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Abstract and Applied Analysis
Volume 2011 (2011), Article ID 563924, 15 pages
http://dx.doi.org/10.1155/2011/563924
Research Article

On Second-Order Duality for Minimax Fractional Programming Problems with Generalized Convexity

1Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, P.O. Box 728, Dhahran 31261, Saudi Arabia
2Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India

Received 27 May 2011; Accepted 12 August 2011

Academic Editor: H. B. Thompson

Copyright © 2011 Izhar Ahmad. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We focus our study on a discussion of duality relationships of a minimax fractional programming problem with its two types of second-order dual models under the second-order generalized convexity type assumptions. Results obtained in this paper naturally unify and extend some previously known results on minimax fractional programming in the literature.

1. Introduction

Fractional programming is an interesting subject applicable to many types of optimization problems such as portfolio selection, production, and information theory and numerous decision making problems in management science. More specifically, it can be used in engineering and economics to minimize a ratio of physical or economical functions, or both, such as cost/time, cost/volume, and cost/benefit, in order to measure the efficiency or productivity of the system (see Stancu-Minasian [1]).

Minimax type functions arise in the design of electronic circuits; however, minimax fractional problems appear in the formulation of discrete and continuous rational approximation problems with respect to the Chebyshev norm [2], continuous rational games [3], multiobjective programming [4, 5], and engineering design as well as some portfolio selection problems discussed by Bajona-Xandri and Martinez-Legaz [6].

In this paper, we consider the minimax fractional programming problemminimize𝜙(𝑥)=sup𝑦𝑌𝑓(𝑥,𝑦),(𝑥,𝑦)subjectto𝑔(𝑥)0,𝑥𝑅𝑛,(1.1) where 𝑌 is a compact subset of 𝑅𝑙 and 𝑓(,)𝑅𝑛×𝑅𝑙𝑅,  (,)𝑅𝑛×𝑅𝑙𝑅, and 𝑔()𝑅𝑛𝑅𝑚 are twice continuously differentiable functions on 𝑅𝑛×𝑅𝑙, 𝑅𝑛×𝑅𝑙, and 𝑅𝑛, respectively. It is assumed that, for each (𝑥,𝑦) in 𝑅𝑛×𝑅𝑙, 𝑓(𝑥,𝑦)0 and (𝑥,𝑦)>0.

For the case of convex differentiable minimax fractional programming, Yadav and Mukherjee [7] formulated two dual models for (1.1) and derived duality theorems. Chandra and Kumar [8] pointed out certain omissions and inconsistencies in the dual formulation of Yadav and Mukherjee [7]; they constructed two modified dual problems for (1.1) and proved appropriate duality results. Liu and Wu [9, 10] and Ahmad [11] obtained sufficient optimality conditions and duality theorems for (1.1) assuming the functions involved to be generalized convex.

Second-order duality provides tighter bounds for the value of the objective function when approximations are used. For more details, one can consult ([12, page 93]). One more advantage of second-order duality, when applicable, is that, if a feasible point in the primal is given and first-order duality does not apply, then we can use second order duality to provide a lower bound of the value of the primal (see [13]).

Mangasarian [14] first formulated the second-order dual for a nonlinear programming problem and established second-order duality results under certain inequalities. Mond [12] reproved second-order duality results assuming rather simple inequalities. Subsequently, Bector and Chandra [15] formulated a second-order dual for a fractional programming problem and obtained usual duality results under the assumptions [14] by naming these as convex/concave functions.

Based upon the ideas of Bector et al. [16] and Rueda et al. [17], Yang and Hou [18] proposed a new concept of generalized convexity and discussed sufficient optimality conditions for (1.1) and duality results for its corresponding dual. Recently, Husain et al. [19] formulated two types of second-order dual models to (1.1) and discussed appropriate duality results involving 𝜂-convexity/generalized 𝜂-convexity assumptions.

In this paper, we are inspired by Chandra and Kumar [8], Bector et al. [16], Liu [20], and Husain et al. [19] to discuss weak, strong, and strict converse duality theorems connecting (1.1) with its two types of second-order duals by using second-order generalized convexity type assumptions [21].

2. Notations and Preliminaries

Let 𝑆={𝑥𝑅𝑛𝑔(𝑥)0} denote the set of all feasible solutions of (1.1). For each (𝑥,𝑦)𝑅𝑛×𝑅𝑙, we define 𝐽(𝑥)=𝑗𝑀𝑔𝑗,(𝑥)=0(2.1) where 𝑀={1,2,,𝑚},𝑌(𝑥)=𝑦𝑌𝑓(𝑥,𝑦)=sup𝑧𝑌,(𝑓(𝑥,𝑧)𝐾(𝑥)=𝑠,𝑡,̃𝑦)×𝑅𝑠+×𝑅𝑙𝑠𝑡1𝑠𝑛+1,𝑡=1,𝑡2,,𝑡𝑠𝑅𝑠+with𝑠𝑖=1𝑡𝑖=1,̃𝑦=𝑦1,𝑦2,.𝑦𝑠with𝑦𝑖.𝑌(𝑥),𝑖=1,2,,𝑠(2.2)

Definition 2.1. A functional 𝑋×𝑋×𝑅𝑛𝑅, where 𝑋𝑅𝑛 is said to be sublinear in its third argument, if 𝑥,𝑥𝑋,(i)(𝑥,𝑥;𝑎1+𝑎2)(𝑥,𝑥;𝑎1)+(𝑥,𝑥;𝑎2)𝑎1,𝑎2𝑅𝑛, (ii)(𝑥,𝑥;𝛼𝑎)=𝛼(𝑥,𝑥;𝑎)𝛼𝑅+,𝑎𝑅𝑛. By (ii), it is clear that (𝑥,𝑥;0𝑎)=0.

Definition 2.2. A point 𝑥𝑆 is said to optimal solution of (1.1) if 𝜙(𝑥)𝜙(𝑥) for each 𝑥𝑆.

The following theorem [8] will be needed in the subsequent analysis.

Theorem 2.3 (necessary conditions). Let 𝑥 be a solution (local or global) of (1.1), and let 𝑔𝑗(𝑥),𝑗𝐽(𝑥) be linearly independent. Then there exist (𝑠,𝑡,𝑦)𝐾(𝑥),𝜆𝑅+, and 𝜇𝑅𝑚+ such that 𝑠𝑖=1𝑡𝑖𝑓𝑥,𝑦𝑖𝜆𝑥,𝑦𝑖+𝑚𝑗=1𝜇𝑗𝑔𝑗𝑥𝑓𝑥=0,,𝑦𝑖𝜆𝑥,𝑦𝑖=0,𝑖=1,2,,𝑠,𝑚𝑗=1𝜇𝑗𝑔𝑗𝑥𝑡=0,𝑖0,𝑠𝑖=1𝑡𝑖=1,𝑦𝑖𝑥𝑌,𝑖=1,2,,𝑠.(2.3) Throughout the paper, we assume that is a sublinear functional. For 𝛽=1,2,,𝑟 let 𝑏,𝑏0,𝑏𝛽𝑋×𝑋𝑅+,𝜙,𝜙0,𝜙𝛽𝑅𝑅,𝜌,𝜌0,𝜌𝛽 be real numbers, and let 𝜃𝑅𝑛×𝑅𝑛𝑅.

3. First Duality Model

In this section, we discuss usual duality results for the following dual [19]:max𝑠,𝑡,𝑦𝐾(𝑧)sup(𝑧,𝜇,𝜆,𝑝)𝐻1𝑠,𝑡,𝑦𝜆,(3.1) where 𝐻1(𝑠,𝑡,𝑦) denotes the set of all (𝑧,𝜇,𝜆,𝑝)𝑅𝑛×𝑅𝑚+×𝑅+×𝑅𝑛 satisfying𝑠𝑖=1𝑡𝑖𝑓𝑧,𝑦𝑖𝜆𝑧,𝑦𝑖+2𝑠𝑖=1𝑡𝑖𝑓𝑧,𝑦𝑖𝜆𝑧,𝑦𝑖𝑝+𝑚𝑗=1𝜇𝑗𝑔𝑗(𝑧)+2𝑚𝑗=1𝜇𝑗𝑔𝑗(𝑧)𝑝=0,(3.2)𝑠𝑖=1𝑡𝑖𝑓𝑧,𝑦𝑖𝜆𝑧,𝑦𝑖12𝑝𝑇2𝑠𝑖=1𝑡𝑖𝑓𝑧,𝑦𝑖𝜆𝑧,𝑦𝑖𝑝0,(3.3)𝑚𝑗=1𝜇𝑗𝑔𝑗1(𝑧)2𝑝𝑇2𝑚𝑗=1𝜇𝑗𝑔𝑗(𝑧)𝑝0.(3.4) If, for a triplet (𝑠,𝑡,𝑦)𝐾(𝑧), the set 𝐻1(𝑠,𝑡,𝑦)=, then we define the supremum over it to be .

Remark 3.1. If 𝑃=0, then (3.1) becomes the dual considered in [9].

Theorem 3.2 (weak duality). Let 𝑥 and (𝑧,𝜇,𝜆,𝑠,𝑡,𝑦,𝑝) be the feasible solutions of (1.1) and (3.1), respectively. Suppose that there exist ,𝜃,𝜙,𝑏 and 𝜌 such that 𝑏(𝑥,𝑧)𝜙𝑠𝑖=1𝑡𝑖𝑓𝑥,𝑦𝑖𝜆𝑥,𝑦𝑖𝑠𝑖=1𝑡𝑖𝑓𝑧,𝑦𝑖𝜆𝑧,𝑦𝑖𝑚𝑗=1𝜇𝑗𝑔𝑗+1(𝑧)2𝑝𝑇2𝑠𝑖=1𝑡𝑖𝑓𝑧,𝑦𝑖𝜆𝑧,𝑦𝑖1𝑝+2𝑝𝑇𝑚𝑗=1𝜇𝑗𝑔𝑗(𝑧)𝑝<0𝑥,𝑧;𝑠𝑖=1𝑡𝑖𝑓𝑧,𝑦𝑖𝜆𝑧,𝑦𝑖+𝑚𝑗=1𝜇𝑗𝑔𝑗(𝑧)+2𝑠𝑖=1𝑡𝑖𝑓𝑧,𝑦𝑖𝜆𝑧,𝑦𝑖𝑝+2𝑚𝑗=1𝜇𝑗𝑔𝑗(𝑧)𝑝<𝜌𝜃(𝑥,𝑧)2.(3.5) Further assume that 𝑎<0𝜙(𝑎)<0,(3.6)𝑏(𝑥,𝑧)>0,(3.7)𝜌0.(3.8) Then sup𝑦𝑌𝑓(𝑥,𝑦)(𝑥,𝑦)𝜆.(3.9)

Proof. Suppose contrary to the result that sup𝑦𝑌𝑓(𝑥,𝑦)(𝑥,𝑦)<𝜆.(3.10) Thus, we have 𝑓𝑥,𝑦𝑖𝜆𝑥,𝑦𝑖<0,𝑦𝑖𝑌(𝑥),𝑖=1,2,,𝑠.(3.11) It follows from 𝑡𝑖0,𝑖=1,2,,𝑠, that 𝑡𝑖𝑓𝑥,𝑦𝑖𝜆𝑥,𝑦𝑖0,(3.12) with at least one strict inequality since 𝑡=(𝑡1,𝑡2,,𝑡𝑠)0. Taking summation over 𝑖, we have 𝑠𝑖=1𝑡𝑖𝑓𝑥,𝑦𝑖𝜆𝑥,𝑦𝑖<0,(3.13) which together with (3.3) gives 𝑠𝑖=1𝑡𝑖𝑓𝑥,𝑦𝑖𝜆𝑥,𝑦𝑖<0𝑠𝑖=1𝑡𝑖𝑓𝑧,𝑦𝑖𝜆𝑧,𝑦𝑖12𝑝𝑇2𝑠𝑖=1𝑡𝑖𝑓𝑧,𝑦𝑖𝜆𝑧,𝑦𝑖𝑝.(3.14) The above inequality along with (3.4) implies 𝑠𝑖=1𝑡𝑖𝑓𝑥,𝑦𝑖𝜆𝑥,𝑦𝑖𝑠𝑖=1𝑡𝑖𝑓𝑧,𝑦𝑖𝜆𝑧,𝑦𝑖𝑚𝑗=1𝜇𝑗𝑔𝑗+1(𝑧)2𝑝𝑇2𝑠𝑖=1𝑡𝑖𝑓𝑧,𝑦𝑖𝜆𝑧,𝑦𝑖1𝑝+2𝑝𝑇2𝑚𝑗=1𝜇𝑗𝑔𝑗(𝑧)𝑝<0.(3.15) Using (3.6) and (3.7), it follows from (3.15) that 𝑏(𝑥,𝑧)𝜙𝑠𝑖=1𝑡𝑖𝑓𝑥,𝑦𝑖𝜆𝑥,𝑦𝑖𝑠𝑖=1𝑡𝑖𝑓𝑧,𝑦𝑖𝜆𝑧,𝑦𝑖𝑚𝑗=1𝜇𝑗𝑔𝑗+1(𝑧)2𝑝𝑇2𝑠𝑖=1𝑡𝑖𝑓𝑧,𝑦𝑖𝜆𝑧,𝑦𝑖1𝑝+2𝑝𝑇2𝑚𝑗=1𝜇𝑗𝑔𝑗(𝑧)𝑝<0,(3.16) which along with (3.5) and (3.8) yields 𝑥,𝑧;𝑠𝑖=1𝑡𝑖𝑓𝑧,𝑦𝑖𝜆𝑧,𝑦𝑖+𝑚𝑗=1𝜇𝑗𝑔𝑗(𝑧)+2𝑠𝑖=1𝑡𝑖𝑓𝑧,𝑦𝑖𝜆𝑧,𝑦𝑖𝑝+2𝑚𝑗=1𝜇𝑗𝑔𝑗(𝑧)𝑝<0,(3.17) which contradicts (3.2) since (𝑥,𝑧;0)=0.

Theorem 3.3 (strong duality). Assume that 𝑥 is an optimal solution of (1.1) and 𝑔𝑗(𝑥),𝑗𝐽(𝑥) are linearly independent. Then there exist (𝑠,𝑡,𝑦)𝐾(𝑥) and (𝑥,𝜇,𝜆,𝑝=0)𝐻1(𝑠,𝑡,𝑦) such that (𝑥,𝜇,𝜆,𝑠,𝑡,𝑦,𝑝=0) is a feasible solution of (3.1) and the two objectives have the same values. Further, if the assumptions of weak duality (Theorem 3.2) hold for all feasible solutions (𝑧,𝜇,𝜆,𝑠,𝑡,𝑦,𝑝) of (3.1), then (𝑥,𝜇,𝜆,𝑠,𝑡,𝑦,𝑝=0) is an optimal solution of (3.1).

Proof. Since 𝑥 is an optimal solution of (1.1) and 𝑔𝑗(𝑥),𝑗𝐽(𝑥) are linearly independent, then, by Theorem 2.3, there exist (𝑠,𝑡,𝑦)𝐾(𝑥) and (𝑥,𝜇,𝜆,𝑝=0)𝐻1(𝑠,𝑡,𝑦) such that (𝑥,𝜇,𝜆,𝑠,𝑡,𝑦,𝑝=0) is a feasible solution of (3.1) and the two objectives have the same values. Optimality of (𝑥,𝜇,𝜆,𝑠,𝑡,𝑦,𝑝=0) for (3.1) thus follows from weak duality (Theorem 3.2).

Theorem 3.4 (Strict converse duality). Let 𝑥 and (𝑧,𝜇,𝜆,𝑠,𝑡,𝑦,𝑝) be the optimal solutions of (1.1) and (3.1), respectively. Suppose that 𝑔𝑗(𝑥),𝑗𝐽(𝑥) are linearly independent and there exist ,𝜃,𝜙,𝑏 and 𝜌 such that 𝑏𝑥,𝑧𝜙𝑠𝑖=1𝑡𝑖𝑓𝑥,𝑦𝑖𝜆𝑥,𝑦𝑖𝑠𝑖=1𝑡𝑖𝑓𝑧,𝑦𝑖𝜆𝑧,𝑦𝑖𝑚𝑗=1𝜇𝑗𝑔𝑗𝑧+12𝑝𝑇2𝑠𝑖=1𝑡𝑖𝑓𝑧,𝑦𝑖𝜆𝑧,𝑦𝑖𝑝+12𝑝𝑇2𝑚𝑗=1𝜇𝑗𝑔𝑗𝑧𝑝𝑥0,𝑧;𝑠𝑖=1𝑡𝑖𝑓𝑧,𝑦𝑖𝜆𝑧,𝑦𝑖+𝑚𝑗=1𝜇𝑗𝑔𝑗𝑧+2𝑠𝑖=1𝑡𝑖𝑓𝑧,𝑦𝑖𝜆𝑧,𝑦𝑖𝑝+2𝑚𝑗=1𝜇𝑗𝑔𝑗𝑧𝑝𝜃𝑥<𝜌,𝑧2.(3.18) Further Assume 𝑏𝑥𝑎<0𝜙(𝑎)0,(3.19),𝑧>0,(3.20)𝜌0.(3.21) Then 𝑧=𝑥, that is, 𝑧 is an optimal solution of (1.1).

Proof. Suppose contrary to the result that 𝑧𝑥. Since 𝑥 and (𝑧,𝜇,𝜆,𝑠,𝑡,𝑦,𝑝) are optimal solutions of (1.1) and (3.1), respectively, and 𝑔𝑗(𝑥),𝑗𝐽(𝑥) are linearly independent, therefore, from strong duality (Theorem 3.3), we reach sup𝑦𝑌𝑓𝑥,𝑦(𝑥,𝑦)=𝜆.(3.22) Thus, we have 𝑓𝑥,𝑦𝑖𝜆𝑥,𝑦𝑖0,𝑦𝑖𝑥𝑌,𝑖=1,2,,𝑠.(3.23) Now, proceeding as in Theorem 3.2, we get 𝑠𝑖=1𝑡𝑖𝑓𝑥,𝑦𝑖𝜆𝑥,𝑦𝑖𝑠𝑖=1𝑡𝑖𝑓𝑧,𝑦𝑖𝜆𝑧,𝑦𝑖𝑚𝑗=1𝜇𝑗𝑔𝑗𝑧+12𝑝𝑇2𝑠𝑖=1𝑡𝑖𝑓𝑧,𝑦𝑖𝜆𝑧,𝑦𝑖𝑝+12𝑝𝑇2𝑚𝑗=1𝜇𝑗𝑔𝑗𝑧𝑝<0.(3.24) Using (3.19) and (3.20), it follows from (3.24) that 𝑏𝑥,𝑧𝜙𝑠𝑖=1𝑡𝑖𝑓𝑥,𝑦𝑖𝜆𝑥,𝑦𝑖𝑠𝑖=1𝑡𝑖𝑓𝑧,𝑦𝑖𝜆𝑧,𝑦𝑖𝑚𝑗=1𝜇𝑗𝑔𝑗𝑧+12𝑝𝑇2𝑠𝑖=1𝑡𝑖𝑓𝑧,𝑦𝑖𝜆𝑧,𝑦𝑖𝑝+12𝑝𝑇2𝑚𝑗=1𝜇𝑗𝑔𝑗𝑧𝑝0,(3.25) which along with (3.18) and (3.21) implies 𝑥,𝑧;𝑠𝑖=1𝑡𝑖𝑓𝑧,𝑦𝑖𝜆𝑧,𝑦𝑖+𝑚𝑗=1𝜇𝑗𝑔𝑗𝑧+2𝑠𝑖=1𝑡𝑖𝑓𝑧,𝑦𝑖𝜆𝑧,𝑦𝑖𝑝+2𝑚𝑗=1𝜇𝑗𝑔𝑗𝑧𝑝<0,(3.26) which contradicts (3.2) since (𝑥,𝑧;0)=0.

4. Second Duality Model

This section deals with duality theorems for the following second-order dual to (1.1):max(𝑠,𝑡,𝑦)𝐾(𝑧)sup(𝑧,𝜇,𝜆,𝑝)𝐻2(𝑠,𝑡,𝑦)𝜆,(4.1) where 𝐻2(𝑠,𝑡,𝑦) denotes the set of all (𝑧,𝜇,𝜆,𝑝)𝑅𝑛×𝑅𝑚+×𝑅+×𝑅𝑛 satisfying𝑠𝑖=1𝑡𝑖𝑓𝑧,𝑦𝑖𝜆𝑧,𝑦𝑖+2𝑠𝑖=1𝑡𝑖𝑓𝑧,𝑦𝑖𝜆𝑧,𝑦𝑖𝑝+𝑚𝑗=1𝜇𝑗𝑔𝑗(𝑧)+2𝑚𝑗=1𝜇𝑗𝑔𝑗(𝑧)𝑝=0,(4.2)𝑠𝑖=1𝑡𝑖𝑓𝑧,𝑦𝑖𝜆𝑧,𝑦𝑖+𝑗𝐽𝜇𝑗𝑔𝑗1(𝑧)2𝑝𝑇2𝑠𝑖=1𝑡𝑖𝑓𝑧,𝑦𝑖𝜆𝑧,𝑦𝑖+𝑗𝐽𝜇𝑗𝑔𝑗(𝑧)𝑝0,(4.3)𝑗𝐽𝛼𝜇𝑗𝑔𝑗1(𝑧)2𝑝𝑇2𝑗𝐽𝛼𝜇𝑗𝑔𝑗(𝑧)𝑝0,𝛼=1,2,,𝑟,(4.4) where 𝐽𝛼𝑀,𝛼=0,1,2,,𝑟, with 𝑟𝛼=0𝐽𝛼=𝑀 and 𝐽𝛼𝐽𝛽=, if 𝛼𝛽.

If, for a triplet (𝑠,𝑡,𝑦)𝐾(𝑧), the set 𝐻2(𝑠,𝑡,𝑦)=, then we define the supremum over it to be .

Theorem 4.1 (weak duality). Let 𝑥 and (𝑧,𝜇,𝜆,𝑠,𝑡,𝑦,𝑝) be the feasible solutions of (1.1) and (4.1), respectively. Suppose that there exist ,𝜃,𝜙0,𝑏0,𝜌0 and 𝜙𝛽,𝑏𝛽,𝜌𝛽,𝛽=1,2,,𝑟 such that 𝑏0(𝑥,𝑧)𝜙0𝑠𝑖=1𝑡𝑖𝑓𝑥,𝑦𝑖𝜆𝑥,𝑦𝑖𝑠𝑖=1𝑡𝑖𝑓𝑧,𝑦𝑖𝜆𝑧,𝑦𝑖𝑗𝐽0𝜇𝑗𝑔𝑗(+1𝑧)2𝑝𝑇2𝑠𝑖=1𝑡𝑖𝑓𝑧,𝑦𝑖𝜆𝑧,𝑦𝑖+𝑗𝐽0𝜇𝑗𝑔𝑗𝑝(𝑧)<0𝑥,𝑧;𝑠𝑖=1𝑡𝑖𝑓𝑧,𝑦𝑖𝜆𝑧,𝑦𝑖+2𝑠𝑖=1𝑡𝑖𝑓𝑧,𝑦𝑖𝜆𝑧,𝑦𝑖𝑝+𝑗𝐽0𝜇𝑗𝑔𝑗(𝑧)+2𝑗𝐽0𝜇𝑗𝑔𝑗(𝑧)𝑝<𝜌0𝜃(𝑥,𝑧)2,(4.5)𝑏𝛼(𝑥,𝑧)𝜙𝛼𝑗𝐽𝛼𝜇𝑗𝑔𝑗1(𝑧)2𝑝𝑇2𝑗𝐽𝛼𝜇𝑗𝑔𝑗(𝑧)𝑝0𝑥,𝑧;𝑗𝐽𝛼𝜇𝑗𝑔𝑗(𝑧)+2𝑗𝐽𝛼𝜇𝑗𝑔𝑗(𝑧)𝑝𝜌𝛼(𝜃𝑥,𝑧)2,𝛼=1,2,,𝑟.(4.6) Further assume that 𝑎0𝜙𝛼(𝑎)0,𝛼=1,2,,𝑟,(4.7)𝑎<0𝜙0(𝑏𝑎)<0,(4.8)0(𝑥,𝑧)>0,𝑏𝛼𝜌(𝑥,𝑧)0,𝛼=1,2,,𝑟,(4.9)0+𝑟𝛼=1𝜌𝛼0.s(4.10) Then sup𝑦𝑌𝑓(𝑥,𝑦)(𝑥,𝑦)𝜆.(4.11)

Proof. Suppose contrary to the result that sup𝑦𝑌𝑓(𝑥,𝑦)(𝑥,𝑦)<𝜆.(4.12) Thus, we have 𝑓𝑥,𝑦𝑖𝜆𝑥,𝑦𝑖<0,𝑦𝑖𝑌(𝑥),𝑖=1,2,,𝑠.(4.13) It follows from 𝑡𝑖0,𝑖=1,2,,𝑠, that 𝑡𝑖𝑓𝑥,𝑦𝑖𝜆𝑥,𝑦𝑖0,(4.14) with at least one strict inequality since 𝑡=(𝑡1,𝑡2,,𝑡𝑠)0. Taking summation over 𝑖, we have 𝑠𝑖=1𝑡𝑖𝑓𝑥,𝑦𝑖𝜆𝑥,𝑦𝑖<0,(4.15) which together with (4.3) implies 𝑠𝑖=1𝑡𝑖𝑓𝑥,𝑦𝑖𝜆𝑥,𝑦𝑖<0𝑠𝑖=1𝑡𝑖𝑓𝑧,𝑦𝑖𝜆𝑧,𝑦𝑖+𝑗𝐽0𝜇𝑗𝑔𝑗1(𝑧)2𝑝𝑇2𝑠𝑖=1𝑡𝑖𝑓𝑧,𝑦𝑖𝜆𝑧,𝑦𝑖+𝑗𝐽0𝜇𝑗𝑔𝑗(𝑧)𝑝.(4.16) Using (4.8) and (4.9), it follows from (4.16) that 𝑏0(𝑥,𝑧)𝜙0𝑠𝑖=1𝑡𝑖𝑓𝑥,𝑦𝑖𝜆𝑥,𝑦𝑖𝑠𝑖=1𝑡𝑖𝑓𝑧,𝑦𝑖𝜆𝑧,𝑦𝑖𝑗𝐽0𝜇𝑗𝑔𝑗(+1𝑧)2𝑝𝑇2𝑠𝑖=1𝑡𝑖𝑓𝑧,𝑦𝑖𝜆𝑧,𝑦𝑖+𝑗𝐽0𝜇𝑗𝑔𝑗𝑝(𝑧)<0,(4.17) which by (4.5) implies 𝑥,𝑧;𝑠𝑖=1𝑡𝑖𝑓𝑧,𝑦𝑖𝜆𝑧,𝑦𝑖+2𝑠𝑖=1𝑡𝑖𝑓𝑧,𝑦𝑖𝜆𝑧,𝑦𝑖𝑝+𝑗𝐽0𝜇𝑗𝑔𝑗(𝑧)+2𝑗𝐽0𝜇𝑗𝑔𝑗(𝑧)𝑝<𝜌0𝜃(𝑥,𝑧)2.(4.18) Also, inequality (4.4) along with (4.7) and (4.9) yields 𝑏𝛼(𝑥,𝑧)𝜙𝛼𝑗𝐽𝛼𝜇𝑗𝑔𝑗(1𝑧)2𝑝𝑇2𝑗𝐽𝛼𝜇𝑗𝑔𝑗(𝑧)𝑝0,𝛼=1,2,,𝑟.(4.19) From (4.6) and the above inequality, we have 𝑥,𝑧;𝑗𝐽𝛼𝜇𝑗𝑔𝑗(𝑧)+2𝑗𝐽𝛼𝜇𝑗𝑔𝑗(𝑧)𝑝𝜌𝛼𝜃(𝑥,𝑧)2,𝛼=1,2,,𝑟.(4.20) On adding (4.18) and (4.20) and making use of the sublinearity of with (4.10), we obtain 𝑥,𝑧;𝑠𝑖=1𝑡𝑖𝑓𝑧,𝑦𝑖𝜆𝑧,𝑦𝑖+2𝑠𝑖=1𝑡𝑖𝑓𝑧,𝑦𝑖𝜆𝑧,𝑦𝑖𝑝+𝑚𝑗=1𝜇𝑗𝑔𝑗(𝑧)+2𝑚𝑗=1𝜇𝑗𝑔𝑗(𝑧)𝑝<0,(4.21) which contradicts (4.2) since (𝑥,𝑧;0)=0.

The proof of the following theorem is similar to that of Theorem 3.3 and, hence, is omitted.

Theorem 4.2 (strong duality). Assume that 𝑥 is an optimal solution of (1.1) and 𝑔𝑗(𝑥),𝑗𝐽(𝑥), are linearly independent. Then there exist (𝑠,𝑡,𝑦)𝐾(𝑥) and (𝑥,𝜇,𝜆,𝑝=0)𝐻2(𝑠,𝑡,𝑦) such that (𝑥,𝜇,𝜆,𝑠,𝑡,𝑦,𝑝=0) is a feasible solution of (4.1) and the two objectives have the same values. Further, if the assumptions of weak duality (Theorem 4.1) hold for all feasible solutions (𝑧,𝜇,𝜆,𝑠,𝑡,𝑦,𝑝) of (4.1), then (𝑥,𝜇,𝜆,𝑠,𝑡,𝑦,𝑝=0) is an optimal solution of (4.1).

Theorem 4.3 (strict converse duality). Let 𝑥 and (𝑧,𝜇,𝜆,𝑠,𝑡,𝑦,𝑝) be the optimal solutions of (1.1) and (4.1), respectively. Suppose that 𝑔𝑗(𝑥),𝑗𝐽(𝑥) are linearly independent and there exist ,𝜃,𝜙0,𝑏0,𝜌0 and 𝜙𝛽,𝑏𝛽,𝜌𝛽,𝛽=1,2,,𝑟 such that 𝑏0𝑥,𝑧𝜙0𝑠𝑖=1𝑡𝑖𝑓𝑥,𝑦𝑖𝜆𝑥,𝑦𝑖𝑠𝑖=1𝑡𝑖𝑓𝑧,𝑦𝑖𝜆𝑧,𝑦𝑖𝑗𝐽0𝜇𝑗𝑔𝑗𝑧+12𝑝𝑇2𝑠𝑖=1𝑡𝑖𝑓𝑧,𝑦𝑖𝜆𝑧,𝑦𝑖+𝑗𝐽0𝜇𝑗𝑔𝑗𝑧𝑝𝑥0,𝑧;𝑠𝑖=1𝑡𝑖𝑓𝑧,𝑦𝑖𝜆𝑧,𝑦𝑖+2𝑠𝑖=1𝑡𝑖𝑓𝑧,𝑦𝑖𝜆𝑧,𝑦𝑖𝑝+𝑗𝐽0𝜇𝑗𝑔𝑗𝑧+2𝑗𝐽0𝜇𝑗𝑔𝑗𝑧𝑝<𝜌0𝜃𝑥,𝑧2(4.22)𝑏𝛼𝑥,𝑧𝜙𝛼𝑗𝐽𝛼𝜇𝑗𝑔𝑗𝑧12𝑝𝑇2𝑗𝐽𝛼𝜇𝑗𝑔𝑗𝑧𝑝𝑥0,𝑧;𝑗𝐽𝛼𝜇𝑗𝑔𝑗𝑧+2𝑗𝐽𝛼𝜇𝑗𝑔𝑗𝑧𝑝𝜌𝛼𝜃𝑥,𝑧2,𝛼=1,2,,𝑟.(4.23) Further assume that 𝑎0𝜙𝛼(𝑎)0,𝛼=1,2,,𝑟,(4.24)𝑎<0𝜙0𝑏(𝑎)0,(4.25)0𝑥,𝑧>0,𝑏𝛼𝑥,𝑧𝜌0,𝛼=1,2,,𝑟,(4.26)0+𝑟𝛼=1𝜌𝛼0.(4.27) Then 𝑧=𝑥, that is, 𝑧 is an optimal solution of (1.1).

Proof. Suppose contrary to the result that 𝑧𝑥. Since 𝑥 and (𝑧,𝜇,𝜆,𝑠,𝑡,𝑦,𝑝) are optimal solutions of (1.1) and (4.1), respectively, and 𝑔𝑗(𝑥),𝑗𝐽(𝑥) are linearly independent, therefore, from strong duality (Theorem 4.2), we reach sup𝑦𝑌𝑓𝑥,𝑦(𝑥,𝑦)=𝜆.(4.28) Thus, we have 𝑓𝑥,𝑦𝑖𝜆𝑥,𝑦𝑖0,𝑦𝑖𝑥𝑌,𝑖=1,2,,𝑠.(4.29) Now, proceeding as in Theorem 4.1, we get 𝑠𝑖=1𝑡𝑖𝑓𝑥,𝑦𝑖𝜆𝑥,𝑦𝑖𝑠𝑖=1𝑡𝑖𝑓𝑧,𝑦𝑖𝜆𝑧,𝑦𝑖𝑗𝐽0𝜇𝑗𝑔𝑗𝑧+12𝑝𝑇2𝑠𝑖=1𝑡𝑖𝑓𝑧,𝑦𝑖𝜆𝑧,𝑦𝑖+𝑗𝐽0𝜇𝑗𝑔𝑗𝑧𝑝<0.(4.30) Using (4.25) and (4.26), it follows from (4.30) that 𝑏0𝑥,𝑧𝜙0𝑠𝑖=1𝑡𝑖𝑓𝑥,𝑦𝑖𝜆𝑥,𝑦𝑖𝑠𝑖=1𝑡𝑖𝑓𝑧,𝑦𝑖𝜆𝑧,𝑦𝑖𝑗𝐽0𝜇𝑗𝑔𝑗𝑧+12𝑝𝑇2𝑠𝑖=1𝑡𝑖𝑓𝑧,𝑦𝑖𝜆𝑧,𝑦𝑖+𝑗𝐽0𝜇𝑗𝑔𝑗𝑧𝑝0,(4.31) which by (4.22) implies 𝑥,𝑧;𝑠𝑖=1𝑡𝑖𝑓𝑧,𝑦𝑖𝜆𝑧,𝑦𝑖+2𝑠𝑖=1𝑡𝑖𝑓𝑧,𝑦𝑖𝜆𝑧,𝑦𝑖𝑝+𝑗𝐽0𝜇𝑗𝑔𝑗𝑧+2𝑗𝐽0𝜇𝑗𝑔𝑗𝑧𝑝<𝜌0𝜃𝑥,𝑧2.(4.32) Also, inequality (4.4) along with (4.24) and (4.26) yields 𝑏𝛼𝑥,𝑧𝜙𝛼𝑗𝐽𝛼𝜇𝑗𝑔𝑗𝑧12𝑝𝑇2𝑗𝐽𝛼𝜇𝑗𝑔𝑗𝑧𝑝0,𝛼=1,2,,𝑟.(4.33) From (4.23) and the above inequality, we have 𝑥,𝑧;𝑗𝐽𝛼𝜇𝑗𝑔𝑗𝑧+2𝑗𝐽𝛼𝜇𝑗𝑔𝑗𝑧𝑝𝜌𝛼𝜃(𝑥,𝑧)2,𝛼=1,2,,𝑟.(4.34) On adding (4.32) and (4.34) and making use of the sublinearity of with (4.27), we obtain 𝑥,𝑧;𝑠𝑖=1𝑡i𝑓𝑧,𝑦𝑖𝜆𝑧,𝑦𝑖+2𝑠𝑖=1𝑡𝑖𝑓𝑧,𝑦𝑖𝜆𝑧,𝑦𝑖𝑝+𝑚𝑗=1𝜇𝑗𝑔𝑗𝑧+2𝑚𝑗=1𝜇𝑗𝑔𝑗𝑧𝑝<0,(4.35) which contradicts (4.2) since (𝑥,𝑧;0)=0.

5. Conclusion and Further Developments

In this paper, we have established weak, strong, and strict converse duality theorems for a class of minimax fractional programming problems in the frame work of second-order generalized convexity. The second-order duality results developed in this paper can be further extended for the following nondifferentiable minimax fractional programming problem [22, 23]:minimize𝜓(𝑥)=sup𝑦𝑌𝑥𝑓(𝑥,𝑦)+𝑇𝐵𝑥1/2𝑥(𝑥,𝑦)𝑇𝐷𝑥1/2,subjectto𝑔(𝑥)0,𝑥𝑅𝑛,(5.1) where 𝑌 is a compact subset of 𝑅𝑙, 𝐵 and 𝐷 are 𝑛×𝑛 positive semidefinite symmetric matrices, and 𝑓(,)𝑅𝑛×𝑅𝑙𝑅,   (,)𝑅𝑛×𝑅𝑙𝑅, and 𝑔()𝑅𝑛𝑅𝑚 are twice continuously differentiable functions on 𝑅𝑛×𝑅𝑙, 𝑅𝑛×𝑅𝑙, and 𝑅𝑛, respectively.

The question arises as to whether the second-order fractional duality results developed in this paper hold for the following complex nondifferentiable minimax fractional problem:minimizeΨ(𝜉)=sup𝜈𝑊𝑧Re𝑓(𝜉,𝜈)+𝑇𝐵𝑧1/2𝑧Re(𝜉,𝜈)𝑇𝐷𝑧1/2,subjectto𝑔(𝜉)𝑆,𝜉𝒞2𝑛,(5.2) where 𝜉=(𝑧,𝑧),𝜈=(𝜔,𝜔) for 𝑧𝒞𝑛,𝜔𝒞𝑙,𝑓(,)𝒞2𝑛×𝒞2𝑙𝒞 and (,)𝒞2𝑛×𝒞2𝑙𝒞 are analytic with respect to 𝜔, 𝑊 ia a specified compact subset in 𝒞2𝑙,𝑆 is a polyhedral cone in 𝒞𝑚, and 𝑔𝒞2𝑛𝒞𝑚 is analytic. Also 𝐵,𝐷𝒞𝑛×𝑛 are positive semidefinite Hermitian matrices.

Acknowledgments

This paper is supported by Fast Track Project no. FT100023 of King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia. The author is thankful to the referee for his/her valuable suggestions to improve the presentation of the paper.

References

  1. I. M. Stancu-Minasian, Fractional programming, vol. 409 of Mathematics and its Applications, Kluwer Academic Publishers Group, Dordrecht, The Netherlands, 1997.
  2. I. Barrodale, “Best rational approximation and strict quasiconvexity,” SIAM Journal on Numerical Analysis, vol. 10, pp. 8–12, 1973. View at Publisher · View at Google Scholar
  3. R. G. Schroeder, “Linear programming solutions to ratio games,” Operations Research, vol. 18, pp. 300–305, 1970. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. A. Soyster, B. Lev, and D. Toof, “Conservative linear programming with mixed multiple objectives,” Omega, vol. 5, no. 2, pp. 193–205, 1977. View at Publisher · View at Google Scholar · View at Scopus
  5. T. Weir, “A dual for a multiple objective fractional programming problem,” Journal of Information & Optimization Sciences, vol. 7, no. 3, pp. 261–269, 1986. View at Google Scholar · View at Zentralblatt MATH
  6. C. Bajona-Xandri and J. E. Martinez-Legaz, “Lower subdifferentiability in minimax fractional programming,” Optimization, vol. 45, no. 1–4, pp. 1–12, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. S. R. Yadav and R. N. Mukherjee, “Duality for fractional minimax programming problems,” Australian Mathematical Society—Series B, vol. 31, no. 4, pp. 484–492, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. S. Chandra and V. Kumar, “Duality in fractional minimax programming,” Australian Mathematical Society, vol. 58, no. 3, pp. 376–386, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. J. C. Liu and C. S. Wu, “On minimax fractional optimality conditions with invexity,” Journal of Mathematical Analysis and Applications, vol. 219, no. 1, pp. 21–35, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. J. C. Liu and C. S. Wu, “On minimax fractional optimality conditions with (F,ρ)-convexity,” Journal of Mathematical Analysis and Applications, vol. 219, no. 1, pp. 36–51, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. I. Ahmad, “Optimality conditions and duality in fractional minimax programming involving generalized ρ−invexity,” International Journal of Statistics and Management System, vol. 19, pp. 165–180, 2003. View at Google Scholar
  12. B. Mond, “Second order duality for nonlinear programs,” Opsearch, vol. 11, no. 2-3, pp. 90–99, 1974. View at Google Scholar
  13. M. A. Hanson, “Second order invexity and duality in mathematical programming,” Opsearch, vol. 30, pp. 313–320, 1993. View at Google Scholar · View at Zentralblatt MATH
  14. O. L. Mangasarian, “Second and higher order duality in nonlinear programming,” Journal of Mathematical Analysis and Applications, vol. 51, no. 3, pp. 607–620, 1975. View at Publisher · View at Google Scholar
  15. C. R. Bector and S. Chandra, “Generalized-bonvexity and higher order duality for fractional programming,” Opsearch, vol. 24, no. 3, pp. 143–154, 1987. View at Google Scholar · View at Zentralblatt MATH
  16. C. R. Bector, S. Chandra, and I. Husain, “Second order duality for a minimax programming problem,” Opsearch, vol. 28, pp. 249–263, 1991. View at Google Scholar · View at Zentralblatt MATH
  17. N. G. Rueda, M. A. Hanson, and C. Singh, “Optimality and duality with generalized convexity,” Journal of Optimization Theory and Applications, vol. 86, no. 2, pp. 491–500, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. X. M. Yang and S. H. Hou, “On minimax fractional optimality and duality with generalized convexity,” Journal of Global Optimization, vol. 31, no. 2, pp. 235–252, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. Z. Husain, I. Ahmad, and S. Sharma, “Second order duality for minmax fractional programming,” Optimization Letters, vol. 3, no. 2, pp. 277–286, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. J. C. Liu, “Second order duality for minimax programming,” Utilitas Mathematica, vol. 56, pp. 53–63, 1999. View at Google Scholar · View at Zentralblatt MATH
  21. Z. Husain, A. Jayswal, and I. Ahmad, “Second order duality for nondifferentiable minimax programming problems with generalized convexity,” Journal of Global Optimization, vol. 44, no. 4, pp. 593–608, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  22. I. Ahmad, S. K. Gupta, N. R. Kailey, and R. P. Agarwal, “Duality in nondifferentiable minimax fractional programming with B-(p,r)-invexity,” Journal of Inequalities and Applications. In press.
  23. I. Ahmad and Z. Husain, “Duality in nondifferentiable minimax fractional programming with generalized convexity,” Applied Mathematics and Computation, vol. 176, no. 2, pp. 545–551, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH