Higher-order duality in nondifferentiable minimax fractional programming involving generalized convexity

Abstract

A higher-order dual for a non-differentiable minimax fractional programming problem is formulated. Using the generalized higher-order η-convexity assumptions on the functions involved, weak, strong and strict converse duality theorems are established in order to relate the primal and dual problems. Results obtained in this paper naturally unify and extend some previously known results on non-differentiable minimax fractional programming in the literature.

MSC:26A51, 90C32, 49N15.

1 Introduction

The problem to be considered in the present analysis is the following nondifferentiable minimax fractional problem:

$$\begin{array}{r}\text{Minimize }\underset{y\in Y}{sup}\frac{f(x,y)+{(x^{T}Bx)}^{1/2}}{g(x,y)-{(x^{T}Cx)}^{1/2}},\\ \text{subject to }h(x)\le 0,x\in R^n,\end{array}$$

(NFP)

where Y is a compact subset of $R^l$, $f(\cdot ,\cdot )$, $g(\cdot ,\cdot ):R^n\times R^l\to R$, and $h(\cdot ):R^n\to R^m$ are continuously differentiable functions. B and C are $n\times n$ positive semidefinite symmetric matrices. It is assumed that for each $(x,y)$ in $R^n\times R^l$, $f(x,y)+{(x^{T}Bx)}^{\frac{1}{2}}\ge 0$ and $g(x,y)-{(x^{T}Cx)}^{\frac{1}{2}}>0$.

In an earlier work, Schmittendorf [1] established necessary and sufficient optimality conditions for the following minimax programming problem:

where Y is a compact subset of $R^l$, the functions $f(\cdot ,\cdot ):R^n\times R^l\to R$, and $h(\cdot ):R^n\to R^m$ are in $C^1$.

Tanimoto [2] applied the necessary conditions in [1] to formulate a dual problem and discussed the duality results, which were extended to fractional analogue of (P₁) by several authors [3–12]. Liu [13] proposed the second-order duality theorems for (P₁) under generalized second-order B-invex functions. Mishra and Rueda [14] and Ahmad, Husain and Sarita [15] discussed the second-order duality results for the following nondifferentiable minimax programming problems:

where Y is a compact subset of $R^l$, $f(\cdot ,\cdot ):R^n\times R^l\to R$, and $h(\cdot ):R^n\to R^m$ are twice differentiable functions. B is an $n\times n$ positive semidefinite symmetric matrix. Ahmad, Husain and Sharma [16] formulated a unified higher-order dual of (P₂) and established weak, strong and strict converse duality theorems under higher-order $(F,\alpha ,\rho ,d)$-Type I assumptions. Recently, Jayswal and Stancu-Minasian [17] obtained higher-order duality results for (P₂).

In this paper, we formulate a higher-order dual of (P) and establish weak, strong and strict converse duality theorems under generalized higher-order η-convexity assumptions. More precisely, this paper is an extension of second-order duality results of Hu, Chen and Jian [18] to a class of higher-order duality and it also presents an answer to a question raised in [17].

2 Notation and preliminaries

Let $\mathcal{X}=\{x\in R^n:h(x)\le 0\}$ denote the set of all feasible solutions of (NFP). Any point $x\in \mathcal{X}$ is called a feasible point of (NFP). For each $(x,y)\in R^n\times R^l$, we define

$$\psi (x,y)=\frac{f(x,y)+{(x^{T}Bx)}^{1/2}}{g(x,y)-{(x^{T}Cx)}^{1/2}}$$

such that for each $(x,y)\in \mathcal{X}\times Y$,

$$f(x,y)+{\left(x^{T}Bx\right)}^{1/2}\ge 0\text{and}g(x,y)-{\left(x^{T}Cx\right)}^{1/2}>0.$$

For each $x\in \mathcal{X}$, we define

$$J(x)=\{j\in J:h_j(x)=0\},$$

where

Since f and g are continuously differentiable and Y is compact in $R^l$, it follows that for each $x^{\ast }\in \mathcal{X}$, $Y(x^{\ast })\ne \mathrm{\varnothing }$, and for any ${\overline{y}}_i\in Y(x^{\ast })$, we have a positive constant

$${\lambda }_{\circ }=\psi (x^{\ast },{\overline{y}}_i)=\frac{f(x^{\ast },{\overline{y}}_i)+{({x^{\ast }}^{T}B{x}^{\ast })}^{1/2}}{g(x^{\ast },{\overline{y}}_i)-{({x^{\ast }}^{T}C{x}^{\ast })}^{1/2}}.$$

Lemma 2.1 (Generalized Schwarz inequality)

Let A be a positive-semidefinite matrix of order n. Then, for all $x,w\in R^n$,

$$x^{T}Aw\le {\left(x^{T}Ax\right)}^{\frac{1}{2}}{\left(w^{T}Aw\right)}^{\frac{1}{2}}.$$

The equality $Ax=\xi Aw$ holds for some $\xi \ge 0$. Clearly, if ${(w^{T}Aw)}^{\frac{1}{2}}\le 1$, we have

$$x^{T}Aw\le {\left(x^{T}Ax\right)}^{\frac{1}{2}}.$$

We will use the following definitions.

Let $\varphi :R^n\to R$ and $k:R^n\times R^n\to R$ be differentiable functions.

Definition 2.1 [19]

A function ϕ is said to be higher-order η-convex if there exists a certain mapping $\eta :R^n\times R^n\to R^n$ such that for all $x,p\in R^n$, we have

$$\varphi (x)-\varphi (\overline{x})-k(\overline{x},p)+p^T{▽}_{p}k(\overline{x},p)\ge \eta {(x,\overline{x})}^T{▽}_{p}k(\overline{x},p).$$

Definition 2.2 [19]

A function ϕ is said to be higher-order (strictly) η-pseudoconvex if there exists a certain mapping $\eta :R^n\times R^n\to R^n$ such that for all $x,p\in R^n$, we have

Definition 2.3 [19]

A function ϕ is said to be higher-order (strictly) η-quasiconvex if there exists a certain mapping $\eta :R^n\times R^n\to R^n$ such that for all $x,p\in R^n$, we have

3 Higher-order duality model

In this section, we formulate the higher-order dual for (NFP) and derive duality results.

$$\underset{(s,t,\tilde{y})\in S(z)}{max}\underset{(z,\mu ,\lambda ,v,w,p)\in L(s,t,\tilde{y})}{sup}\lambda ,$$

(NMD)

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

(3.1)

(3.2)

(3.3)

(3.4)

where $J_{\alpha }\subseteq M=\{1,2,\dots ,m\}$, $\alpha =0,1,2,\dots ,r$ with ${\bigcup }_{\alpha =0}^r{J}_{\alpha }=M$ and $J_{\alpha }\cap J\beta =\mathrm{\varnothing }$ if $\alpha \ne \beta$. If for a triplet $(s,t,\tilde{y})\in S(z)$, the set $L(s,t,\tilde{y})=\mathrm{\varnothing }$, then we define the supremum over it to be ∞.

Remark 3.1 (i) Let $J_0=\mathrm{\varnothing }$, $F(z,{\overline{y}}_i,p)=p^T▽f(z,{\overline{y}}_i)+\frac{1}{2}{p}^T{▽}^{2}f(z,{\overline{y}}_i)p$, $G(z,{\overline{y}}_i,p)=p^T▽g(z,{\overline{y}}_i)+\frac{1}{2}{p}^T{▽}^{2}g(z,{\overline{y}}_i)p$, $i=1,2,\dots ,s$ and $H_j(z,p)=p^T▽h_j(z)+\frac{1}{2}{p}^T{▽}^2{h}_j(z)p$, $j=1,2,\dots ,m$. Then (NMD) reduces to the second-order dual in [18, 20]. If, in addition, $p=0$, then we get the dual formulated by Ahmad, Gupta, Kailey and Agarwal [5].

(ii) If $J_0=\mathrm{\varnothing }$, then the above dual becomes the dual formulated in [21].

Theorem 3.1 (Weak duality)

Let x and $(z,\mu ,\lambda ,s,t,v,w,\tilde{y},p)$ be feasible solutions of (NFP) and (NMD) respectively. Assume that

• (i) $[{\sum }_{i=1}^s{t}_i\{f(\cdot ,{\overline{y}}_i)+{(\cdot )}^{T}Bw-\lambda (g(\cdot ,{\overline{y}}_i)-{(\cdot )}^{T}Cv)\}+{\sum }_{j\in J_0}{\mu }_j{h}_j(\cdot )]$ is higher-order η-pseudoconvex at z,

• (ii) ${\sum }_{j\in J_{\alpha }}{\mu }_j{h}_j(\cdot )$, $\alpha =1,2,\dots ,r$, is higher-order η-quasiconvex at z.

Then

$$\underset{y\in Y}{sup}\frac{f(x,y)+{(x^{t}Bx)}^{1/2}}{g(x,y)-{(x^{t}Cx)}^{1/2}}\ge \lambda .$$

Proof Suppose to the contrary that

$$\underset{y\in Y}{sup}\frac{f(x,y)+{(x^{T}Bx)}^{1/2}}{g(x,y)-{(x^{T}Cx)}^{1/2}}<\lambda .$$

Then we have

$$f(x,{\overline{y}}_i)+{\left(x^{T}Bx\right)}^{1/2}-\lambda (g(x,{\overline{y}}_i)-{\left(x^{T}Cx\right)}^{1/2})<0,\text{for all }{\overline{y}}_i\in Y,i=1,2,\dots ,s.$$

It follows from $t_i\ge 0$, $i=1,2,\dots ,s$, that

$$t_i[f(x,{\overline{y}}_i)+{\left(x^{T}Bx\right)}^{1/2}-\lambda (g(x,{\overline{y}}_i)-{\left(x^{T}Cx\right)}^{1/2})]\le 0,i=1,2,\dots ,s,$$

with at least one strict inequality since $t=(t_1,t_2,\dots ,t_s)\ne 0$. Taking summation over i and using ${\sum }_{i=1}^s{t}_i=1$, we have

$$\sum _{i=1}^s{t}_i[f(x,{\overline{y}}_i)+{\left(x^{T}Bx\right)}^{1/2}-\lambda (g(x,{\overline{y}}_i)-{\left(x^{T}Cx\right)}^{1/2})]<0.$$

It follows from the generalized Schwarz inequality and (3.4) that

$$\sum _{i=1}^s{t}_i[f(x,{\overline{y}}_i)+x^{T}Bw-\lambda (g(x,{\overline{y}}_i)-x^{T}Cv)]<0.$$

(3.5)

By the feasibility of x for (NFP)and $\mu \ge 0$, we obtain

$$\sum _{j\in J_0}{\mu }_j{h}_j(x)\le 0.$$

(3.6)

The above inequality with (3.5) gives

$$\sum _{i=1}^s{t}_i[f(x,{\overline{y}}_i)+x^{T}Bw-\lambda (g(x,{\overline{y}}_i)-x^{T}Cv)]+\sum _{j\in J_0}{\mu }_j{h}_j(x)<0.$$

(3.7)

Now, the feasibility of x for (NFP), $\mu \ge 0$ and (3.3) yields

$$\sum _{j\in J_{\alpha }}{\mu }_j{h}_j(x)\le 0\le \sum _{j\in J_{\alpha }}{\mu }_j[h_j(z)+H_j(z,p)-p^T{▽}_p{H}_j(z,p)],\alpha =1,2,\dots ,r.$$

(3.8)

The inequality (3.8) and Hypothesis (ii) gives

$${\eta }^T(x,z)[\sum _{j\in J_{\alpha }}{\mu }_j{▽}_p{H}_j(z,p)]\le 0,\alpha =1,2,\dots ,r.$$

(3.9)

From (3.1) and (3.9), we have

(3.10)

which by virtue of Hypothesis (i) yields

This contradicts (3.7). □

Theorem 3.2 (Strong duality)

Let $x^{\ast }$ be an optimal solution of (NFP) and let $\mathrm{\nabla }{h}_j(x^{\ast })$, $j\in J(x^{\ast })$ be linearly independent. Assume that

Then there exist $(s^{\ast },t^{\ast },{\tilde{y}}^{\ast })\in S$ and $(x^{\ast },{\mu }^{\ast },{\lambda }^{\ast },v^{\ast },w^{\ast },p^{\ast })\in L(s^{\ast },t^{\ast },{\tilde{y}}^{\ast })$ such that $(x^{\ast },{\mu }^{\ast },{\lambda }^{\ast },v^{\ast },w^{\ast },s^{\ast },t^{\ast },{\tilde{y}}^{\ast },p^{\ast }=0)$ is a feasible solution of (NMD) and the two objectives have the same values. Furthermore, if the assumptions of weak duality (Theorem  3.1) hold for all feasible solutions of (NFP) and (NMD), then $(x^{\ast },{\mu }^{\ast },{\lambda }^{\ast },v^{\ast },w^{\ast },s^{\ast },t^{\ast },{\tilde{y}}^{\ast },p^{\ast }=0)$ is an optimal solution of (NMD).

Proof Since $x^{\ast }$ is an optimal solution of (NFP) and $\mathrm{\nabla }{h}_j(x^{\ast })$, $j\in J(x^{\ast })$ are linearly independent, by the necessary conditions obtained in [13], there exist $(s^{\ast },t^{\ast },{\tilde{y}}^{\ast })\in S$ and $(x^{\ast },{\mu }^{\ast },{\lambda }^{\ast },v^{\ast },w^{\ast },p^{\ast })\in L(s^{\ast },t^{\ast },{\tilde{y}}^{\ast })$ such that $(x^{\ast },{\mu }^{\ast },{\lambda }^{\ast },v^{\ast },w^{\ast },s^{\ast },t^{\ast },{\tilde{y}}^{\ast },p^{\ast }=0)$ is a feasible solution of (NMD) and the problems (NFP) and (NMD) have the same objectives values and

$${\lambda }^{\ast }=\frac{f(x^{\ast },{\overline{y}}_i^{\ast })+{({x^{\ast }}^{T}B{x}^{\ast })}^{1/2}}{g(x^{\ast },{\overline{y}}_i^{\ast })-{({x^{\ast }}^{T}C{x}^{\ast })}^{1/2}}.$$

 □

Theorem 3.3 (Strict converse duality)

Let $x^{\ast }$ and $(z^{\ast },{\mu }^{\ast },{\lambda }^{\ast },s^{\ast },t^{\ast },v^{\ast },w^{\ast },{\tilde{y}}^{\ast },p^{\ast })$ be the optimal solutions of (NFP) and (NMD), respectively. Assume that

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

• (ii) $[{\sum }_{i=1}^s{t}_i^{\ast }\{f(\cdot ,{\overline{y}}_i^{\ast })+{(\cdot )}^{T}B{w}^{\ast }-{\lambda }^{\ast }(g(\cdot ,{\overline{y}}_i^{\ast })-{(\cdot )}^{T}C{v}^{\ast })\}+{\sum }_{j\in J_0}{\mu }_j^{\ast }{h}_j(\cdot )]$ is higher-order strictly η-pseudoconvex at $z^{\ast }$,

• (iii) ${\sum }_{j\in J_{\alpha }}{\mu }_j^{\ast }{h}_j(\cdot )$, $\alpha =1,2,\dots ,r$, is higher-order η-quasiconvex at $z^{\ast }$.

Then, $z^{\ast }=x^{\ast }$; that is, $z^{\ast }$ is an optimal solution of (NFP).

Proof We will assume that $z^{\ast }\ne x^{\ast }$ and reach a contradiction. From the strong duality theorem (Theorem 3.2), it follows that

$$\underset{y\in Y}{sup}\frac{f(x^{\ast },{\tilde{y}}^{\ast })+{(x^{\ast T}B{x}^{\ast })}^{1/2}}{g(x^{\ast },{\tilde{y}}^{\ast })-{(x^{\ast T}C{x}^{\ast })}^{1/2}}={\lambda }^{\ast }.$$

(3.11)

Now, proceeding as in Theorem 3.1, we get

$$\sum _{i=1}^s{t}_i^{\ast }[f(x^{\ast },{\overline{y}}_i^{\ast })+x^{\ast T}B{w}^{\ast }-{\lambda }^{\ast }(g(x^{\ast },{\overline{y}}_i^{\ast })-x^{\ast T}C{v}^{\ast })]+\sum _{j\in J_0}{\mu }_j^{\ast }{h}_j\left(x^{\ast }\right)<0$$

(3.12)

and

(3.13)

By Hypothesis (ii), (3.2) and (3.13), we have

$$\sum _{i=1}^s{t}_i^{\ast }[f(x^{\ast },{\overline{y}}_i^{\ast })+x^{\ast T}B{w}^{\ast }-{\lambda }^{\ast }(g(x^{\ast },{\overline{y}}_i^{\ast })-x^{\ast T}C{v}^{\ast })]+\sum _{j\in J_0}{\mu }_j^{\ast }{h}_j\left(x^{\ast }\right)>0,$$

which contradicts (3.12). Hence the results. □

4 Concluding remarks

The notion of higher-order invexity is adopted, which includes many other generalized convexity concepts in mathematical programming as special cases. If we take $J_0=\mathrm{\varnothing }$, $F(z,{\overline{y}}_i,p)=p^T▽f(z,{\overline{y}}_i)+\frac{1}{2}{p}^T{▽}^{2}f(z,{\overline{y}}_i)p$, $G(z,{\overline{y}}_i,p)=p^T▽g(z,{\overline{y}}_i)+\frac{1}{2}{p}^T{▽}^{2}g(z,{\overline{y}}_i)p$, $i=1,2,\dots ,s$ and $H_j(z,p)=p^T▽h_j(z)+\frac{1}{2}{p}^T{▽}^2{h}_j(z)p$, $j=1,2,\dots ,m$ in Theorems 3.1-3.3, then we get Theorems 3.1-3.3 in [18].

The presented results in this paper can be further extended to the following related class of nondifferentiable minimax fractional programming problems:

$$\begin{array}{r}Min\underset{\nu \in W}{sup}\frac{Re[\varphi (\xi ,\nu )+{(z^{H}Bz)}^{1/2}]}{Re[\psi (\xi ,\nu )-{(z^{H}Dz)}^{1/2}]}\\ \text{subject to }-g(\xi )\in S^{\circ },\xi \in C^{2n},\end{array}$$

(CP)

where $\xi =(z,\overline{z})$, $\nu =(\omega ,\overline{\omega })$ for $z\in C^n$, $\omega \in C^l$. $\varphi (\cdot ,\cdot ):C^{2n}\times C^{2l}\to C$ and $\psi (\cdot ,\cdot ):C^{2n}\times C^{2l}\to C$ are analytic with respect to ξ, W is a specified compact subset in $C^{2l}$, $S^{\circ }$ is a polyhedral cone in $C^m$ and $g:C^{2n}\to C^m$ is analytic. Also B, $D\in C^{n\times n}$ are positive semidefinite Hermitian matrices.