- Research
- Open Access

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

- SK Gupta
^{1}Email author, - D Dangar
^{1}and - Sumit Kumar
^{2}

**2012**:187

https://doi.org/10.1186/1029-242X-2012-187

© Gupta et al.; licensee Springer 2012

**Received:**12 April 2012**Accepted:**6 August 2012**Published:**31 August 2012

## 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 [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].

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

**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

**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

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

*α*-univex but not invex, since for $x=3$, $u=2$, and $p=1$, we obtain

**Lemma 2.1** (Generalized Schwartz inequality)

*Let*

*B*

*be a positive semidefinite matrix of order*

*n*.

*Then*,

*for all*$x,w\in {R}^{n}$,

*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*

*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

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

## 3 Model I

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*

*Proof*Assume on contrary to the result that

*x*implies

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

which contradicts (3.8), hence the result. □

## 4 Model II

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*

*Proof*Assume on contrary to the result that

*α*-univexity of ${\psi}_{2}(\cdot )$ at

*z*, we get

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.

## Declarations

### Acknowledgements

The authors wish to thank anonymous reviewers for their constructive and valuable suggestions which have considerably improved the presentation of the paper. The second author is also thankful to the Ministry of Human Resource Development, New Delhi (India) for financial support.

## Authors’ Affiliations

## References

- Schmitendorf WE: Necessary conditions and sufficient conditions for static minmax problems.
*J. Math. Anal. Appl.*1977, 57: 683–693. 10.1016/0022-247X(77)90255-4MathSciNetView ArticleMATHGoogle Scholar - Ahmad I: Optimality conditions and duality in fractional minimax programming involving generalized
*ρ*-invexity.*Int. J. Manag. Syst.*2003, 19: 165–180.Google Scholar - Ahmad I: On second-order duality for minimax fractional programming problems with generalized convexity.
*Abstr. Appl. Anal.*2011. doi:10.1155/2011/563924Google Scholar - Ahmad I, Gupta SK, Kailey N, Agarwal RP:Duality in nondifferentiable minimax fractional programming with $B\text{-}(p,r)$-invexity.
*J. Inequal. Appl.*2011. doi:10.1186/1029–242X-2011–75Google Scholar - Ahmad I, Husain Z: Duality in nondifferentiable minimax fractional programming with generalized convexity.
*Appl. Math. Comput.*2006, 176: 545–551. 10.1016/j.amc.2005.10.002MathSciNetView ArticleMATHGoogle Scholar - Ahmad I, Husain Z: Optimality conditions and duality in nondifferentiable minimax fractional programming with generalized convexity.
*J. Optim. Theory Appl.*2006, 129: 255–275. 10.1007/s10957-006-9057-0MathSciNetView ArticleMATHGoogle Scholar - Chandra S, Kumar V: Duality in fractional minimax programming problem.
*J. Aust. Math. Soc. A*1995, 58: 376–386. 10.1017/S1446788700038362MathSciNetView ArticleMATHGoogle Scholar - Husain Z, Ahmad I, Sharma S: Second order duality for minmax fractional programming.
*Optim. Lett.*2009, 3: 277–286. 10.1007/s11590-008-0107-4MathSciNetView ArticleMATHGoogle Scholar - Hu Q, Chen Y, Jian J: Second-order duality for non-differentiable minimax fractional programming.
*Int. J. Comput. Math.*2012, 89: 11–16. 10.1080/00207160.2011.631529MathSciNetView ArticleMATHGoogle Scholar - Hu Q, Yang G, Jian J: On second order duality for minimax fractional programming.
*Nonlinear Anal.*2011, 12: 3509–3514. 10.1016/j.nonrwa.2011.06.011MathSciNetView ArticleMATHGoogle Scholar - Jayswal A: Non-differentiable minimax fractional programming with generalized
*α*-univexity.*J. Comput. Appl. Math.*2008, 214: 121–135. 10.1016/j.cam.2007.02.007MathSciNetView ArticleMATHGoogle Scholar - Lai HC, Lee JC: On duality theorems for a nondifferentiable minimax fractional programming.
*J. Comput. Appl. Math.*2002, 146: 115–126. 10.1016/S0377-0427(02)00422-3MathSciNetView ArticleMATHGoogle Scholar - Lai HC, Liu JC, Tanaka K: Necessary and sufficient conditions for minimax fractional programming.
*J. Math. Anal. Appl.*1999, 230: 311–328. 10.1006/jmaa.1998.6204MathSciNetView ArticleMATHGoogle Scholar - Liu JC, Wu CS: On minimax fractional optimality conditions with invexity.
*J. Math. Anal. Appl.*1998, 219: 21–35. 10.1006/jmaa.1997.5786MathSciNetView ArticleMATHGoogle Scholar - Liu JC, Wu CS, Sheu RL: Duality for fractional minimax programming.
*Optimization*1997, 41: 117–133. 10.1080/02331939708844330MathSciNetView ArticleMATHGoogle Scholar - Sharma S, Gulati TR: Second order duality in minmax fractional programming with generalized univexity.
*J. Glob. Optim.*2012, 52: 161–169. 10.1007/s10898-011-9694-1MathSciNetView ArticleMATHGoogle Scholar - Yang XM, Hou SH: On minimax fractional optimality and duality with generalized convexity.
*J. Glob. Optim.*2005, 31: 235–252. 10.1007/s10898-004-5698-4MathSciNetView ArticleMATHGoogle Scholar - Du D-Z, Pardalos PM:
*Minimax and Applications*. Kluwer Academic, Dordrecht; 1995.View ArticleMATHGoogle Scholar - Mangasarian OL: Second- and higher-order duality in nonlinear programming.
*J. Math. Anal. Appl.*1975, 51: 607–620. 10.1016/0022-247X(75)90111-0MathSciNetView ArticleGoogle Scholar - Hanson MA: Second order invexity and duality in mathematical programming.
*Opsearch*1993, 30: 311–320.Google Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.