Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
VolumeΒ 2011, Article IDΒ 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.Β Thompson

Copyright Β© 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 problemminimizeπœ™(π‘₯)=supπ‘¦βˆˆπ‘Œπ‘“(π‘₯,𝑦),β„Ž(π‘₯,𝑦)subjectto𝑔(π‘₯)≀0,π‘₯βˆˆπ‘…π‘›,(1.1) where π‘Œ is a compact subset of 𝑅𝑙 and 𝑓(β‹…,β‹…)βˆΆπ‘…π‘›Γ—π‘…π‘™β†’π‘…,β€‰β€‰β„Ž(β‹…,β‹…)βˆΆπ‘…π‘›Γ—π‘…π‘™β†’π‘…, and 𝑔(β‹…)βˆΆπ‘…π‘›β†’π‘…π‘š are twice continuously differentiable functions on 𝑅𝑛×𝑅𝑙, 𝑅𝑛×𝑅𝑙, and 𝑅𝑛, respectively. It is assumed that, for each (π‘₯,𝑦) in 𝑅𝑛×𝑅𝑙, 𝑓(π‘₯,𝑦)β‰₯0 and β„Ž(π‘₯,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 πœ‚-convexity/generalized πœ‚-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={π‘₯βˆˆπ‘…π‘›βˆΆπ‘”(π‘₯)≀0} denote the set of all feasible solutions of (1.1). For each (π‘₯,𝑦)βˆˆπ‘…π‘›Γ—π‘…π‘™, we define 𝐽(π‘₯)=π‘—βˆˆπ‘€βˆΆπ‘”π‘—ξ€Ύ,(π‘₯)=0(2.1) where 𝑀={1,2,…,π‘š},ξ‚»π‘Œ(π‘₯)=π‘¦βˆˆπ‘ŒβˆΆπ‘“(π‘₯,𝑦)=supπ‘§βˆˆπ‘Œξ‚Ό,ξƒ―(𝑓(π‘₯,𝑧)𝐾(π‘₯)=𝑠,𝑑,̃𝑦)βˆˆβ„•Γ—π‘…π‘ +Γ—π‘…π‘™π‘ ξ€·π‘‘βˆΆ1≀𝑠≀𝑛+1,𝑑=1,𝑑2,…,π‘‘π‘ ξ€Έβˆˆπ‘…π‘ +with𝑠𝑖=1𝑑𝑖=1,̃𝑦=𝑦1,𝑦2,….𝑦𝑠with𝑦𝑖.βˆˆπ‘Œ(π‘₯),𝑖=1,2,…,𝑠(2.2)

Definition 2.1. A functional β„±βˆΆπ‘‹Γ—π‘‹Γ—π‘…π‘›β†’π‘…, where π‘‹βŠ†π‘…π‘› is said to be sublinear in its third argument, if βˆ€π‘₯,π‘₯βˆˆπ‘‹,(i)β„±(π‘₯,π‘₯;π‘Ž1+π‘Ž2)≀ℱ(π‘₯,π‘₯;π‘Ž1)+β„±(π‘₯,π‘₯;π‘Ž2)βˆ€π‘Ž1,π‘Ž2βˆˆπ‘…π‘›, (ii)β„±(π‘₯,π‘₯;π›Όπ‘Ž)=𝛼ℱ(π‘₯,π‘₯;π‘Ž)βˆ€π›Όβˆˆπ‘…+,π‘Žβˆˆπ‘…π‘›. By (ii), it is clear that β„±(π‘₯,π‘₯;0π‘Ž)=0.

Definition 2.2. A point π‘₯βˆˆπ‘† is said to optimal solution of (1.1) if πœ™(π‘₯)β‰₯πœ™(π‘₯) for each π‘₯βˆˆπ‘†.

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

Theorem 2.3 (necessary conditions). Let π‘₯βˆ— be a solution (local or global) of (1.1), and let βˆ‡π‘”π‘—(π‘₯βˆ—),π‘—βˆˆπ½(π‘₯βˆ—) be linearly independent. Then there exist (π‘ βˆ—,π‘‘βˆ—,π‘¦βˆ—)∈𝐾(π‘₯βˆ—),πœ†βˆ—βˆˆπ‘…+, and πœ‡βˆ—βˆˆπ‘…π‘š+ such that βˆ‡π‘ βˆ—ξ“π‘–=1π‘‘βˆ—π‘–ξ€·π‘“ξ€·π‘₯βˆ—,π‘¦βˆ—π‘–ξ€Έβˆ’πœ†βˆ—β„Žξ€·π‘₯βˆ—,π‘¦βˆ—π‘–ξ€Έξ€Έ+βˆ‡π‘šξ“π‘—=1πœ‡βˆ—π‘—π‘”π‘—ξ€·π‘₯βˆ—ξ€Έπ‘“ξ€·π‘₯=0,βˆ—,π‘¦βˆ—π‘–ξ€Έβˆ’πœ†βˆ—β„Žξ€·π‘₯βˆ—,π‘¦βˆ—π‘–ξ€Έ=0,𝑖=1,2,…,π‘ βˆ—,π‘šξ“π‘—=1πœ‡βˆ—π‘—π‘”π‘—ξ€·π‘₯βˆ—ξ€Έπ‘‘=0,βˆ—π‘–β‰₯0,π‘ βˆ—ξ“π‘–=1π‘‘βˆ—π‘–=1,π‘¦βˆ—π‘–ξ€·π‘₯βˆˆπ‘Œβˆ—ξ€Έ,𝑖=1,2,…,π‘ βˆ—.(2.3) Throughout the paper, we assume that β„± is a sublinear functional. For 𝛽=1,2,…,π‘Ÿ let 𝑏,𝑏0,π‘π›½βˆΆπ‘‹Γ—π‘‹β†’π‘…+,πœ™,πœ™0,πœ™π›½βˆΆπ‘…β†’π‘…,𝜌,𝜌0,πœŒπ›½ be real numbers, and let πœƒβˆΆπ‘…π‘›Γ—π‘…π‘›β†’π‘….

3. First Duality Model

In this section, we discuss usual duality results for the following dual [19]:max𝑠,𝑑,π‘¦ξ€ΈβˆˆπΎ(𝑧)sup(𝑧,πœ‡,πœ†,𝑝)∈𝐻1𝑠,𝑑,π‘¦ξ€Έπœ†,(3.1) where 𝐻1(𝑠,𝑑,𝑦) denotes the set of all (𝑧,πœ‡,πœ†,𝑝)βˆˆπ‘…π‘›Γ—π‘…π‘š+×𝑅+×𝑅𝑛 satisfyingβˆ‡π‘ ξ“π‘–=1𝑑𝑖𝑓𝑧,π‘¦π‘–ξ€Έξ€·βˆ’πœ†β„Žπ‘§,𝑦𝑖+βˆ‡2𝑠𝑖=1𝑑𝑖𝑓𝑧,π‘¦π‘–ξ€Έξ€·βˆ’πœ†β„Žπ‘§,𝑦𝑖𝑝+βˆ‡π‘šξ“π‘—=1πœ‡π‘—π‘”π‘—(𝑧)+βˆ‡2π‘šξ“π‘—=1πœ‡π‘—π‘”π‘—(𝑧)𝑝=0,(3.2)𝑠𝑖=1𝑑𝑖𝑓𝑧,π‘¦π‘–ξ€Έξ€·βˆ’πœ†β„Žπ‘§,π‘¦π‘–βˆ’1ξ€Έξ€Έ2π‘π‘‡βˆ‡2𝑠𝑖=1𝑑𝑖𝑓𝑧,π‘¦π‘–ξ€Έξ€·βˆ’πœ†β„Žπ‘§,𝑦𝑖𝑝β‰₯0,(3.3)π‘šξ“π‘—=1πœ‡π‘—π‘”π‘—1(𝑧)βˆ’2π‘π‘‡βˆ‡2π‘šξ“π‘—=1πœ‡π‘—π‘”π‘—(𝑧)𝑝β‰₯0.(3.4) If, for a triplet (𝑠,𝑑,𝑦)∈𝐾(𝑧), the set 𝐻1(𝑠,𝑑,𝑦)=βˆ…, then we define the supremum over it to be βˆ’βˆž.

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

Theorem 3.2 (weak duality). Let π‘₯ and (𝑧,πœ‡,πœ†,𝑠,𝑑,𝑦,𝑝) be the feasible solutions of (1.1) and (3.1), respectively. Suppose that there exist β„±,πœƒ,πœ™,𝑏 and 𝜌 such that 𝑏(π‘₯,𝑧)πœ™π‘ ξ“π‘–=1𝑑𝑖𝑓π‘₯,π‘¦π‘–ξ€Έξ€·βˆ’πœ†β„Žπ‘₯,π‘¦π‘–βˆ’ξ€Έξ€Έπ‘ ξ“π‘–=1𝑑𝑖𝑓𝑧,π‘¦π‘–ξ€Έξ€·βˆ’πœ†β„Žπ‘§,π‘¦π‘–βˆ’ξ€Έξ€Έπ‘šξ“π‘—=1πœ‡π‘—π‘”π‘—+1(𝑧)2π‘π‘‡βˆ‡2𝑠𝑖=1𝑑𝑖𝑓𝑧,π‘¦π‘–ξ€Έξ€·βˆ’πœ†β„Žπ‘§,𝑦𝑖1𝑝+2π‘π‘‡π‘šξ“π‘—=1πœ‡π‘—π‘”π‘—ξƒ­ξƒ©(𝑧)𝑝<0βŸΉβ„±π‘₯,𝑧;βˆ‡π‘ ξ“π‘–=1𝑑𝑖𝑓𝑧,π‘¦π‘–ξ€Έξ€·βˆ’πœ†β„Žπ‘§,𝑦𝑖+βˆ‡π‘šξ“π‘—=1πœ‡π‘—π‘”π‘—(𝑧)+βˆ‡2𝑠𝑖=1𝑑𝑖𝑓𝑧,π‘¦π‘–ξ€Έξ€·βˆ’πœ†β„Žπ‘§,𝑦𝑖𝑝+βˆ‡2π‘šξ“π‘—=1πœ‡π‘—π‘”π‘—ξƒͺβ€–(𝑧)𝑝<βˆ’πœŒβ€–πœƒ(π‘₯,𝑧)2.(3.5) Further assume that π‘Ž<0βŸΉπœ™(π‘Ž)<0,(3.6)𝑏(π‘₯,𝑧)>0,(3.7)𝜌β‰₯0.(3.8) Then supπ‘¦βˆˆπ‘Œπ‘“(π‘₯,𝑦)β„Ž(π‘₯,𝑦)β‰₯πœ†.(3.9)

Proof. Suppose contrary to the result that supπ‘¦βˆˆπ‘Œπ‘“(π‘₯,𝑦)β„Ž(π‘₯,𝑦)<πœ†.(3.10) Thus, we have 𝑓π‘₯,π‘¦π‘–ξ€Έξ€·βˆ’πœ†β„Žπ‘₯,𝑦𝑖<0,βˆ€π‘¦π‘–βˆˆπ‘Œ(π‘₯),𝑖=1,2,…,𝑠.(3.11) It follows from 𝑑𝑖β‰₯0,𝑖=1,2,…,𝑠, that 𝑑𝑖𝑓π‘₯,π‘¦π‘–ξ€Έξ€·βˆ’πœ†β„Žπ‘₯,𝑦𝑖≀0,(3.12) with at least one strict inequality since 𝑑=(𝑑1,𝑑2,…,𝑑𝑠)β‰ 0. Taking summation over 𝑖, we have 𝑠𝑖=1𝑑𝑖𝑓π‘₯,π‘¦π‘–ξ€Έξ€·βˆ’πœ†β„Žπ‘₯,𝑦𝑖<0,(3.13) which together with (3.3) gives 𝑠𝑖=1𝑑𝑖𝑓π‘₯,π‘¦π‘–ξ€Έξ€·βˆ’πœ†β„Žπ‘₯,𝑦𝑖≀<0𝑠𝑖=1𝑑𝑖𝑓𝑧,π‘¦π‘–ξ€Έξ€·βˆ’πœ†β„Žπ‘§,π‘¦π‘–βˆ’1ξ€Έξ€Έ2π‘π‘‡βˆ‡2𝑠𝑖=1𝑑𝑖𝑓𝑧,π‘¦π‘–ξ€Έξ€·βˆ’πœ†β„Žπ‘§,𝑦𝑖𝑝.(3.14) The above inequality along with (3.4) implies 𝑠𝑖=1𝑑𝑖𝑓π‘₯,π‘¦π‘–ξ€Έξ€·βˆ’πœ†β„Žπ‘₯,π‘¦π‘–βˆ’ξ€Έξ€Έπ‘ ξ“π‘–=1𝑑𝑖𝑓𝑧,π‘¦π‘–ξ€Έξ€·βˆ’πœ†β„Žπ‘§,π‘¦π‘–βˆ’ξ€Έξ€Έπ‘šξ“π‘—=1πœ‡π‘—π‘”π‘—+1(𝑧)2π‘π‘‡βˆ‡2𝑠𝑖=1𝑑𝑖𝑓𝑧,π‘¦π‘–ξ€Έξ€·βˆ’πœ†β„Žπ‘§,𝑦𝑖1𝑝+2π‘π‘‡βˆ‡2π‘šξ“π‘—=1πœ‡π‘—π‘”π‘—(𝑧)𝑝<0.(3.15) Using (3.6) and (3.7), it follows from (3.15) that 𝑏(π‘₯,𝑧)πœ™π‘ ξ“π‘–=1𝑑𝑖𝑓π‘₯,π‘¦π‘–ξ€Έξ€·βˆ’πœ†β„Žπ‘₯,π‘¦π‘–βˆ’ξ€Έξ€Έπ‘ ξ“π‘–=1𝑑𝑖𝑓𝑧,π‘¦π‘–ξ€Έξ€·βˆ’πœ†β„Žπ‘§,π‘¦π‘–βˆ’ξ€Έξ€Έπ‘šξ“π‘—=1πœ‡π‘—π‘”π‘—+1(𝑧)2π‘π‘‡βˆ‡2𝑠𝑖=1𝑑𝑖𝑓𝑧,π‘¦π‘–ξ€Έξ€·βˆ’πœ†β„Žπ‘§,𝑦𝑖1𝑝+2π‘π‘‡βˆ‡2π‘šξ“π‘—=1πœ‡π‘—π‘”π‘—ξƒ­(𝑧)𝑝<0,(3.16) which along with (3.5) and (3.8) yields ℱπ‘₯,𝑧;βˆ‡π‘ ξ“π‘–=1𝑑𝑖𝑓𝑧,π‘¦π‘–ξ€Έξ€·βˆ’πœ†β„Žπ‘§,𝑦𝑖+βˆ‡π‘šξ“π‘—=1πœ‡π‘—π‘”π‘—(𝑧)+βˆ‡2𝑠𝑖=1𝑑𝑖𝑓𝑧,π‘¦π‘–ξ€Έξ€·βˆ’πœ†β„Žπ‘§,𝑦𝑖𝑝+βˆ‡2π‘šξ“π‘—=1πœ‡π‘—π‘”π‘—ξƒͺ(𝑧)𝑝<0,(3.17) which contradicts (3.2) since β„±(π‘₯,𝑧;0)=0.

Theorem 3.3 (strong duality). Assume that π‘₯βˆ— is an optimal solution of (1.1) and βˆ‡π‘”π‘—(π‘₯βˆ—),π‘—βˆˆπ½(π‘₯βˆ—) are linearly independent. Then there exist (π‘ βˆ—,π‘‘βˆ—,π‘¦βˆ—)∈𝐾(π‘₯βˆ—) and (π‘₯βˆ—,πœ‡βˆ—,πœ†βˆ—,π‘βˆ—=0)∈𝐻1(π‘ βˆ—,π‘‘βˆ—,π‘¦βˆ—) such that (π‘₯βˆ—,πœ‡βˆ—,πœ†βˆ—,π‘ βˆ—,π‘‘βˆ—,π‘¦βˆ—,π‘βˆ—=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,πœ‡,πœ†,𝑠,𝑑,𝑦,𝑝) of (3.1), then (π‘₯βˆ—,πœ‡βˆ—,πœ†βˆ—,π‘ βˆ—,π‘‘βˆ—,π‘¦βˆ—,π‘βˆ—=0) is an optimal solution of (3.1).

Proof. Since π‘₯βˆ— is an optimal solution of (1.1) and βˆ‡π‘”π‘—(π‘₯βˆ—),π‘—βˆˆπ½(π‘₯βˆ—) are linearly independent, then, by Theorem 2.3, there exist (π‘ βˆ—,π‘‘βˆ—,π‘¦βˆ—)∈𝐾(π‘₯βˆ—) and (π‘₯βˆ—,πœ‡βˆ—,πœ†βˆ—,π‘βˆ—=0)∈𝐻1(π‘ βˆ—,π‘‘βˆ—,π‘¦βˆ—) such that (π‘₯βˆ—,πœ‡βˆ—,πœ†βˆ—,π‘ βˆ—,π‘‘βˆ—,π‘¦βˆ—,π‘βˆ—=0) is a feasible solution of (3.1) and the two objectives have the same values. Optimality of (π‘₯βˆ—,πœ‡βˆ—,πœ†βˆ—,π‘ βˆ—,π‘‘βˆ—,π‘¦βˆ—,π‘βˆ—=0) for (3.1) thus follows from weak duality (Theorem 3.2).

Theorem 3.4 (Strict converse duality). Let π‘₯βˆ— and (π‘§βˆ—,πœ‡βˆ—,πœ†βˆ—,π‘ βˆ—,π‘‘βˆ—,π‘¦βˆ—,π‘βˆ—) be the optimal solutions of (1.1) and (3.1), respectively. Suppose that βˆ‡π‘”π‘—(π‘₯βˆ—),π‘—βˆˆπ½(π‘₯βˆ—) are linearly independent and there exist β„±,πœƒ,πœ™,𝑏 and 𝜌 such that 𝑏π‘₯βˆ—,π‘§βˆ—ξ€Έπœ™βŽ‘βŽ’βŽ’βŽ£π‘ βˆ—ξ“π‘–=1π‘‘βˆ—π‘–ξ€·π‘“ξ€·π‘₯βˆ—,π‘¦βˆ—π‘–ξ€Έβˆ’πœ†βˆ—β„Žξ€·π‘₯βˆ—,π‘¦βˆ—π‘–βˆ’ξ€Έξ€Έπ‘ βˆ—ξ“π‘–=1π‘‘βˆ—π‘–ξ€·π‘“ξ€·π‘§βˆ—,π‘¦βˆ—π‘–ξ€Έβˆ’πœ†βˆ—β„Žξ€·π‘§βˆ—,π‘¦βˆ—π‘–βˆ’ξ€Έξ€Έπ‘šξ“π‘—=1πœ‡βˆ—π‘—π‘”π‘—ξ€·π‘§βˆ—ξ€Έ+12π‘βˆ—π‘‡βˆ‡2π‘ βˆ—ξ“π‘–=1π‘‘βˆ—π‘–ξ€·π‘“ξ€·π‘§βˆ—,π‘¦βˆ—π‘–ξ€Έβˆ’πœ†βˆ—β„Žξ€·π‘§βˆ—,π‘¦βˆ—π‘–π‘ξ€Έξ€Έβˆ—+12π‘βˆ—π‘‡βˆ‡2π‘šξ“π‘—=1πœ‡βˆ—π‘—π‘”π‘—ξ€·π‘§βˆ—ξ€Έπ‘βˆ—βŽ€βŽ₯βŽ₯βŽ¦βŽ›βŽœβŽœβŽπ‘₯≀0βŸΉβ„±βˆ—,π‘§βˆ—;βˆ‡π‘ βˆ—ξ“π‘–=1π‘‘βˆ—π‘–ξ€·π‘“ξ€·π‘§βˆ—,π‘¦βˆ—π‘–ξ€Έβˆ’πœ†βˆ—β„Žξ€·π‘§βˆ—,π‘¦βˆ—π‘–ξ€Έξ€Έ+βˆ‡π‘šξ“π‘—=1πœ‡βˆ—π‘—π‘”π‘—ξ€·π‘§βˆ—ξ€Έ+βˆ‡2π‘ βˆ—ξ“π‘–=1π‘‘βˆ—π‘–ξ€·π‘“ξ€·π‘§βˆ—,π‘¦βˆ—π‘–ξ€Έβˆ’πœ†βˆ—β„Žξ€·π‘§βˆ—,π‘¦βˆ—π‘–π‘ξ€Έξ€Έβˆ—+βˆ‡2π‘šξ“π‘—=1πœ‡βˆ—π‘—π‘”π‘—ξ€·π‘§βˆ—ξ€Έπ‘βˆ—βŽžβŽŸβŽŸβŽ β€–β€–πœƒξ€·π‘₯<βˆ’πœŒβˆ—,π‘§βˆ—ξ€Έβ€–β€–2.(3.18) Further Assume 𝑏π‘₯π‘Ž<0βŸΉπœ™(π‘Ž)≀0,(3.19)βˆ—,π‘§βˆ—ξ€Έ>0,(3.20)𝜌β‰₯0.(3.21) Then π‘§βˆ—=π‘₯βˆ—, that is, π‘§βˆ— is an optimal solution of (1.1).

Proof. Suppose contrary to the result that π‘§βˆ—β‰ π‘₯βˆ—. Since π‘₯βˆ— and (π‘§βˆ—,πœ‡βˆ—,πœ†βˆ—,π‘ βˆ—,π‘‘βˆ—,π‘¦βˆ—,π‘βˆ—) are optimal solutions of (1.1) and (3.1), respectively, and βˆ‡π‘”π‘—(π‘₯βˆ—),π‘—βˆˆπ½(π‘₯βˆ—) are linearly independent, therefore, from strong duality (Theorem 3.3), we reach supπ‘¦βˆ—βˆˆπ‘Œπ‘“ξ€·π‘₯βˆ—,π‘¦βˆ—ξ€Έβ„Ž(π‘₯βˆ—,π‘¦βˆ—)=πœ†βˆ—.(3.22) Thus, we have 𝑓π‘₯βˆ—,π‘¦βˆ—π‘–ξ€Έβˆ’πœ†βˆ—β„Žξ€·π‘₯βˆ—,π‘¦βˆ—π‘–ξ€Έβ‰€0,βˆ€π‘¦βˆ—π‘–ξ€·π‘₯βˆˆπ‘Œβˆ—ξ€Έ,𝑖=1,2,…,π‘ βˆ—.(3.23) Now, proceeding as in Theorem 3.2, we get π‘ βˆ—ξ“π‘–=1π‘‘βˆ—π‘–ξ€·π‘“ξ€·π‘₯βˆ—,π‘¦βˆ—π‘–ξ€Έβˆ’πœ†βˆ—β„Žξ€·π‘₯βˆ—,π‘¦βˆ—π‘–βˆ’ξ€Έξ€Έπ‘ βˆ—ξ“π‘–=1π‘‘βˆ—π‘–ξ€·π‘“ξ€·π‘§βˆ—,π‘¦βˆ—π‘–ξ€Έβˆ’πœ†βˆ—β„Žξ€·π‘§βˆ—,π‘¦βˆ—π‘–βˆ’ξ€Έξ€Έπ‘šξ“π‘—=1πœ‡βˆ—π‘—π‘”π‘—ξ€·π‘§βˆ—ξ€Έ+12π‘βˆ—π‘‡βˆ‡2π‘ βˆ—ξ“π‘–=1π‘‘βˆ—π‘–ξ€·π‘“ξ€·π‘§βˆ—,π‘¦βˆ—π‘–ξ€Έβˆ’πœ†βˆ—β„Žξ€·π‘§βˆ—,π‘¦βˆ—π‘–π‘ξ€Έξ€Έβˆ—+12π‘βˆ—π‘‡βˆ‡2π‘šξ“π‘—=1πœ‡βˆ—π‘—π‘”π‘—ξ€·π‘§βˆ—ξ€Έπ‘βˆ—<0.(3.24) Using (3.19) and (3.20), it follows from (3.24) that 𝑏π‘₯βˆ—,π‘§βˆ—ξ€Έπœ™βŽ‘βŽ’βŽ’βŽ£π‘ βˆ—ξ“π‘–=1π‘‘βˆ—π‘–ξ€·π‘“ξ€·π‘₯βˆ—,π‘¦βˆ—π‘–ξ€Έβˆ’πœ†βˆ—β„Žξ€·π‘₯βˆ—,π‘¦βˆ—π‘–βˆ’ξ€Έξ€Έπ‘ βˆ—ξ“π‘–=1π‘‘βˆ—π‘–ξ€·π‘“ξ€·π‘§βˆ—,π‘¦βˆ—π‘–ξ€Έβˆ’πœ†βˆ—β„Žξ€·π‘§βˆ—,π‘¦βˆ—π‘–βˆ’ξ€Έξ€Έπ‘šξ“π‘—=1πœ‡βˆ—π‘—π‘”π‘—ξ€·π‘§βˆ—ξ€Έ+12π‘βˆ—π‘‡βˆ‡2π‘ βˆ—ξ“π‘–=1π‘‘βˆ—π‘–ξ€·π‘“ξ€·π‘§βˆ—,π‘¦βˆ—π‘–ξ€Έβˆ’πœ†βˆ—β„Žξ€·π‘§βˆ—,π‘¦βˆ—π‘–π‘ξ€Έξ€Έβˆ—+12π‘βˆ—π‘‡βˆ‡2π‘šξ“π‘—=1πœ‡βˆ—π‘—π‘”π‘—ξ€·π‘§βˆ—ξ€Έπ‘βˆ—βŽ€βŽ₯βŽ₯βŽ¦β‰€0,(3.25) which along with (3.18) and (3.21) implies β„±βŽ›βŽœβŽœβŽπ‘₯βˆ—,π‘§βˆ—;βˆ‡π‘ βˆ—ξ“π‘–=1π‘‘βˆ—π‘–ξ€·π‘“ξ€·π‘§βˆ—,π‘¦βˆ—π‘–ξ€Έβˆ’πœ†βˆ—β„Žξ€·π‘§βˆ—,π‘¦βˆ—π‘–ξ€Έξ€Έ+βˆ‡π‘šξ“π‘—=1πœ‡βˆ—π‘—π‘”π‘—ξ€·π‘§βˆ—ξ€Έ+βˆ‡2π‘ βˆ—ξ“π‘–=1π‘‘βˆ—π‘–ξ€·π‘“ξ€·π‘§βˆ—,π‘¦βˆ—π‘–ξ€Έβˆ’πœ†βˆ—β„Žξ€·π‘§βˆ—,π‘¦βˆ—π‘–π‘ξ€Έξ€Έβˆ—+βˆ‡2π‘šξ“π‘—=1πœ‡βˆ—π‘—π‘”π‘—ξ€·π‘§βˆ—ξ€Έπ‘βˆ—βŽžβŽŸβŽŸβŽ <0,(3.26) which contradicts (3.2) since β„±(π‘₯βˆ—,π‘§βˆ—;0)=0.

4. Second Duality Model

This section deals with duality theorems for the following second-order dual to (1.1):max(𝑠,𝑑,𝑦)∈𝐾(𝑧)sup(𝑧,πœ‡,πœ†,𝑝)∈𝐻2(𝑠,𝑑,𝑦)πœ†,(4.1) where 𝐻2(𝑠,𝑑,𝑦) denotes the set of all (𝑧,πœ‡,πœ†,𝑝)βˆˆπ‘…π‘›Γ—π‘…π‘š+×𝑅+×𝑅𝑛 satisfyingβˆ‡π‘ βˆ‘π‘–=1𝑑𝑖𝑓𝑧,π‘¦π‘–ξ€Έξ€·βˆ’πœ†β„Žπ‘§,𝑦𝑖+βˆ‡2π‘ βˆ‘π‘–=1𝑑𝑖𝑓𝑧,π‘¦π‘–ξ€Έξ€·βˆ’πœ†β„Žπ‘§,𝑦𝑖𝑝+βˆ‡π‘šβˆ‘π‘—=1πœ‡π‘—π‘”π‘—(𝑧)+βˆ‡2π‘šβˆ‘π‘—=1πœ‡π‘—π‘”π‘—(𝑧)𝑝=0,(4.2)π‘ βˆ‘π‘–=1𝑑𝑖𝑓𝑧,π‘¦π‘–ξ€Έξ€·βˆ’πœ†β„Žπ‘§,𝑦𝑖+βˆ‘ξ€Έξ€Έπ‘—βˆˆπ½βˆ˜πœ‡π‘—π‘”π‘—βˆ’1(𝑧)2π‘π‘‡βˆ‡2ξƒ¬π‘ βˆ‘π‘–=1𝑑𝑖𝑓𝑧,π‘¦π‘–ξ€Έξ€·βˆ’πœ†β„Žπ‘§,𝑦𝑖+βˆ‘ξ€Έξ€Έπ‘—βˆˆπ½βˆ˜πœ‡π‘—π‘”π‘—(𝑧)𝑝β‰₯0,(4.3)π‘—βˆˆπ½π›Όπœ‡π‘—π‘”π‘—1(𝑧)βˆ’2π‘π‘‡βˆ‡2ξ“π‘—βˆˆπ½π›Όπœ‡π‘—π‘”π‘—(𝑧)𝑝β‰₯0,𝛼=1,2,…,π‘Ÿ,(4.4) where π½π›ΌβŠ†π‘€,𝛼=0,1,2,…,π‘Ÿ, with β‹ƒπ‘Ÿπ›Ό=0𝐽𝛼=𝑀 and π½π›Όβˆ©π½π›½=βˆ…, if 𝛼≠𝛽.

If, for a triplet (𝑠,𝑑,𝑦)∈𝐾(𝑧), the set 𝐻2(𝑠,𝑑,𝑦)=βˆ…, then we define the supremum over it to be βˆ’βˆž.

Theorem 4.1 (weak duality). Let π‘₯ and (𝑧,πœ‡,πœ†,𝑠,𝑑,𝑦,𝑝) be the feasible solutions of (1.1) and (4.1), respectively. Suppose that there exist β„±,πœƒ,πœ™0,𝑏0,𝜌0 and πœ™π›½,𝑏𝛽,πœŒπ›½,𝛽=1,2,…,π‘Ÿ such that 𝑏0(π‘₯,𝑧)πœ™0βŽ‘βŽ’βŽ’βŽ£π‘ ξ“π‘–=1𝑑𝑖𝑓π‘₯,π‘¦π‘–ξ€Έξ€·βˆ’πœ†β„Žπ‘₯,π‘¦π‘–βˆ’ξ€Έξ€Έπ‘ ξ“π‘–=1𝑑𝑖𝑓𝑧,π‘¦π‘–ξ€Έξ€·βˆ’πœ†β„Žπ‘§,π‘¦π‘–βˆ’ξ“ξ€Έξ€Έπ‘—βˆˆπ½0πœ‡π‘—π‘”π‘—(+1𝑧)2π‘π‘‡βˆ‡2βŽ›βŽœβŽœβŽπ‘ ξ“π‘–=1𝑑𝑖𝑓𝑧,π‘¦π‘–ξ€Έξ€·βˆ’πœ†β„Žπ‘§,𝑦𝑖+ξ“ξ€Έξ€Έπ‘—βˆˆπ½0πœ‡π‘—π‘”π‘—βŽžβŽŸβŽŸβŽ π‘βŽ€βŽ₯βŽ₯βŽ¦ξƒ©(𝑧)<0βŸΉβ„±π‘₯,𝑧;βˆ‡π‘ ξ“π‘–=1𝑑𝑖𝑓𝑧,π‘¦π‘–ξ€Έξ€·βˆ’πœ†β„Žπ‘§,𝑦𝑖+βˆ‡2𝑠𝑖=1𝑑𝑖𝑓𝑧,π‘¦π‘–ξ€Έξ€·βˆ’πœ†β„Žπ‘§,𝑦𝑖𝑝+βˆ‡π‘—βˆˆπ½0πœ‡π‘—π‘”π‘—(𝑧)+βˆ‡2ξ“π‘—βˆˆπ½0πœ‡π‘—π‘”π‘—(βŽžβŽŸβŽŸβŽ π‘§)𝑝<βˆ’πœŒ0β€–πœƒ(π‘₯,𝑧)β€–2,(4.5)βˆ’π‘π›Ό(π‘₯,𝑧)πœ™π›ΌβŽ‘βŽ’βŽ’βŽ£ξ“π‘—βˆˆπ½π›Όπœ‡π‘—π‘”π‘—1(𝑧)βˆ’2π‘π‘‡βˆ‡2ξ“π‘—βˆˆπ½π›Όπœ‡π‘—π‘”π‘—βŽ€βŽ₯βŽ₯βŽ¦βŽ›βŽœβŽœβŽξ“(𝑧)𝑝≀0βŸΉβ„±π‘₯,𝑧;βˆ‡π‘—βˆˆπ½π›Όπœ‡π‘—π‘”π‘—(𝑧)+βˆ‡2ξ“π‘—βˆˆπ½π›Όπœ‡π‘—π‘”π‘—(βŽžβŽŸβŽŸβŽ π‘§)π‘β‰€βˆ’πœŒπ›Ό(β€–πœƒπ‘₯,𝑧)β€–2,𝛼=1,2,…,π‘Ÿ.(4.6) Further assume that π‘Žβ‰₯0βŸΉπœ™π›Ό(π‘Ž)β‰₯0,𝛼=1,2,…,π‘Ÿ,(4.7)π‘Ž<0βŸΉπœ™0(π‘π‘Ž)<0,(4.8)0(π‘₯,𝑧)>0,π‘π›ΌπœŒ(π‘₯,𝑧)β‰₯0,𝛼=1,2,…,π‘Ÿ,(4.9)0+π‘Ÿξ“π›Ό=1πœŒπ›Όβ‰₯0.s(4.10) Then supπ‘¦βˆˆπ‘Œπ‘“(π‘₯,𝑦)β„Ž(π‘₯,𝑦)β‰₯πœ†.(4.11)

Proof. Suppose contrary to the result that supπ‘¦βˆˆπ‘Œπ‘“(π‘₯,𝑦)β„Ž(π‘₯,𝑦)<πœ†.(4.12) Thus, we have 𝑓π‘₯,π‘¦π‘–ξ€Έξ€·βˆ’πœ†β„Žπ‘₯,𝑦𝑖<0,βˆ€π‘¦π‘–βˆˆπ‘Œ(π‘₯),𝑖=1,2,…,𝑠.(4.13) It follows from 𝑑𝑖β‰₯0,𝑖=1,2,…,𝑠, that 𝑑𝑖𝑓π‘₯,π‘¦π‘–ξ€Έξ€·βˆ’πœ†β„Žπ‘₯,𝑦𝑖≀0,(4.14) with at least one strict inequality since 𝑑=(𝑑1,𝑑2,…,𝑑𝑠)β‰ 0. Taking summation over 𝑖, we have 𝑠𝑖=1𝑑𝑖𝑓π‘₯,π‘¦π‘–ξ€Έξ€·βˆ’πœ†β„Žπ‘₯,𝑦𝑖<0,(4.15) which together with (4.3) implies 𝑠𝑖=1𝑑𝑖𝑓π‘₯,π‘¦π‘–ξ€Έξ€·βˆ’πœ†β„Žπ‘₯,𝑦𝑖≀<0𝑠𝑖=1𝑑𝑖𝑓𝑧,π‘¦π‘–ξ€Έξ€·βˆ’πœ†β„Žπ‘§,𝑦𝑖+ξ“ξ€Έξ€Έπ‘—βˆˆπ½0πœ‡π‘—π‘”π‘—βˆ’1(𝑧)2π‘π‘‡βˆ‡2βŽ‘βŽ’βŽ’βŽ£π‘ ξ“π‘–=1𝑑𝑖𝑓𝑧,π‘¦π‘–ξ€Έξ€·βˆ’πœ†β„Žπ‘§,𝑦𝑖+ξ“ξ€Έξ€Έπ‘—βˆˆπ½0πœ‡π‘—π‘”π‘—(⎀βŽ₯βŽ₯βŽ¦π‘§)𝑝.(4.16) Using (4.8) and (4.9), it follows from (4.16) that 𝑏0(π‘₯,𝑧)πœ™0βŽ‘βŽ’βŽ’βŽ£π‘ ξ“π‘–=1𝑑𝑖𝑓π‘₯,π‘¦π‘–ξ€Έξ€·βˆ’πœ†β„Žπ‘₯,π‘¦π‘–βˆ’ξ€Έξ€Έπ‘ ξ“π‘–=1𝑑𝑖𝑓𝑧,π‘¦π‘–ξ€Έξ€·βˆ’πœ†β„Žπ‘§,π‘¦π‘–βˆ’ξ“ξ€Έξ€Έπ‘—βˆˆπ½0πœ‡π‘—π‘”π‘—(+1𝑧)2π‘π‘‡βˆ‡2βŽ‘βŽ’βŽ’βŽ£π‘ ξ“π‘–=1𝑑𝑖𝑓𝑧,π‘¦π‘–ξ€Έξ€·βˆ’πœ†β„Žπ‘§,𝑦𝑖+ξ“ξ€Έξ€Έπ‘—βˆˆπ½0πœ‡π‘—π‘”π‘—βŽ€βŽ₯βŽ₯βŽ¦π‘βŽ€βŽ₯βŽ₯⎦(𝑧)<0,(4.17) which by (4.5) implies ℱπ‘₯,𝑧;βˆ‡π‘ ξ“π‘–=1𝑑𝑖𝑓𝑧,π‘¦π‘–ξ€Έξ€·βˆ’πœ†β„Žπ‘§,𝑦𝑖+βˆ‡2𝑠𝑖=1𝑑𝑖𝑓𝑧,π‘¦π‘–ξ€Έξ€·βˆ’πœ†β„Žπ‘§,𝑦𝑖𝑝+βˆ‡π‘—βˆˆπ½0πœ‡π‘—π‘”π‘—(𝑧)+βˆ‡2ξ“π‘—βˆˆπ½0πœ‡π‘—π‘”π‘—βŽžβŽŸβŽŸβŽ (𝑧)𝑝<βˆ’πœŒ0β€–β€–πœƒ(π‘₯,𝑧)2.(4.18) Also, inequality (4.4) along with (4.7) and (4.9) yields βˆ’π‘π›Ό(π‘₯,𝑧)πœ™π›ΌβŽ‘βŽ’βŽ’βŽ£ξ“π‘—βˆˆπ½π›Όπœ‡π‘—π‘”π‘—(1𝑧)βˆ’2π‘π‘‡βˆ‡2ξ“π‘—βˆˆπ½π›Όπœ‡π‘—π‘”π‘—(⎀βŽ₯βŽ₯βŽ¦π‘§)𝑝≀0,𝛼=1,2,…,π‘Ÿ.(4.19) From (4.6) and the above inequality, we have β„±βŽ›βŽœβŽœβŽξ“π‘₯,𝑧;βˆ‡π‘—βˆˆπ½π›Όπœ‡π‘—π‘”π‘—(𝑧)+βˆ‡2ξ“π‘—βˆˆπ½π›Όπœ‡π‘—π‘”π‘—βŽžβŽŸβŽŸβŽ (𝑧)π‘β‰€βˆ’πœŒπ›Όβ€–β€–πœƒ(π‘₯,𝑧)2,𝛼=1,2,…,π‘Ÿ.(4.20) On adding (4.18) and (4.20) and making use of the sublinearity of β„± with (4.10), we obtain ℱπ‘₯,𝑧;βˆ‡π‘ ξ“π‘–=1𝑑𝑖𝑓𝑧,π‘¦π‘–ξ€Έξ€·βˆ’πœ†β„Žπ‘§,𝑦𝑖+βˆ‡2𝑠𝑖=1𝑑𝑖𝑓𝑧,π‘¦π‘–ξ€Έξ€·βˆ’πœ†β„Žπ‘§,𝑦𝑖𝑝+βˆ‡π‘šξ“π‘—=1πœ‡π‘—π‘”π‘—(𝑧)+βˆ‡2π‘šξ“π‘—=1πœ‡π‘—π‘”π‘—ξƒͺ(𝑧)𝑝<0,(4.21) which contradicts (4.2) since β„±(π‘₯,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 π‘₯βˆ— is an optimal solution of (1.1) and βˆ‡π‘”π‘—(π‘₯βˆ—),π‘—βˆˆπ½(π‘₯βˆ—), are linearly independent. Then there exist (π‘ βˆ—,π‘‘βˆ—,π‘¦βˆ—)∈𝐾(π‘₯βˆ—) and (π‘₯βˆ—,πœ‡βˆ—,πœ†βˆ—,π‘βˆ—=0)∈𝐻2(π‘ βˆ—,π‘‘βˆ—,π‘¦βˆ—) such that (π‘₯βˆ—,πœ‡βˆ—,πœ†βˆ—,π‘ βˆ—,π‘‘βˆ—,π‘¦βˆ—,π‘βˆ—=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,πœ‡,πœ†,𝑠,𝑑,𝑦,𝑝) of (4.1), then (π‘₯βˆ—,πœ‡βˆ—,πœ†βˆ—,π‘ βˆ—,π‘‘βˆ—,π‘¦βˆ—,π‘βˆ—=0) is an optimal solution of (4.1).

Theorem 4.3 (strict converse duality). Let π‘₯βˆ— and (π‘§βˆ—,πœ‡βˆ—,πœ†βˆ—,π‘ βˆ—,π‘‘βˆ—,π‘¦βˆ—,π‘βˆ—) be the optimal solutions of (1.1) and (4.1), respectively. Suppose that βˆ‡π‘”π‘—(π‘₯βˆ—),π‘—βˆˆπ½(π‘₯βˆ—) are linearly independent and there exist β„±,πœƒ,πœ™0,𝑏0,𝜌0 and πœ™π›½,𝑏𝛽,πœŒπ›½,𝛽=1,2,…,π‘Ÿ such that 𝑏0ξ€·π‘₯βˆ—,π‘§βˆ—ξ€Έπœ™0βŽ‘βŽ’βŽ’βŽ£π‘ βˆ—ξ“π‘–=1π‘‘βˆ—π‘–ξ€·π‘“ξ€·π‘₯βˆ—,π‘¦βˆ—π‘–ξ€Έβˆ’πœ†βˆ—β„Žξ€·π‘₯βˆ—,π‘¦βˆ—π‘–βˆ’ξ€Έξ€Έπ‘ βˆ—ξ“π‘–=1π‘‘βˆ—π‘–ξ€·π‘“ξ€·π‘§βˆ—,π‘¦βˆ—π‘–ξ€Έβˆ’πœ†βˆ—β„Žξ€·π‘§βˆ—,π‘¦βˆ—π‘–βˆ’ξ“ξ€Έξ€Έπ‘—βˆˆπ½0πœ‡βˆ—π‘—π‘”π‘—ξ€·π‘§βˆ—ξ€Έ+12π‘βˆ—π‘‡βˆ‡2βŽ›βŽœβŽœβŽπ‘ βˆ—ξ“π‘–=1π‘‘βˆ—π‘–ξ€·π‘“ξ€·π‘§βˆ—,π‘¦βˆ—π‘–ξ€Έβˆ’πœ†βˆ—β„Žξ€·π‘§βˆ—,π‘¦βˆ—π‘–+ξ“ξ€Έξ€Έπ‘—βˆˆπ½0πœ‡βˆ—π‘—π‘”π‘—ξ€·π‘§βˆ—ξ€ΈβŽžβŽŸβŽŸβŽ π‘βˆ—βŽ€βŽ₯βŽ₯βŽ¦βŽ›βŽœβŽœβŽπ‘₯≀0βŸΉβ„±βˆ—,π‘§βˆ—;βˆ‡π‘ βˆ—ξ“π‘–=1π‘‘βˆ—π‘–ξ€·π‘“ξ€·π‘§βˆ—,π‘¦βˆ—π‘–ξ€Έβˆ’πœ†βˆ—β„Žξ€·π‘§βˆ—,π‘¦βˆ—π‘–ξ€Έξ€Έ+βˆ‡2π‘ βˆ—ξ“π‘–=1π‘‘βˆ—π‘–ξ€·π‘“ξ€·π‘§βˆ—,π‘¦βˆ—π‘–ξ€Έβˆ’πœ†βˆ—β„Žξ€·π‘§βˆ—,π‘¦βˆ—π‘–π‘ξ€Έξ€Έβˆ—ξ“+βˆ‡π‘—βˆˆπ½0πœ‡βˆ—π‘—π‘”π‘—ξ€·π‘§βˆ—ξ€Έ+βˆ‡2ξ“π‘—βˆˆπ½0πœ‡βˆ—π‘—π‘”π‘—ξ€·π‘§βˆ—ξ€Έπ‘βˆ—βŽžβŽŸβŽŸβŽ <βˆ’πœŒ0β€–β€–πœƒξ€·π‘₯βˆ—,π‘§βˆ—ξ€Έβ€–β€–2(4.22)βˆ’π‘π›Όξ€·π‘₯βˆ—,π‘§βˆ—ξ€Έπœ™π›ΌβŽ‘βŽ’βŽ’βŽ£ξ“π‘—βˆˆπ½π›Όπœ‡βˆ—π‘—π‘”π‘—ξ€·π‘§βˆ—ξ€Έβˆ’12π‘βˆ—π‘‡βˆ‡2ξ“π‘—βˆˆπ½π›Όπœ‡βˆ—π‘—π‘”π‘—ξ€·π‘§βˆ—ξ€Έπ‘βˆ—βŽ€βŽ₯βŽ₯βŽ¦βŽ›βŽœβŽœβŽπ‘₯≀0βŸΉβ„±βˆ—,π‘§βˆ—ξ“;βˆ‡π‘—βˆˆπ½π›Όπœ‡βˆ—π‘—π‘”π‘—ξ€·π‘§βˆ—ξ€Έ+βˆ‡2ξ“π‘—βˆˆπ½π›Όπœ‡βˆ—π‘—π‘”π‘—ξ€·π‘§βˆ—ξ€Έπ‘βˆ—βŽžβŽŸβŽŸβŽ β‰€βˆ’πœŒπ›Όβ€–β€–πœƒξ€·π‘₯βˆ—,π‘§βˆ—ξ€Έβ€–β€–2,𝛼=1,2,…,π‘Ÿ.(4.23) Further assume that π‘Žβ‰₯0βŸΉπœ™π›Ό(π‘Ž)β‰₯0,𝛼=1,2,…,π‘Ÿ,(4.24)π‘Ž<0βŸΉπœ™0𝑏(π‘Ž)≀0,(4.25)0ξ€·π‘₯βˆ—,π‘§βˆ—ξ€Έ>0,𝑏𝛼π‘₯βˆ—,π‘§βˆ—ξ€ΈπœŒβ‰₯0,𝛼=1,2,…,π‘Ÿ,(4.26)0+π‘Ÿξ“π›Ό=1πœŒπ›Όβ‰₯0.(4.27) Then π‘§βˆ—=π‘₯βˆ—, that is, π‘§βˆ— is an optimal solution of (1.1).

Proof. Suppose contrary to the result that π‘§βˆ—β‰ π‘₯βˆ—. Since π‘₯βˆ— and (π‘§βˆ—,πœ‡βˆ—,πœ†βˆ—,π‘ βˆ—,π‘‘βˆ—,π‘¦βˆ—,π‘βˆ—) are optimal solutions of (1.1) and (4.1), respectively, and βˆ‡π‘”π‘—(π‘₯βˆ—),π‘—βˆˆπ½(π‘₯βˆ—) are linearly independent, therefore, from strong duality (Theorem 4.2), we reach supπ‘¦βˆ—βˆˆπ‘Œπ‘“ξ€·π‘₯βˆ—,π‘¦βˆ—ξ€Έβ„Ž(π‘₯βˆ—,π‘¦βˆ—)=πœ†βˆ—.(4.28) Thus, we have 𝑓π‘₯βˆ—,π‘¦βˆ—π‘–ξ€Έβˆ’πœ†βˆ—β„Žξ€·π‘₯βˆ—,π‘¦βˆ—π‘–ξ€Έβ‰€0,βˆ€π‘¦βˆ—π‘–ξ€·π‘₯βˆˆπ‘Œβˆ—ξ€Έ,𝑖=1,2,…,π‘ βˆ—.(4.29) Now, proceeding as in Theorem 4.1, we get π‘ βˆ—ξ“π‘–=1π‘‘βˆ—π‘–ξ€·π‘“ξ€·π‘₯βˆ—,π‘¦βˆ—π‘–ξ€Έβˆ’πœ†βˆ—β„Žξ€·π‘₯βˆ—,π‘¦βˆ—π‘–βˆ’ξ€Έξ€Έπ‘ βˆ—ξ“π‘–=1π‘‘βˆ—π‘–ξ€·π‘“ξ€·π‘§βˆ—,π‘¦βˆ—π‘–ξ€Έβˆ’πœ†βˆ—β„Žξ€·π‘§βˆ—,π‘¦βˆ—π‘–βˆ’ξ“ξ€Έξ€Έπ‘—βˆˆπ½0πœ‡βˆ—π‘—π‘”π‘—ξ€·π‘§βˆ—ξ€Έ+12π‘βˆ—π‘‡βˆ‡2βŽ›βŽœβŽœβŽπ‘ βˆ—ξ“π‘–=1π‘‘βˆ—π‘–ξ€·π‘“ξ€·π‘§βˆ—,π‘¦βˆ—π‘–ξ€Έβˆ’πœ†βˆ—β„Žξ€·π‘§βˆ—,π‘¦βˆ—π‘–+ξ“ξ€Έξ€Έπ‘—βˆˆπ½0πœ‡βˆ—π‘—π‘”π‘—ξ€·π‘§βˆ—ξ€ΈβŽžβŽŸβŽŸβŽ π‘βˆ—<0.(4.30) Using (4.25) and (4.26), it follows from (4.30) that 𝑏0ξ€·π‘₯βˆ—,π‘§βˆ—ξ€Έπœ™0βŽ‘βŽ’βŽ’βŽ£π‘ βˆ—ξ“π‘–=1π‘‘βˆ—π‘–ξ€·π‘“ξ€·π‘₯βˆ—,π‘¦βˆ—π‘–ξ€Έβˆ’πœ†βˆ—β„Žξ€·π‘₯βˆ—,π‘¦βˆ—π‘–βˆ’ξ€Έξ€Έπ‘ βˆ—ξ“π‘–=1π‘‘βˆ—π‘–ξ€·π‘“ξ€·π‘§βˆ—,π‘¦βˆ—π‘–ξ€Έβˆ’πœ†βˆ—β„Žξ€·π‘§βˆ—,π‘¦βˆ—π‘–βˆ’ξ“ξ€Έξ€Έπ‘—βˆˆπ½0πœ‡βˆ—π‘—π‘”π‘—ξ€·π‘§βˆ—ξ€Έ+12π‘βˆ—π‘‡βˆ‡2βŽ›βŽœβŽœβŽπ‘ βˆ—ξ“π‘–=1π‘‘βˆ—π‘–ξ€·π‘“ξ€·π‘§βˆ—,π‘¦βˆ—π‘–ξ€Έβˆ’πœ†βˆ—β„Žξ€·π‘§βˆ—,π‘¦βˆ—π‘–+ξ“ξ€Έξ€Έπ‘—βˆˆπ½0πœ‡βˆ—π‘—π‘”π‘—ξ€·π‘§βˆ—ξ€ΈβŽžβŽŸβŽŸβŽ π‘βˆ—βŽ€βŽ₯βŽ₯βŽ¦β‰€0,(4.31) which by (4.22) implies β„±βŽ›βŽœβŽœβŽπ‘₯βˆ—,π‘§βˆ—;βˆ‡π‘ βˆ—ξ“π‘–=1π‘‘βˆ—π‘–ξ€·π‘“ξ€·π‘§βˆ—,π‘¦βˆ—π‘–ξ€Έβˆ’πœ†βˆ—β„Žξ€·π‘§βˆ—,π‘¦βˆ—π‘–ξ€Έξ€Έ+βˆ‡2π‘ βˆ—ξ“π‘–=1π‘‘βˆ—π‘–ξ€·π‘“ξ€·π‘§βˆ—,π‘¦βˆ—π‘–ξ€Έβˆ’πœ†βˆ—β„Žξ€·π‘§βˆ—,π‘¦βˆ—π‘–π‘ξ€Έξ€Έβˆ—ξ“+βˆ‡π‘—βˆˆπ½0πœ‡βˆ—π‘—π‘”π‘—ξ€·π‘§βˆ—ξ€Έ+βˆ‡2ξ“π‘—βˆˆπ½0πœ‡βˆ—π‘—π‘”π‘—ξ€·π‘§βˆ—ξ€Έπ‘βˆ—βŽžβŽŸβŽŸβŽ <βˆ’πœŒ0β€–β€–πœƒξ€·π‘₯βˆ—,π‘§βˆ—ξ€Έβ€–β€–2.(4.32) Also, inequality (4.4) along with (4.24) and (4.26) yields βˆ’π‘π›Όξ€·π‘₯βˆ—,π‘§βˆ—ξ€Έπœ™π›ΌβŽ‘βŽ’βŽ’βŽ£ξ“π‘—βˆˆπ½π›Όπœ‡βˆ—π‘—π‘”π‘—ξ€·π‘§βˆ—ξ€Έβˆ’12π‘βˆ—π‘‡βˆ‡2ξ“π‘—βˆˆπ½π›Όπœ‡βˆ—π‘—π‘”π‘—ξ€·π‘§βˆ—ξ€Έπ‘βˆ—βŽ€βŽ₯βŽ₯βŽ¦β‰€0,𝛼=1,2,…,π‘Ÿ.(4.33) From (4.23) and the above inequality, we have β„±βŽ›βŽœβŽœβŽπ‘₯βˆ—,π‘§βˆ—ξ“;βˆ‡π‘—βˆˆπ½π›Όπœ‡βˆ—π‘—π‘”π‘—ξ€·π‘§βˆ—ξ€Έ+βˆ‡2ξ“π‘—βˆˆπ½π›Όπœ‡βˆ—π‘—π‘”π‘—ξ€·π‘§βˆ—ξ€Έπ‘βˆ—βŽžβŽŸβŽŸβŽ β‰€βˆ’πœŒπ›Όβ€–πœƒ(π‘₯βˆ—,π‘§βˆ—)β€–2,𝛼=1,2,…,π‘Ÿ.(4.34) On adding (4.32) and (4.34) and making use of the sublinearity of β„± with (4.27), we obtain β„±βŽ›βŽœβŽœβŽπ‘₯βˆ—,π‘§βˆ—;βˆ‡π‘ βˆ—ξ“π‘–=1π‘‘βˆ—iξ€·π‘“ξ€·π‘§βˆ—,π‘¦βˆ—π‘–ξ€Έβˆ’πœ†βˆ—β„Žξ€·π‘§βˆ—,π‘¦βˆ—π‘–ξ€Έξ€Έ+βˆ‡2π‘ βˆ—ξ“π‘–=1π‘‘βˆ—π‘–ξ€·π‘“ξ€·π‘§βˆ—,π‘¦βˆ—π‘–ξ€Έβˆ’πœ†βˆ—β„Žξ€·π‘§βˆ—,π‘¦βˆ—π‘–π‘ξ€Έξ€Έβˆ—+βˆ‡π‘šξ“π‘—=1πœ‡βˆ—π‘—π‘”π‘—ξ€·π‘§βˆ—ξ€Έ+βˆ‡2π‘šξ“π‘—=1πœ‡βˆ—π‘—π‘”π‘—ξ€·π‘§βˆ—ξ€Έπ‘βˆ—ξƒͺ<0,(4.35) which contradicts (4.2) since β„±(π‘₯βˆ—,π‘§βˆ—;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]:minimizeπœ“(π‘₯)=supπ‘¦βˆˆπ‘Œξ€·π‘₯𝑓(π‘₯,𝑦)+𝑇𝐡π‘₯1/2ξ€·π‘₯β„Ž(π‘₯,𝑦)βˆ’π‘‡ξ€Έπ·π‘₯1/2,subjectto𝑔(π‘₯)≀0,π‘₯βˆˆπ‘…π‘›,(5.1) where π‘Œ is a compact subset of 𝑅𝑙, 𝐡 and 𝐷 are 𝑛×𝑛 positive semidefinite symmetric matrices, and 𝑓(β‹…,β‹…)βˆΆπ‘…π‘›Γ—π‘…π‘™β†’π‘…, β€‰β€‰β„Ž(β‹…,β‹…)βˆΆπ‘…π‘›Γ—π‘…π‘™β†’π‘…, and 𝑔(β‹…)βˆΆπ‘…π‘›β†’π‘…π‘š 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:minimizeΞ¨(πœ‰)=supπœˆβˆˆπ‘Šξ‚ƒξ€·π‘§Re𝑓(πœ‰,𝜈)+𝑇𝐡𝑧1/2𝑧Reβ„Ž(πœ‰,𝜈)βˆ’π‘‡ξ€Έπ·π‘§1/2ξ‚„,subjecttoβˆ’π‘”(πœ‰)βˆˆπ‘†,πœ‰βˆˆπ’ž2𝑛,(5.2) where πœ‰=(𝑧,𝑧),𝜈=(πœ”,πœ”) for π‘§βˆˆπ’žπ‘›,πœ”βˆˆπ’žπ‘™,𝑓(β‹…,β‹…)βˆΆπ’ž2π‘›Γ—π’ž2π‘™β†’π’ž and β„Ž(β‹…,β‹…)βˆΆπ’ž2π‘›Γ—π’ž2π‘™β†’π’ž are analytic with respect to πœ”, π‘Š ia a specified compact subset in π’ž2𝑙,𝑆 is a polyhedral cone in π’žπ‘š, and π‘”βˆΆπ’ž2π‘›β†’π’žπ‘š is analytic. Also 𝐡,π·βˆˆπ’žπ‘›Γ—π‘› 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, β€œBest rational approximation and strict quasiconvexity,” SIAM Journal on Numerical Analysis, vol. 10, pp. 8–12, 1973. View at Publisher Β· View at Google Scholar
  3. R. G. Schroeder, β€œLinear programming solutions to ratio games,” Operations Research, vol. 18, pp. 300–305, 1970. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  4. A. Soyster, B. Lev, and D. Toof, β€œConservative linear programming with mixed multiple objectives,” Omega, vol. 5, no. 2, pp. 193–205, 1977. View at Publisher Β· View at Google Scholar Β· View at Scopus
  5. T. Weir, β€œA 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 Β· View at Zentralblatt MATH
  6. C. Bajona-Xandri and J. E. Martinez-Legaz, β€œLower subdifferentiability in minimax fractional programming,” Optimization, vol. 45, no. 1–4, pp. 1–12, 1999. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  7. S. R. Yadav and R. N. Mukherjee, β€œDuality for fractional minimax programming problems,” Australian Mathematical Society—Series B, vol. 31, no. 4, pp. 484–492, 1990. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  8. S. Chandra and V. Kumar, β€œDuality in fractional minimax programming,” Australian Mathematical Society, vol. 58, no. 3, pp. 376–386, 1995. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  9. J. C. Liu and C. S. Wu, β€œOn minimax fractional optimality conditions with invexity,” Journal of Mathematical Analysis and Applications, vol. 219, no. 1, pp. 21–35, 1998. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  10. J. C. Liu and C. S. Wu, β€œOn 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 Β· View at Google Scholar Β· View at Zentralblatt MATH
  11. I. Ahmad, β€œOptimality 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, β€œSecond order duality for nonlinear programs,” Opsearch, vol. 11, no. 2-3, pp. 90–99, 1974. View at Google Scholar
  13. M. A. Hanson, β€œSecond order invexity and duality in mathematical programming,” Opsearch, vol. 30, pp. 313–320, 1993. View at Google Scholar Β· View at Zentralblatt MATH
  14. O. L. Mangasarian, β€œSecond and higher order duality in nonlinear programming,” Journal of Mathematical Analysis and Applications, vol. 51, no. 3, pp. 607–620, 1975. View at Publisher Β· View at Google Scholar
  15. C. R. Bector and S. Chandra, β€œGeneralized-bonvexity and higher order duality for fractional programming,” Opsearch, vol. 24, no. 3, pp. 143–154, 1987. View at Google Scholar Β· View at Zentralblatt MATH
  16. C. R. Bector, S. Chandra, and I. Husain, β€œSecond order duality for a minimax programming problem,” Opsearch, vol. 28, pp. 249–263, 1991. View at Google Scholar Β· View at Zentralblatt MATH
  17. N. G. Rueda, M. A. Hanson, and C. Singh, β€œOptimality and duality with generalized convexity,” Journal of Optimization Theory and Applications, vol. 86, no. 2, pp. 491–500, 1995. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  18. X. M. Yang and S. H. Hou, β€œOn minimax fractional optimality and duality with generalized convexity,” Journal of Global Optimization, vol. 31, no. 2, pp. 235–252, 2005. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  19. Z. Husain, I. Ahmad, and S. Sharma, β€œSecond order duality for minmax fractional programming,” Optimization Letters, vol. 3, no. 2, pp. 277–286, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  20. J. C. Liu, β€œSecond order duality for minimax programming,” Utilitas Mathematica, vol. 56, pp. 53–63, 1999. View at Google Scholar Β· View at Zentralblatt MATH
  21. Z. Husain, A. Jayswal, and I. Ahmad, β€œSecond 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 Β· View at Google Scholar Β· View at Zentralblatt MATH
  22. I. Ahmad, S. K. Gupta, N. R. Kailey, and R. P. Agarwal, β€œDuality in nondifferentiable minimax fractional programming with B-(p,r)-invexity,” Journal of Inequalities and Applications. In press.
  23. I. Ahmad and Z. Husain, β€œDuality in nondifferentiable minimax fractional programming with generalized convexity,” Applied Mathematics and Computation, vol. 176, no. 2, pp. 545–551, 2006. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH