\sum }_{j=1}^m{\mu }_j{g}_j(z)}{{\sum }_{i=1}^s{t}_i(h(z,{\tilde{y}}_i)-{(z^{T}Dz)}^{1/2})}.$$

Proof Assume on contrary to the result that

$$\underset{\tilde{y}\in Y}{sup}\frac{f(x,\tilde{y})+{(x^{T}Bx)}^{1/2}}{h(x,\tilde{y})-{(x^{T}Dx)}^{1/2}}<\frac{{\sum }_{i=1}^s{t}_i(f(z,{\tilde{y}}_i)+{(z^{T}Bz)}^{1/2})+{\sum }_{j=1}^m{\mu }_j{g}_j(z)}{{\sum }_{i=1}^s{t}_i(h(z,{\tilde{y}}_i)-{(z^{T}Dz)}^{1/2})}$$

or

Using $t_i\ge 0$, $i=1,2,\dots ,s$ and (4.3) in above, we have

(4.4)

Now,

$$\begin{array}{rcl}{\psi }_2(x)& =& [\sum _{i=1}^s{t}_i(f(x,{\tilde{y}}_i)+x^{T}Bw)+\sum _{j=1}^m{\mu }_j{g}_j(x)]\left[\sum _{i=1}^s{t}_i(h(z,{\tilde{y}}_i)-z^{T}Dv)\right]\\ -\left[\sum _{i=1}^s{t}_i(h(x,{\tilde{y}}_i)-x^{T}Dv)\right][\sum _{i=1}^s{t}_i(f(z,{\tilde{y}}_i)+z^{T}Bw)+\sum _{j=1}^m{\mu }_j{g}_j(z)]\\ \le & [\sum _{i=1}^s{t}_i(f(x,{\tilde{y}}_i)+{\left(x^{T}Bx\right)}^{1/2})+\sum _{j=1}^m{\mu }_j{g}_j(x)]\left[\sum _{i=1}^s{t}_i(h(z,{\tilde{y}}_i)-z^{T}Dv)\right]\\ -\left[\sum _{i=1}^s{t}_i(h(x,{\tilde{y}}_i)-{\left(x^{T}Dx\right)}^{1/2})\right][\sum _{i=1}^s{t}_i(f(z,{\tilde{y}}_i)+z^{T}Bw)+\sum _{j=1}^m{\mu }_j{g}_j(z)]\\ \left(\text{from Lemma 2.1 and (4.3)}\right)\\ <& \sum _{i=1}^s{t}_i(h(z,{\tilde{y}}_i)-z^{T}Dv)\sum _{j=1}^m{\mu }_j{g}_j(x)\left(\text{using (4.4)}\right)\\ \le & 0(\text{since }\sum _{i=1}^s{t}_i(h(z,{\tilde{y}}_i)-z^{T}Dv)>0\text{ and }\sum _{j=1}^m{\mu }_j{g}_j(x)\le 0).\end{array}$$

Hence,

$${\psi }_2(x)<0={\psi }_2(z).$$

(4.5)

Now, by the second-order α-univexity of ${\psi }_2(\cdot )$ at z, we get

$$b_0{\varphi }_0[{\psi }_2(x)-{\psi }_2(z)+\frac{1}{2}{p}^T{\mathrm{\nabla }}^2{\psi }_2(z)p]\ge {\eta }^T(x,z)\alpha (x,z)\{\mathrm{\nabla }{\psi }_2(z)+{\mathrm{\nabla }}^2{\psi }_2(z)p\}$$

which using (4.1) and hypothesis (ii) give

$${\psi }_2(x)-{\psi }_2(z)+\frac{1}{2}{p}^T{\mathrm{\nabla }}^2{\psi }_2(z)p\ge 0.$$

This from (4.2) follows that

$${\psi }_2(x)\ge {\psi }_2(z)$$

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$\mathrm{\nabla }{g}_j(x^{\ast })$, $j\in J(x^{\ast })$be linearly independent. Then there exist$(s^{\ast },t^{\ast },{\tilde{y}}^{\ast })\in K(x^{\ast })$and$(x^{\ast },{\mu }^{\ast },w^{\ast },v^{\ast },p^{\ast }=0)\in H_2(s^{\ast },t^{\ast },{\tilde{y}}^{\ast })$, such that$(x^{\ast },{\mu }^{\ast },w^{\ast },v^{\ast },s^{\ast },t^{\ast },{\tilde{y}}^{\ast },p^{\ast }=0)$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$(x,\mu ,w,v,s,t,\tilde{y},p)$of (DM2), then$(x^{\ast },{\mu }^{\ast },w^{\ast },v^{\ast },s^{\ast },t^{\ast },{\tilde{y}}^{\ast },p^{\ast }=0)$is an optimal solution of (DM2).

Theorem 4.3 (Strict converse duality)

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

• (i) ${\psi }_2(\cdot )$ is strictly second-order α-univex at z,

• (ii) $\{\mathrm{\nabla }{g}_j(x^{\ast }),j\in J(x^{\ast })\}$ are linearly independent,

• (iii) ${\varphi }_0(a)>0\Rightarrow a>0$ and $b_0(x^{\ast },z^{\ast })>0$.

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.