International Scholarly Research Notices

International Scholarly Research Notices / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 491941 | 19 pages | https://doi.org/10.5402/2011/491941

Optimality and Duality for Nondifferentiable Minimax Fractional Programming with Generalized Convexity

Academic Editor: H. C. So
Received15 Mar 2011
Accepted02 May 2011
Published28 Jun 2011

Abstract

We establish several sufficient optimality conditions for a class of nondifferentiable minimax fractional programming problems from a view point of generalized convexity. Subsequently, these optimality criteria are utilized as a basis for constructing dual models, and certain duality results have been derived in the framework of generalized convex functions. Our results extend and unify some known results on minimax fractional programming problems.

1. Introduction

Several authors have been interested in the optimality conditions and duality results for minimax programming problems. Necessary and sufficient conditions for generalized minimax programming were developed first by Schmitendorf [1]. Tanimoto [2] defined a dual problem and derived duality theorems for convex minimax programming problems using Schmitendorf's results.

Yadav and Mukherjee [3] also employed the optimality conditions of Schmitendorf [1] to construct the two dual problems and derived duality theorems for differentiable fractional minimax programming problems. Chandra and Kumar [4] pointed out that the formulation of Yadav and Mukherjee [3] has some omissions and inconsistencies, and they constructed two new dual problems and proved duality theorems for differentiable fractional minimax programming. Liu et al. [5, 6], Liang and Shi [7], and Yang and Hou [8] paid much attention on minimax fractional programming problem and established sufficient optimality conditions and duality results.

Lai et al. [9] derived necessary and sufficient conditions for nondifferentiable minimax fractional problem with generalized convexity and applied these optimality conditions to construct one parametric dual model and also discussed duality theorems. Lai and Lee [10] obtained duality theorems for two parameter-free dual models of a nondifferentiable minimax fractional programming problem, involving generalized convexity assumptions. Ahmad and Husain [11, 12] established sufficient optimality conditions and duality theorems for nondifferentiable minimax fractional programming problem under (๐น,๐›ผ,๐œŒ,๐‘‘) convexity assumptions, thus extending the results of Lai et al. [9] and Lai and Lee [10]. Jayswal [13] discussed the optimality conditions and duality results for nondifferentiable minimax fractional programming under ๐›ผ-univexity. Yuan et al. [14] introduced the concept of generalized (๐ถ,๐›ผ,๐œŒ,๐‘‘)-convexity and focused their study on a nondifferentiable minimax fractional programming problems. Recently, Jayswal and Kumar [15] established sufficient optimality conditions and duality theorems for a class of nondifferentiable minimax fractional programming involving (๐ถ,๐›ผ,๐œŒ,๐‘‘)-convexity.

In the present paper, we discuss the sufficient optimality conditions for a nondifferentiable minimax fractional programming problem from a view point of generalized convexity. Subsequently, we apply the optimality conditions to formulate a dual problem, and we prove weak, strong and strict converse duality theorems involving generalized convexity.

The paper is organized as follows. In Section 2, we present a few definitions and notations and recall a set of necessary optimality conditions for a nondifferentiable minimax fractional programming problem which will be needed in the sequel. In Section 3, we discussed sufficient optimality conditions with somewhat limited structures of generalized convexity. Furthermore, a dual problem is formulated and duality results are presented in Section 4. Finally, in Section 5, we summarize our main results and also point out some additional research opportunities arising from certain modifications of the principal problem model considered in this paper.

2. Notations and Preliminaries

Let ๐‘…๐‘› denote the ๐‘›-dimensional Euclidean space and let ๐‘…๐‘›+ be its nonnegative orthant.

In this paper, we consider the following nondifferentiable minimax fractional programming problem:min๐‘ฅโˆˆ๐‘…๐‘›sup๐‘ฆโˆˆ๐‘Œ๐‘“(๐‘ฅ,๐‘ฆ)+โŸจ๐‘ฅ,๐ด๐‘ฅโŸฉ1/2โ„Ž(๐‘ฅ,๐‘ฆ)โˆ’โŸจ๐‘ฅ,๐ต๐‘ฅโŸฉ1/2subjectto๐‘”(๐‘ฅ)โ‰ค0,(P) where ๐‘“,โ„Žโˆถ๐‘…๐‘›ร—๐‘…๐‘šโ†’๐‘… and ๐‘”โˆถ๐‘…๐‘›โ†’๐‘…๐‘ are continuous differentiable functions, ๐‘Œ is a compact subset of ๐‘…๐‘š, and ๐ด and ๐ต are ๐‘›ร—๐‘› positive semidefinite matrices. The problem (P) is nondifferentiable programming problem if either ๐ด or ๐ต is nonzero. If ๐ด and ๐ต are null matrices, then the problem (P) is a usual minimax fractional programming problem which was studied by Liang and Shi [7] and Yang and Hou [8].

Let โ„‘๐‘ƒ = {๐‘ฅโˆˆ๐‘…๐‘›โˆถ๐‘”(๐‘ฅ)โ‰ค0} be the set of all feasible solutions of (P). For each (๐‘ฅ,๐‘ฆ)โˆˆ๐‘…๐‘›ร—๐‘…๐‘š, we define๐œ™(๐‘ฅ,๐‘ฆ)=๐‘“(๐‘ฅ,๐‘ฆ)+โŸจ๐‘ฅ,๐ด๐‘ฅโŸฉ1/2โ„Ž(๐‘ฅ,๐‘ฆ)โˆ’โŸจ๐‘ฅ,๐ต๐‘ฅโŸฉ1/2.(2.1) Assume that for each (๐‘ฅ,๐‘ฆ)โˆˆ๐‘…๐‘›ร—๐‘Œ, ๐‘“(๐‘ฅ,๐‘ฆ)+โŸจ๐‘ฅ,๐ด๐‘ฅโŸฉ1/2โ‰ฅ0, and โ„Ž(๐‘ฅ,๐‘ฆ)โˆ’โŸจ๐‘ฅ,๐ต๐‘ฅโŸฉ1/2>0.

Denote๎ƒฏ๐‘Œ(๐‘ฅ)=๐‘“๎€ท๐‘ฆโˆˆ๐‘Œโˆถ๐‘ฅ,๐‘ฆ๎€ธ+โŸจ๐‘ฅ,๐ด๐‘ฅโŸฉ1/2โ„Ž๎€ท๐‘ฅ,๐‘ฆ๎€ธโˆ’โŸจ๐‘ฅ,๐ต๐‘ฅโŸฉ1/2=sup๐‘งโˆˆ๐‘Œ๐‘“(๐‘ฅ,๐‘ง)+โŸจ๐‘ฅ,๐ด๐‘ฅโŸฉ1/2โ„Ž(๐‘ฅ,๐‘ง)โˆ’โŸจ๐‘ฅ,๐ต๐‘ฅโŸฉ1/2๎ƒฐ,๎€ฝ๐ฝ={1,2,โ€ฆ,๐‘},๐ฝ(๐‘ฅ)=๐‘—โˆˆ๐ฝโˆถ๐‘”๐‘—๎€พ,๎‚ป(๐‘ฅ)=0๐พ(๐‘ฅ)=(๐‘ ,๐‘ก,ฬƒ๐‘ฆ)โˆˆ๐‘ร—๐‘…๐‘ +ร—๐‘…๐‘š๐‘ ๎€ท๐‘กโˆถ1โ‰ฆ๐‘ โ‰ฆ๐‘›+1,๐‘ก=1,๐‘ก2,โ€ฆ,๐‘ก๐‘ ๎€ธโˆˆ๐‘…๐‘ +with๐‘ ๎“๐‘–=1๐‘ก๐‘–๎€ท=1,ฬƒ๐‘ฆ=๐‘ฆ1,๐‘ฆ2โ€ฆ,๐‘ฆ๐‘ ๎€ธ,๐‘ฆ๐‘–โˆˆ๎ƒฐ.๐‘Œ(๐‘ฅ),๐‘–=1,2,โ€ฆ,๐‘ (2.2)

Since ๐‘“ and โ„Ž are continuously differentiable and ๐‘Œ is a compact subset of ๐‘…๐‘š, it follows that for each ๐‘ฅโˆ—โˆˆโ„‘๐‘ƒ, ๐‘Œ(๐‘ฅโˆ—)โ‰ ๐œ™. Thus, for any ๐‘ฆ๐‘–โˆˆ๐‘Œ(๐‘ฅโˆ—), we have a positive constant ๐‘ฃโˆ—=๐œ™(๐‘ฅโˆ—,๐‘ฆ๐‘–).

Definition 2.1. A functional ๐นโˆถ๐‘‹ร—๐‘‹ร—๐‘…๐‘›โ†’๐‘…(where๐‘‹โŠ†๐‘…๐‘›) is said to be sublinear in its third argument, if for all (๐‘ฅ,๐‘ฅ0)โˆˆ๐‘‹ร—๐‘‹, ๐น๎€ท๐‘ฅ,๐‘ฅ0;๐‘Ž1+๐‘Ž2๎€ธ๎€ทโ‰ค๐น๐‘ฅ,๐‘ฅ0;๐‘Ž1๎€ธ๎€ท+๐น๐‘ฅ,๐‘ฅ0;๐‘Ž2๎€ธ,โˆ€๐‘Ž1,๐‘Ž2โˆˆ๐‘…๐‘›,๐น๎€ท๐‘ฅ,๐‘ฅ0๎€ธ๎€ท;๐›ผ๐‘Ž=๐›ผ๐น๐‘ฅ,๐‘ฅ0๎€ธ;๐‘Ž,โˆ€๐›ผโˆˆ๐‘…,๐›ผโ‰ฅ0,โˆ€๐‘Žโˆˆ๐‘…๐‘›.(2.3) The following result from Lai and Lee [10] is needed in the sequel.

Lemma 2.2. Let ๐‘ฅโˆ— be an optimal solution for (P) satisfying โŸจ๐‘ฅโˆ—,๐ด๐‘ฅโˆ—โŸฉ>0,โŸจ๐‘ฅโˆ—,๐ต๐‘ฅโˆ—โŸฉ>0 and let โˆ‡๐‘”๐‘—(๐‘ฅโˆ—), ๐‘—โˆˆ๐ฝ(๐‘ฅโˆ—) be linearly independent, then there exist (๐‘ ,๐‘กโˆ—,ฬƒ๐‘ฆ)โˆˆ๐พ(๐‘ฅโˆ—), ๐‘ฃโˆ—โˆˆ๐‘…+, ๐‘ข,๐‘ฃโˆˆ๐‘…๐‘› and ๐œ‡โˆ—โˆˆ๐‘…๐‘+ such that ๐‘ ๎“๐‘–=1๐‘กโˆ—๐‘–๎€ท๎€ท๐‘ฅโˆ‡๐‘“โˆ—,๐‘ฆ๐‘–๎€ธ+๐ด๐‘ขโˆ’๐‘ฃโˆ—๎€ท๎€ท๐‘ฅโˆ‡โ„Žโˆ—,๐‘ฆ๐‘–๎€ธ+โˆ’๐ต๐‘ฃ๎€ธ๎€ธ๐‘๎“๐‘—=1๐œ‡โˆ—๐‘—โˆ‡๐‘”๐‘—๎€ท๐‘ฅโˆ—๎€ธ=0,(2.4)๐‘“๎€ท๐‘ฅโˆ—,๐‘ฆ๐‘–๎€ธ+โŸจ๐‘ฅโˆ—,๐ด๐‘ฅโˆ—โŸฉ1/2โˆ’๐‘ฃโˆ—๎‚€โ„Ž๎€ท๐‘ฅโˆ—,๐‘ฆ๐‘–๎€ธโˆ’โŸจ๐‘ฅโˆ—,๐ต๐‘ฅโˆ—โŸฉ1/2๎‚=0,๐‘–=1,2,โ€ฆ,๐‘ ,(2.5)๐‘๎“๐‘—=1๐œ‡โˆ—๐‘—๐‘”๐‘—๎€ท๐‘ฅโˆ—๎€ธ=0,(2.6)๐‘กโˆ—๐‘–โˆˆ๐‘…๐‘ +,๐‘ ๎“๐‘–=1๐‘กโˆ—๐‘–=1,๐‘ฆ๐‘–๎€ท๐‘ฅโˆˆ๐‘Œโˆ—๎€ธ,๐‘–=1,2,โ€ฆ,๐‘ ,(2.7)โŸจ๐‘ข,๐ด๐‘ขโŸฉโ‰ค1,โŸจ๐‘ฃ,๐ต๐‘ฃโŸฉโ‰ค1,โŸจ๐‘ฅโˆ—,๐ด๐‘ขโŸฉ=โŸจ๐‘ฅโˆ—,๐ด๐‘ฅโˆ—โŸฉ1/2,โŸจ๐‘ฅโˆ—,๐ต๐‘ฃโŸฉ=โŸจ๐‘ฅโˆ—,๐ต๐‘ฅโˆ—โŸฉ1/2.(2.8)

It should be noted that both the matrices ๐ด and ๐ต are positive definite at the solution ๐‘ฅโˆ— in the above Lemma. If one of โŸจ๐ด๐‘ฅโˆ—,๐‘ฅโˆ—โŸฉ and โŸจ๐ต๐‘ฅโˆ—,๐‘ฅโˆ—โŸฉ is zero, or both ๐ด and ๐ต are singular at ๐‘ฅโˆ—, then for (๐‘ ,๐‘กโˆ—,ฬƒ๐‘ฆ)โˆˆ๐พ(๐‘ฅโˆ—), we can take ๐‘ฬƒ๐‘ฆ(๐‘ฅโˆ—) defined in Lai and Lee [10] by๐‘ฬƒ๐‘ฆ๎€ท๐‘ฅโˆ—๎€ธ=๎€ฝ๐‘งโˆˆ๐‘…๐‘›โˆถ๎ซโˆ‡๐‘”๐‘—๎€ท๐‘ฅโˆ—๎€ธ๎ฌ๎€ท๐‘ฅ,๐‘งโ‰ค0,๐‘—โˆˆ๐ฝโˆ—๎€ธ๎€พwithanyoneofthefollowing(๐‘–)-(๐‘–๐‘–๐‘–)holds(i)โŸจ๐ด๐‘ฅโˆ—,๐‘ฅโˆ—โŸฉ>0,โŸจ๐ต๐‘ฅโˆ—,๐‘ฅโˆ—โŸน๎„”โŸฉ=0๐‘ ๎“๐‘–=1๐‘กโˆ—๐‘–๎€ท๐‘ฅโˆ‡๐‘“โˆ—,๐‘ฆ๐‘–๎€ธ+๐ด๐‘ฅโˆ—โŸจ๐ด๐‘ฅโˆ—,๐‘ฅโˆ—โŸฉ1/2โˆ’๐‘ฃโˆ—๎€ท๐‘ฅโˆ‡โ„Žโˆ—,๐‘ฆ๐‘–๎€ธ๎„•+๐‘ฃ,๐‘ง๎‚ฌ๎‚€โˆ—2๐ต๎‚๎‚ญ๐‘ง,๐‘ง1/2<0,(ii)โŸจ๐ด๐‘ฅโˆ—,๐‘ฅโˆ—โŸฉ=0,โŸจ๐ต๐‘ฅโˆ—,๐‘ฅโˆ—โŸน๎„”โŸฉ>0๐‘ ๎“๐‘–=1๐‘กโˆ—๐‘–๎ƒฉ๎€ท๐‘ฅโˆ‡๐‘“โˆ—,๐‘ฆ๐‘–๎€ธโˆ’๐‘ฃโˆ—๎ƒฉ๎€ท๐‘ฅโˆ‡โ„Žโˆ—,๐‘ฆ๐‘–๎€ธโˆ’๐ต๐‘ฅโˆ—โŸจ๐ต๐‘ฅโˆ—,๐‘ฅโˆ—โŸฉ1/2๎„•๎ƒช๎ƒช,๐‘ง+โŸจ๐ต๐‘ง,๐‘งโŸฉ1/2<0,(iii)โŸจ๐ด๐‘ฅโˆ—,๐‘ฅโˆ—โŸฉ=0,โŸจ๐ต๐‘ฅโˆ—,๐‘ฅโˆ—โŸน๎„”โŸฉ=0๐‘ ๎“๐‘–=1๐‘กโˆ—๐‘–๎€ท๎€ท๐‘ฅโˆ‡๐‘“โˆ—,๐‘ฆ๐‘–๎€ธโˆ’๐‘ฃโˆ—๎€ท๐‘ฅโˆ‡โ„Žโˆ—,๐‘ฆ๐‘–๎„•+๐‘ฃ๎€ธ๎€ธ,๐‘ง๎ซ๎€ทโˆ—๐ต๎€ธ๎ฌ๐‘ง,๐‘ง1/2+โŸจ๐ต๐‘ง,๐‘งโŸฉ1/2<0.(2.9) If we take the condition ๐‘ฬƒ๐‘ฆ(๐‘ฅโˆ—)=๐œ™ in Lemma 2.2, then the result of Lemma 2.2 still holds.

3. Sufficient Optimality Conditions

In this section, we present three sets of sufficient optimality conditions for (P) in the framework of generalized convexity.

Let ๐นโˆถ๐‘‹ร—๐‘‹ร—๐‘…๐‘›โ†’๐‘… be sublinear functional, ๐œ™0,๐œ™1โˆถ๐‘…โ†’๐‘…, ๐œƒโˆถ๐‘…๐‘›ร—๐‘…๐‘›โ†’๐‘…๐‘›, and ๐‘0,๐‘1โˆถ๐‘‹ร—๐‘‹โ†’๐‘…+. Let ๐œŒ0, ๐œŒ1 be real numbers.

Theorem 3.1. Let ๐‘ฅโˆ—โˆˆโ„‘๐‘ƒ be a feasible solution for (P), and there exist ๐‘ฃโˆ—โˆˆ๐‘…+,(๐‘ ,๐‘กโˆ—,ฬƒ๐‘ฆ)โˆˆ๐พ(๐‘ฅโˆ—), ๐‘ข,๐‘ฃโˆˆ๐‘…๐‘›, and ๐œ‡โˆ—โˆˆ๐‘…๐‘+ satisfying (2.4)โ€“(2.8). Suppose that there exist ๐น,๐œƒ,๐œ™0,๐‘0,๐œŒ0 and ๐œ™1,๐‘1,๐œŒ1 such that ๐น๎ƒฉ๐‘ฅ,๐‘ฅโˆ—;๐‘ ๎“๐‘–=1๐‘กโˆ—๐‘–๎€ท๐‘ฅ๎€ท๎€ทโˆ‡๐‘“โˆ—,๐‘ฆ๐‘–๎€ธ๎€ธ+๐ด๐‘ขโˆ’๐‘ฃโˆ—๎€ท๎€ท๐‘ฅโˆ‡โ„Žโˆ—,๐‘ฆ๐‘–๎€ธ๎ƒชโˆ’๐ต๐‘ฃ๎€ธ๎€ธโ‰ฅโˆ’๐œŒ0โ€–โ€–๐œƒ๎€ท๐‘ฅ,๐‘ฅโˆ—๎€ธโ€–โ€–2โŸน๐‘0๎€ท๐‘ฅ,๐‘ฅโˆ—๎€ธ๐œ™0๎ƒฌ๐‘ ๎“๐‘–=1๐‘กโˆ—๐‘–๎€ท๐‘“๎€ท๐‘ฅ,๐‘ฆ๐‘–๎€ธ+โŸจ๐‘ฅ,๐ด๐‘ขโŸฉโˆ’๐‘ฃโˆ—๎€ทโ„Ž๎€ท๐‘ฅ,๐‘ฆ๐‘–๎€ธโˆ’โˆ’โŸจ๐‘ฅ,๐ต๐‘ฃโŸฉ๎€ธ๎€ธ๐‘ ๎“๐‘–=1๐‘กโˆ—๐‘–๎€ท๐‘“๎€ท๐‘ฅโˆ—,๐‘ฆ๐‘–๎€ธ+โŸจ๐‘ฅโˆ—,๐ด๐‘ขโŸฉโˆ’๐‘ฃโˆ—๎€ทโ„Ž๎€ท๐‘ฅโˆ—,๐‘ฆ๐‘–๎€ธโˆ’โŸจ๐‘ฅโˆ—๎ƒญ,๐ต๐‘ฃโŸฉ๎€ธ๎€ธโ‰ฅ0,(3.1)โˆ’๐‘1๎€ท๐‘ฅ,๐‘ฅโˆ—๎€ธ๐œ™1๎ƒฉ๐‘๎“๐‘—=1๐œ‡โˆ—๐‘—๐‘”๐‘—๎€ท๐‘ฅโˆ—๎€ธ๎ƒช๎ƒฉโ‰ค0โŸน๐น๐‘ฅ,๐‘ฅโˆ—;๐‘๎“๐‘—=1๐œ‡โˆ—๐‘—โˆ‡๐‘”๐‘—๎€ท๐‘ฅโˆ—๎€ธ๎ƒชโ‰คโˆ’๐œŒ1โ€–โ€–๐œƒ๎€ท๐‘ฅ,๐‘ฅโˆ—๎€ธโ€–โ€–2.(3.2) Further, assume that ๐‘Žโ‰ฅ0โŸน๐œ™1๐œ™(๐‘Ž)โ‰ฅ0,(3.3)0(๐‘๐‘Ž)โ‰ฅ0โŸน๐‘Žโ‰ฅ0,(3.4)0๎€ท๐‘ฅ,๐‘ฅโˆ—๎€ธโ‰ฅ0,๐‘1๎€ท๐‘ฅ,๐‘ฅโˆ—๎€ธ๐œŒ>0,(3.5)0+๐œŒ1โ‰ฅ0,(3.6) then ๐‘ฅโˆ— is an optimal solution of (P).

Proof. Suppose to the contrary that ๐‘ฅโˆ— is not an optimal solution of (P), then there exists ๐‘ฅโˆˆโ„‘๐‘ƒ such that sup๐‘ฆโˆˆ๐‘Œ๐‘“(๐‘ฅ,๐‘ฆ)+โŸจ๐‘ฅ,๐ด๐‘ฅโŸฉ1/2โ„Ž(๐‘ฅ,๐‘ฆ)โˆ’โŸจ๐‘ฅ,๐ต๐‘ฅโŸฉ1/2<sup๐‘ฆโˆˆ๐‘Œ๐‘“๎€ท๐‘ฅโˆ—๎€ธ,๐‘ฆ+โŸจ๐‘ฅโˆ—,๐ด๐‘ฅโˆ—โŸฉ1/2โ„Ž(๐‘ฅโˆ—,๐‘ฆ)โˆ’โŸจ๐‘ฅโˆ—,๐ต๐‘ฅโˆ—โŸฉ1/2.(3.7) We note that sup๐‘ฆโˆˆ๐‘Œ๐‘“๎€ท๐‘ฅโˆ—๎€ธ,๐‘ฆ+โŸจ๐‘ฅโˆ—,๐ด๐‘ฅโˆ—โŸฉ1/2โ„Ž(๐‘ฅโˆ—,๐‘ฆ)โˆ’โŸจ๐‘ฅโˆ—,๐ต๐‘ฅโˆ—โŸฉ1/2=๐‘“๎€ท๐‘ฅโˆ—,๐‘ฆ๐‘–๎€ธ+โŸจ๐‘ฅโˆ—,๐ด๐‘ฅโˆ—โŸฉ1/2โ„Ž๎€ท๐‘ฅโˆ—,๐‘ฆ๐‘–๎€ธโˆ’โŸจ๐‘ฅโˆ—,๐ต๐‘ฅโˆ—โŸฉ1/2=๐‘ฃโˆ—,(3.8) for ๐‘ฆ๐‘–โˆˆ๐‘Œ(๐‘ฅโˆ—), ๐‘–=1,2,โ€ฆ,๐‘ , ๐‘“๎€ท๐‘ฅ,๐‘ฆ๐‘–๎€ธ+โŸจ๐‘ฅ,๐ด๐‘ฅโŸฉ1/2โ„Ž๎€ท๐‘ฅ,๐‘ฆ๐‘–๎€ธโˆ’โŸจ๐‘ฅ,๐ต๐‘ฅโŸฉ1/2โ‰คsup๐‘ฆโˆˆ๐‘Œ๐‘“(๐‘ฅ,๐‘ฆ)+โŸจ๐‘ฅ,๐ด๐‘ฅโŸฉ1/2โ„Ž(๐‘ฅ,๐‘ฆ)โˆ’โŸจ๐‘ฅ,๐ต๐‘ฅโŸฉ1/2.(3.9) Thus, we have ๐‘“๎€ท๐‘ฅ,๐‘ฆ๐‘–๎€ธ+โŸจ๐‘ฅ,๐ด๐‘ฅโŸฉ1/2โ„Ž๎€ท๐‘ฅ,๐‘ฆ๐‘–๎€ธโˆ’โŸจ๐‘ฅ,๐ต๐‘ฅโŸฉ1/2<๐‘ฃโˆ—for๐‘–=1,2,โ€ฆ,๐‘ .(3.10) It follows that ๐‘“๎€ท๐‘ฅ,๐‘ฆ๐‘–๎€ธ+โŸจ๐‘ฅ,๐ด๐‘ฅโŸฉ1/2โˆ’๐‘ฃโˆ—๎€ทโ„Ž๎€ท๐‘ฅ,๐‘ฆ๐‘–๎€ธโˆ’โŸจ๐‘ฅ,๐ต๐‘ฅโŸฉ1/2๎€ธ<0,for๐‘–=1,2,โ€ฆ,๐‘ .(3.11) From (2.5), (2.7), (2.8), and (3.11), we get ๐‘ ๎“๐‘–=1๐‘กโˆ—๐‘–๎€ท๐‘“๎€ท๐‘ฅ,๐‘ฆ๐‘–๎€ธ+โŸจ๐‘ฅ,๐ด๐‘ขโŸฉโˆ’๐‘ฃโˆ—๎€ทโ„Ž๎€ท๐‘ฅ,๐‘ฆ๐‘–๎€ธ<โˆ’โŸจ๐‘ฅ,๐ต๐‘ฃโŸฉ๎€ธ๎€ธ๐‘ ๎“๐‘–=1๐‘กโˆ—๐‘–๎€ท๐‘“๎€ท๐‘ฅโˆ—,๐‘ฆ๐‘–๎€ธ+โŸจ๐‘ฅโˆ—,๐ด๐‘ขโŸฉโˆ’๐‘ฃโˆ—๎€ทโ„Ž๎€ท๐‘ฅโˆ—,๐‘ฆ๐‘–๎€ธโˆ’โŸจ๐‘ฅโˆ—.,๐ต๐‘ฃโŸฉ๎€ธ๎€ธ(3.12) On the other hand, from (2.6), (3.3), and (3.5), we have โˆ’๐‘1๎€ท๐‘ฅ,๐‘ฅโˆ—๎€ธ๐œ™1๎ƒฉ๐‘๎“๐‘—=1๐œ‡โˆ—๐‘—๐‘”๐‘—๎€ท๐‘ฅโˆ—๎€ธ๎ƒชโ‰ค0.(3.13) It follows from (3.2) that ๐น๎ƒฉ๐‘ฅ,๐‘ฅโˆ—;๐‘๎“๐‘—=1๐œ‡โˆ—๐‘—โˆ‡๐‘”๐‘—๎€ท๐‘ฅโˆ—๎€ธ๎ƒชโ‰คโˆ’๐œŒ1โ€–โ€–๐œƒ๎€ท๐‘ฅ,๐‘ฅโˆ—๎€ธโ€–โ€–2.(3.14) From (2.4), the sublinearity of ๐น, and (3.6), we get ๐น๎ƒฉ๐‘ฅ,๐‘ฅโˆ—;๐‘ ๎“๐‘–=1๐‘กโˆ—๐‘–๎€ท๎€ท๐‘ฅโˆ‡๐‘“โˆ—,๐‘ฆ๐‘–๎€ธ+๐ด๐‘ขโˆ’๐‘ฃโˆ—๎€ท๎€ท๐‘ฅโˆ‡โ„Žโˆ—,๐‘ฆ๐‘–๎€ธ๎ƒชโˆ’๐ต๐‘ฃ๎€ธ๎€ธโ‰ฅโˆ’๐œŒ0โ€–โ€–๐œƒ๎€ท๐‘ฅ,๐‘ฅโˆ—๎€ธโ€–โ€–2.(3.15) Then by (3.1), we have ๐‘0๎€ท๐‘ฅ,๐‘ฅโˆ—๎€ธ๐œ™0๎ƒฌ๐‘ ๎“๐‘–=1๐‘กโˆ—๐‘–๎€ท๐‘“๎€ท๐‘ฅ,๐‘ฆ๐‘–๎€ธ+โŸจ๐‘ฅ,๐ด๐‘ขโŸฉโˆ’๐‘ฃโˆ—๎€ทโ„Ž๎€ท๐‘ฅ,๐‘ฆ๐‘–๎€ธโˆ’โˆ’โŸจ๐‘ฅ,๐ต๐‘ฃโŸฉ๎€ธ๎€ธ๐‘ ๎“๐‘–=1๐‘กโˆ—๐‘–๎€ท๐‘“๎€ท๐‘ฅโˆ—,๐‘ฆ๐‘–๎€ธ+โŸจ๐‘ฅโˆ—,๐ด๐‘ขโŸฉโˆ’๐‘ฃโˆ—๎€ทโ„Ž๎€ท๐‘ฅโˆ—,๐‘ฆ๐‘–๎€ธโˆ’โŸจ๐‘ฅโˆ—๎ƒญ,๐ต๐‘ฃโŸฉ๎€ธ๎€ธโ‰ฅ0.(3.16) From (3.4), (3.5), and the above inequality, we obtain ๐‘ ๎“๐‘–=1๐‘กโˆ—๐‘–๎€ท๐‘“๎€ท๐‘ฅ,๐‘ฆ๐‘–๎€ธ+โŸจ๐‘ฅ,๐ด๐‘ขโŸฉโˆ’๐‘ฃโˆ—๎€ทโ„Ž๎€ท๐‘ฅ,๐‘ฆ๐‘–๎€ธโˆ’โˆ’โŸจ๐‘ฅ,๐ต๐‘ฃโŸฉ๎€ธ๎€ธ๐‘ ๎“๐‘–=1๐‘กโˆ—๐‘–๎€ท๐‘“๎€ท๐‘ฅโˆ—,๐‘ฆ๐‘–๎€ธ+โŸจ๐‘ฅโˆ—,๐ด๐‘ขโŸฉโˆ’๐‘ฃโˆ—๎€ทโ„Ž๎€ท๐‘ฅโˆ—,๐‘ฆ๐‘–๎€ธโˆ’โŸจ๐‘ฅโˆ—,๐ต๐‘ฃโŸฉ๎€ธ๎€ธโ‰ฅ0,(3.17) which contradicts (3.12). Therefore, ๐‘ฅโˆ— is an optimal solution for (P). This completes the proof.

Remark 3.2. If both ๐ด and B are zero matrices, then Theorem 3.1 above reduces to Theorem 3.1 given in Yang and Hou [8].

Theorem 3.3. Let ๐‘ฅโˆ—โˆˆโ„‘๐‘ƒ be a feasible solution for (P), and there exist ๐‘ฃโˆ—โˆˆ๐‘…+,(๐‘ ,๐‘กโˆ—,ฬƒ๐‘ฆ)โˆˆ๐พ(๐‘ฅโˆ—), ๐‘ข,๐‘ฃโˆˆ๐‘…๐‘›, and ๐œ‡โˆ—โˆˆ๐‘…๐‘+ satisfying (2.4)โ€“(2.8). Suppose that there exist ๐น,๐œƒ,๐œ™0,๐‘0,๐œŒ0 and ๐œ™1,๐‘1,๐œŒ1 such that ๐น๎ƒฉ๐‘ฅ,๐‘ฅโˆ—;๐‘ ๎“๐‘–=1๐‘กโˆ—๐‘–๎€ท๎€ท๐‘ฅโˆ‡๐‘“โˆ—,๐‘ฆ๐‘–๎€ธ+๐ด๐‘ขโˆ’๐‘ฃโˆ—๎€ท๎€ท๐‘ฅโˆ‡โ„Žโˆ—,๐‘ฆ๐‘–๎€ธ๎ƒชโˆ’๐ต๐‘ฃ๎€ธ๎€ธโ‰ฅโˆ’๐œŒ0โ€–โ€–๐œƒ๎€ท๐‘ฅ,๐‘ฅโˆ—๎€ธโ€–โ€–2โŸน๐‘0๎€ท๐‘ฅ,๐‘ฅโˆ—๎€ธ๐œ™0๎ƒฌ๐‘ ๎“๐‘–=1๐‘กโˆ—๐‘–๎€ท๐‘“๎€ท๐‘ฅ,๐‘ฆ๐‘–๎€ธ+โŸจ๐‘ฅ,๐ด๐‘ขโŸฉโˆ’๐‘ฃโˆ—๎€ทโ„Ž๎€ท๐‘ฅ,๐‘ฆ๐‘–๎€ธโˆ’โˆ’โŸจ๐‘ฅ,๐ต๐‘ฃโŸฉ๎€ธ๎€ธ๐‘ ๎“๐‘–=1๐‘กโˆ—๐‘–๎€ท๐‘“๎€ท๐‘ฅโˆ—,๐‘ฆ๐‘–๎€ธ+โŸจ๐‘ฅโˆ—,๐ด๐‘ขโŸฉโˆ’๐‘ฃโˆ—๎€ทโ„Ž๎€ท๐‘ฅโˆ—,๐‘ฆ๐‘–๎€ธโˆ’โŸจ๐‘ฅโˆ—๎ƒญ,๐ต๐‘ฃโŸฉ๎€ธ๎€ธ>0,(3.18) or equivalently, ๐‘0๎€ท๐‘ฅ,๐‘ฅโˆ—๎€ธ๐œ™0๎ƒฌ๐‘ ๎“๐‘–=1๐‘กโˆ—๐‘–๎€ท๐‘“๎€ท๐‘ฅ,๐‘ฆ๐‘–๎€ธ+โŸจ๐‘ฅ,๐ด๐‘ขโŸฉโˆ’๐‘ฃโˆ—๎€ทโ„Ž๎€ท๐‘ฅ,๐‘ฆ๐‘–๎€ธโˆ’โˆ’โŸจ๐‘ฅ,๐ต๐‘ฃโŸฉ๎€ธ๎€ธ๐‘ ๎“๐‘–=1๐‘กโˆ—๐‘–๎€ท๐‘“๎€ท๐‘ฅโˆ—,๐‘ฆ๐‘–๎€ธ+โŸจ๐‘ฅโˆ—,๐ด๐‘ขโŸฉโˆ’๐‘ฃโˆ—๎€ทโ„Ž๎€ท๐‘ฅโˆ—,๐‘ฆ๐‘–๎€ธโˆ’โŸจ๐‘ฅโˆ—๎ƒญ๎ƒฉ,๐ต๐‘ฃโŸฉ๎€ธ๎€ธโ‰ค0โŸน๐น๐‘ฅ,๐‘ฅโˆ—;๐‘ ๎“๐‘–=1๐‘กโˆ—๐‘–๎€ท๎€ท๐‘ฅโˆ‡๐‘“โˆ—,๐‘ฆ๐‘–๎€ธ+๐ด๐‘ขโˆ’๐‘ฃโˆ—๎€ท๎€ท๐‘ฅโˆ‡โ„Žโˆ—,๐‘ฆ๐‘–๎€ธ๎ƒชโˆ’๐ต๐‘ฃ๎€ธ๎€ธ<โˆ’๐œŒ0โ€–โ€–๐œƒ๎€ท๐‘ฅ,๐‘ฅโˆ—๎€ธโ€–โ€–2,(3.19)โˆ’๐‘1๎€ท๐‘ฅ,๐‘ฅโˆ—๎€ธ๐œ™1๎ƒฉ๐‘๎“๐‘—=1๐œ‡โˆ—๐‘—๐‘”๐‘—๎€ท๐‘ฅโˆ—๎€ธ๎ƒช๎ƒฉโ‰ค0โŸน๐น๐‘ฅ,๐‘ฅโˆ—;๐‘๎“๐‘—=1๐œ‡โˆ—๐‘—โˆ‡๐‘”๐‘—๎€ท๐‘ฅโˆ—๎€ธ๎ƒชโ‰คโˆ’๐œŒ1โ€–โ€–๐œƒ๎€ท๐‘ฅ,๐‘ฅโˆ—๎€ธโ€–โ€–2.(3.20) Further, assume that (3.3), (3.5), (3.6), and ๐‘Žโ‰ค0โŸน๐œ™0(๐‘Ž)โ‰ค0,(3.21) are satisfied, then ๐‘ฅโˆ— is an optimal solution of (P).

Proof. Suppose to the contrary that ๐‘ฅโˆ— is not an optimal solution of (P). Following the proof of Theorem 3.1, we get ๐‘ ๎“๐‘–=1๐‘กโˆ—๐‘–๎€ท๐‘“๎€ท๐‘ฅ,๐‘ฆ๐‘–๎€ธ+โŸจ๐‘ฅ,๐ด๐‘ขโŸฉโˆ’๐‘ฃโˆ—๎€ทโ„Ž๎€ท๐‘ฅ,๐‘ฆ๐‘–๎€ธ<โˆ’โŸจ๐‘ฅ,๐ต๐‘ฃโŸฉ๎€ธ๎€ธ๐‘ ๎“๐‘–=1๐‘กโˆ—๐‘–๎€ท๐‘“๎€ท๐‘ฅโˆ—,๐‘ฆ๐‘–๎€ธ+โŸจ๐‘ฅโˆ—,๐ด๐‘ขโŸฉโˆ’๐‘ฃโˆ—๎€ทโ„Ž๎€ท๐‘ฅโˆ—,๐‘ฆ๐‘–๎€ธโˆ’โŸจ๐‘ฅโˆ—.,๐ต๐‘ฃโŸฉ๎€ธ๎€ธ(3.22) Using (3.5), (3.19), (3.21), and (3.22), we have ๐น๎ƒฉ๐‘ฅ,๐‘ฅโˆ—;๐‘ ๎“๐‘–=1๐‘กโˆ—๐‘–๎€ท๎€ท๐‘ฅโˆ‡๐‘“โˆ—,๐‘ฆ๐‘–๎€ธ+๐ด๐‘ขโˆ’๐‘ฃโˆ—๎€ท๎€ท๐‘ฅโˆ‡โ„Žโˆ—,๐‘ฆ๐‘–๎€ธ๎ƒชโˆ’๐ต๐‘ฃ๎€ธ๎€ธ<โˆ’๐œŒ0โ€–โ€–๐œƒ๎€ท๐‘ฅ,๐‘ฅโˆ—๎€ธโ€–โ€–2.(3.23) By (2.6), (3.3), (3.5), and (3.20), it follows that ๐น๎ƒฉ๐‘ฅ,๐‘ฅโˆ—;๐‘๎“๐‘—=1๐œ‡โˆ—๐‘—โˆ‡๐‘”๐‘—๎€ท๐‘ฅโˆ—๎€ธ๎ƒชโ‰คโˆ’๐œŒ1โ€–โ€–๐œƒ๎€ท๐‘ฅ,๐‘ฅโˆ—๎€ธโ€–โ€–2.(3.24) On adding (3.23) and (3.24), and making use of the sublinearity of ๐น and (3.6), we have ๐น๎ƒฉ๐‘ฅ,๐‘ฅโˆ—;๐‘ ๎“๐‘–=1๐‘กโˆ—๐‘–๎€ท๎€ท๐‘ฅโˆ‡๐‘“โˆ—,๐‘ฆ๐‘–๎€ธ+๐ด๐‘ขโˆ’๐‘ฃโˆ—๎€ท๎€ท๐‘ฅโˆ‡โ„Žโˆ—,๐‘ฆ๐‘–๎€ธ๎ƒช๎ƒฉโˆ’๐ต๐‘ฃ๎€ธ๎€ธ+๐น๐‘ฅ,๐‘ฅโˆ—;๐‘๎“๐‘—=1๐œ‡โˆ—๐‘—โˆ‡๐‘”๐‘—๎€ท๐‘ฅโˆ—๎€ธ๎ƒช๎€ท๐œŒโ‰คโˆ’0+๐œŒ1๎€ธโ€–โ€–๐œƒ๎€ท๐‘ฅ,๐‘ฅโˆ—๎€ธโ€–โ€–2๎ƒฉ<0โŸน๐น๐‘ฅ,๐‘ฅโˆ—;๐‘ ๎“๐‘–=1๐‘กโˆ—๐‘–๎ƒฉ๎€ท๐‘ฅโˆ‡๐‘“โˆ—,๐‘ฆ๐‘–๎€ธ+๐ด๐‘ขโˆ’๐‘ฃโˆ—๎€ท๎€ท๐‘ฅโˆ‡โ„Žโˆ—,๐‘ฆ๐‘–๎€ธ๎€ธ+โˆ’๐ต๐‘ฃ๐‘๎“๐‘—=1๐œ‡โˆ—๐‘—โˆ‡๐‘”๐‘—๎€ท๐‘ฅโˆ—๎€ธ๎ƒช๎ƒช<0.(3.25) On the other hand, (2.4) implies ๐น๎ƒฉ๐‘ฅ,๐‘ฅโˆ—;๐‘ ๎“๐‘–=1๐‘กโˆ—๐‘–๎ƒฉ๎€ท๐‘ฅโˆ‡๐‘“โˆ—,๐‘ฆ๐‘–๎€ธ+๐ด๐‘ขโˆ’๐‘ฃโˆ—๎€ท๎€ท๐‘ฅโˆ‡โ„Žโˆ—,๐‘ฆ๐‘–๎€ธ๎€ธ+โˆ’๐ต๐‘ฃ๐‘๎“๐‘—=1๐œ‡โˆ—๐‘—โˆ‡๐‘”๐‘—๎€ท๐‘ฅโˆ—๎€ธ๎ƒช๎ƒช=0.(3.26) Hence we have a contradiction to inequality (3.25). Therefore, ๐‘ฅโˆ— is an optimal solution for (P). This completes the proof.

Theorem 3.4. Let ๐‘ฅโˆ—โˆˆโ„‘๐‘ƒ be a feasible solution for (P) and there exist๐‘ฃโˆ—โˆˆ๐‘…+,(๐‘ ,๐‘กโˆ—,ฬƒ๐‘ฆ)โˆˆ๐พ(๐‘ฅโˆ—), ๐‘ข,๐‘ฃโˆˆ๐‘…๐‘›, and ๐œ‡โˆ—โˆˆ๐‘…๐‘+ satisfying (2.4)โ€“(2.8). Suppose that there exist ๐น,๐œƒ,๐œ™0,๐‘0,๐œŒ0 and ๐œ™1,๐‘1,๐œŒ1 such that ๐‘0๎€ท๐‘ฅ,๐‘ฅโˆ—๎€ธ๐œ™0๎ƒฌ๐‘ ๎“๐‘–=1๐‘กโˆ—๐‘–๎€ท๐‘“๎€ท๐‘ฅ,๐‘ฆ๐‘–๎€ธ+โŸจ๐‘ฅ,๐ด๐‘ขโŸฉโˆ’๐‘ฃโˆ—๎€ทโ„Ž๎€ท๐‘ฅ,๐‘ฆ๐‘–๎€ธโˆ’โˆ’โŸจ๐‘ฅ,๐ต๐‘ฃโŸฉ๎€ธ๎€ธ๐‘ ๎“๐‘–=1๐‘กโˆ—๐‘–๎€ท๐‘“๎€ท๐‘ฅโˆ—,๐‘ฆ๐‘–๎€ธ+โŸจ๐‘ฅโˆ—,๐ด๐‘ขโŸฉโˆ’๐‘ฃโˆ—๎€ทโ„Ž๎€ท๐‘ฅโˆ—,๐‘ฆ๐‘–๎€ธโˆ’โŸจ๐‘ฅโˆ—๎ƒญ๎ƒฉ,๐ต๐‘ฃโŸฉ๎€ธ๎€ธโ‰ค0โŸน๐น๐‘ฅ,๐‘ฅโˆ—;๐‘ ๎“๐‘–=1๐‘กโˆ—๐‘–๎€ท๎€ท๐‘ฅโˆ‡๐‘“โˆ—,๐‘ฆ๐‘–๎€ธ+๐ด๐‘ขโˆ’๐‘ฃโˆ—๎€ท๎€ท๐‘ฅโˆ‡โ„Žโˆ—,๐‘ฆ๐‘–๎€ธ๎ƒชโˆ’๐ต๐‘ฃ๎€ธ๎€ธโ‰คโˆ’๐œŒ0โ€–โ€–๐œƒ๎€ท๐‘ฅ,๐‘ฅโˆ—๎€ธโ€–โ€–2,๐น๎ƒฉ๐‘ฅ,๐‘ฅโˆ—;๐‘๎“๐‘—=1๐œ‡โˆ—๐‘—โˆ‡๐‘”๐‘—๎€ท๐‘ฅโˆ—๎€ธ๎ƒชโ‰ฅโˆ’๐œŒ1โ€–โ€–๐œƒ๎€ท๐‘ฅ,๐‘ฅโˆ—๎€ธโ€–โ€–2โŸนโˆ’๐‘1๎€ท๐‘ฅ,๐‘ฅโˆ—๎€ธ๐œ™1๎ƒฉ๐‘๎“๐‘—=1๐œ‡โˆ—๐‘—๐‘”๐‘—๎€ท๐‘ฅโˆ—๎€ธ๎ƒช>0.(3.27) Further, assume that (3.3), (3.5), (3.6), and (3.21) are satisfied, then ๐‘ฅโˆ— is an optimal solution of (P).

Proof. The proof is similar to that of Theorem 3.3 and hence omitted.

Remark 3.5. (i) If both ๐ด and ๐ต are zero matrices, then Theorems 3.3 and 3.4 above reduce to Theorems 3.3 given in Yang and Hou [8].
(ii) If ๐น(๐‘ฅ,๐‘ข;๐‘Ž)=โŸจ๐œ‚(๐‘ฅ,๐‘ข),๐‘ŽโŸฉ where ๐œ‚ is a function from ๐‘‹ร—๐‘‹โ†’๐‘…๐‘›, โˆ’๐œ™1(โˆ‘๐‘๐‘—=1๐œ‡โˆ—๐‘—๐‘”๐‘—(๐‘ฅโˆ—))=๐œ™1(โˆ‘๐‘๐‘—=1๐œ‡โˆ—๐‘—๐‘”๐‘—(๐‘ฅ)โˆ’๐œ‡โˆ—๐‘—๐‘”๐‘—(๐‘ฅโˆ—)), and ๐œŒ0=๐œŒ1=0, then Theorems 3.3 and 3.4 above reduce to Theorems 1(b) and 1(c) given by Mishra et al. [16].

4. Duality

In this section, we present a dual model to (P) and establish weak, strong, and strict converse duality results.

To unify and extend the dual models, we need to divide {1,2,โ€ฆ,๐‘} into several parts. Let ๐ฝ๐›ผ(0โ‰ค๐›ผโ‰ค๐‘Ÿ) be a partition of {1,2,โ€ฆ๐‘}, that is, ๐ฝ๐›ผโˆฉ๐ฝ๐›ฝ=๐œ™,for๐›ผโ‰ ๐›ฝ,๐‘Ÿ๎š๐›ผ=0๐ฝ๐›ผ={1,2,โ€ฆ,๐‘}.(4.1)

We note that for (P)-optimal ๐‘ฅโˆ—, (2.6) implies๎“๐‘—โˆˆ๐ฝ๐›ผ๐œ‡โˆ—๐‘—๐‘”๐‘—๎€ท๐‘ฅโˆ—๎€ธ=0,๐›ผ=0,1,โ€ฆ,๐‘Ÿ.(4.2) We now recast the necessary condition in Lemma 2.2 in the following form.

Lemma 4.1. Let ๐‘ฅโˆ— be an optimal solution for (P) satisfying โŸจ๐‘ฅโˆ—,๐ด๐‘ฅโˆ—โŸฉ>0, โŸจ๐‘ฅโˆ—,๐ต๐‘ฅโˆ—โŸฉ>0 and let โˆ‡๐‘”๐‘—(๐‘ฅโˆ—), ๐‘—โˆˆ๐ฝ(๐‘ฅโˆ—) be linearly independent, then there exist (๐‘ ,๐‘กโˆ—,ฬƒ๐‘ฆ)โˆˆ๐พ(๐‘ฅโˆ—), ๐‘ข,๐‘ฃโˆˆ๐‘…๐‘› and ๐œ‡โˆ—โˆˆ๐‘…๐‘+ such that ๎ƒฉ๐‘ ๎“๐‘–=1๐‘กโˆ—๐‘–๎‚€โ„Ž๎€ท๐‘ฅโˆ—,๐‘ฆ๐‘–๎€ธโˆ’โŸจ๐‘ฅโˆ—,๐ต๐‘ฅโˆ—โŸฉ1/2๎‚๎ƒชโˆ‡๎ƒฉ๐‘ ๎“๐‘–=1๐‘กโˆ—๐‘–๎€ท๐‘“๎€ท๐‘ฅโˆ—,๐‘ฆ๐‘–๎€ธ๎€ธ++๐ด๐‘ข๐‘๎“๐‘—=1๐œ‡โˆ—๐‘—๐‘”๐‘—๎€ท๐‘ฅโˆ—๎€ธ๎ƒชโˆ’โŽ›โŽœโŽœโŽ๐‘ ๎“๐‘–=1๐‘กโˆ—๐‘–๎‚€๐‘“๎€ท๐‘ฅโˆ—,๐‘ฆ๐‘–๎€ธ+โŸจ๐‘ฅโˆ—,๐ด๐‘ฅโˆ—โŸฉ1/2๎‚+๎“๐‘—โˆˆ๐ฝ0๐œ‡โˆ—๐‘—๐‘”๐‘—๎€ท๐‘ฅโˆ—๎€ธโŽžโŽŸโŽŸโŽ โˆ‡๎ƒฉ๐‘ ๎“๐‘–=1๐‘กโˆ—๐‘–๎€ทโ„Ž๎€ท๐‘ฅโˆ—,๐‘ฆ๐‘–๎€ธ๎€ธ๎ƒช๎“โˆ’๐ต๐‘ฃ=0,(4.3)๐‘—โˆˆ๐ฝ๐›ผ๐œ‡โˆ—๐‘—๐‘”๐‘—๎€ท๐‘ฅโˆ—๎€ธ๐œ‡=0,๐›ผ=1,2,โ€ฆ,๐‘Ÿ,(4.4)โˆ—โˆˆ๐‘…๐‘+๐‘กโˆ—๐‘–โ‰ฅ0,๐‘ ๎“๐‘–=1๐‘กโˆ—๐‘–=1,๐‘ฆ๐‘–๎€ท๐‘ฅโˆˆ๐‘Œโˆ—๎€ธ,๐‘–=1,2,โ€ฆ,๐‘ ,(4.5) where ๐ฝ๐›ผ(0โ‰ค๐›ผโ‰ค๐‘Ÿ) is a partition of {1,2,โ€ฆ๐‘}.

Proof. It suffices to establish (4.3). From (2.4) and (2.5), ๎ƒฉโˆ‡๐‘ ๎“๐‘–=1๐‘กโˆ—๐‘–๎€ท๐‘“๎€ท๐‘ฅโˆ—,๐‘ฆ๐‘–๎€ธ๎€ธ๎ƒชโˆ’๐‘“๎€ท๐‘ฅ+๐ด๐‘ขโˆ—,๐‘ฆ๐‘–๎€ธ+โŸจ๐‘ฅโˆ—,๐ด๐‘ฅโˆ—โŸฉ1/2โ„Ž๎€ท๐‘ฅโˆ—,๐‘ฆ๐‘–๎€ธโˆ’โŸจ๐‘ฅโˆ—,๐ต๐‘ฅโˆ—โŸฉ1/2๎ƒฉโˆ‡๐‘ ๎“๐‘–=1๐‘กโˆ—๐‘–๎€ทโ„Ž๎€ท๐‘ฅโˆ—,๐‘ฆ๐‘–๎€ธ๎€ธ๎ƒชโˆ’๐ต๐‘ฃ+โˆ‡๐‘๎“๐‘—=1๐œ‡โˆ—๐‘—๐‘”๐‘—๎€ท๐‘ฅโˆ—๎€ธ=0,๐‘–=1,2,โ€ฆ,๐‘ .(4.6) Multiply the respective equation above by ๐‘กโˆ—๐‘–(โ„Ž(๐‘ฅโˆ—,๐‘ฆ๐‘–)โˆ’โŸจ๐‘ฅโˆ—,๐ต๐‘ฅโˆ—โŸฉ1/2), ๐‘–=1,2,โ€ฆ,๐‘  and add them together, we have ๎ƒฉ๐‘ ๎“๐‘–=1๐‘กโˆ—๐‘–๎‚€โ„Ž๎€ท๐‘ฅโˆ—,๐‘ฆ๐‘–๎€ธโˆ’โŸจ๐‘ฅโˆ—,๐ต๐‘ฅโˆ—โŸฉ1/2๎‚๎ƒชโˆ‡๎ƒฉ๐‘ ๎“๐‘–=1๐‘กโˆ—๐‘–๎€ท๐‘“๎€ท๐‘ฅโˆ—,๐‘ฆ๐‘–๎€ธ๎€ธ++๐ด๐‘ข๐‘๎“๐‘—=1๐œ‡โˆ—๐‘—๐‘”๐‘—๎€ท๐‘ฅโˆ—๎€ธ๎ƒชโˆ’๎ƒฉ๐‘ ๎“๐‘–=1๐‘กโˆ—๐‘–๎‚€๐‘“๎€ท๐‘ฅโˆ—,๐‘ฆ๐‘–๎€ธ+โŸจ๐‘ฅโˆ—,๐ด๐‘ฅโˆ—โŸฉ1/2๎‚๎ƒชโˆ‡๎ƒฉ๐‘ ๎“๐‘–=1๐‘กโˆ—๐‘–๎€ทโ„Ž๎€ท๐‘ฅโˆ—,๐‘ฆ๐‘–๎€ธ๎€ธ๎ƒชโˆ’๐ต๐‘ฃ=0.(4.7) The above equation together with (2.6) implies that ๎ƒฉ๐‘ ๎“๐‘–=1๐‘กโˆ—๐‘–๎‚€โ„Ž๎€ท๐‘ฅโˆ—,๐‘ฆ๐‘–๎€ธโˆ’โŸจ๐‘ฅโˆ—,๐ต๐‘ฅโˆ—โŸฉ1/2๎‚๎ƒชโˆ‡๎ƒฉ๐‘ ๎“๐‘–=1๐‘กโˆ—๐‘–๎€ท๐‘“๎€ท๐‘ฅโˆ—,๐‘ฆ๐‘–๎€ธ๎€ธ++๐ด๐‘ข๐‘๎“๐‘—=1๐œ‡โˆ—๐‘—๐‘”๐‘—๎€ท๐‘ฅโˆ—๎€ธ๎ƒชโˆ’โŽ›โŽœโŽœโŽ๐‘ ๎“๐‘–=1๐‘กโˆ—๐‘–๎‚€๐‘“๎€ท๐‘ฅโˆ—,๐‘ฆ๐‘–๎€ธ+โŸจ๐‘ฅโˆ—,๐ด๐‘ฅโˆ—โŸฉ1/2๎‚+๎“๐‘—โˆˆ๐ฝ0๐œ‡โˆ—๐‘—๐‘”๐‘—๎€ท๐‘ฅโˆ—๎€ธโŽžโŽŸโŽŸโŽ โˆ‡๎ƒฉ๐‘ ๎“๐‘–=1๐‘กโˆ—๐‘–๎€ทโ„Ž๎€ท๐‘ฅโˆ—,๐‘ฆ๐‘–๎€ธ๎€ธ๎ƒชโˆ’๐ต๐‘ฃ=0.(4.8) Hence, the lemma is established.

Our dual model is as follows: max๎€ท๐‘ ,๐‘ก,ฬƒ๐‘ฆ๎€ธโˆˆ๐พ(๐‘ง)sup๎€ท(๐‘ง,๐œ‡,๐‘ข,๐‘ฃ)โˆˆ๐ป๐‘ ,๐‘ก,ฬƒ๐‘ฆ๎€ธ๎ƒฉโˆ‘๐‘ ๐‘–=1๐‘ก๐‘–๐‘“๎€ท๐‘ง,๐‘ฆ๐‘–๎€ธ+โŸจ๐‘ง,๐ด๐‘งโŸฉ1/2+โˆ‘๐‘—โˆˆ๐ฝ0๐œ‡๐‘—๐‘”๐‘—(๐‘ง)โˆ‘๐‘ ๐‘–=1๐‘ก๐‘–๎€ทโ„Ž๎€ท๐‘ง,๐‘ฆ๐‘–๎€ธโˆ’โŸจ๐‘ง,๐ด๐‘งโŸฉ1/2๎€ธ๎ƒช,(D) where ๐ป(๐‘ ,๐‘ก,ฬƒ๐‘ฆ) denotes the set of all (๐‘ง,๐œ‡,๐‘ข,๐‘ฃ)โˆˆ๐‘…๐‘›ร—๐‘…๐‘›+ร—๐‘…๐‘›ร—๐‘…๐‘› satisfying๎ƒฉ๐‘ ๎“๐‘–=1๐‘ก๐‘–๎€ทโ„Ž๎€ท๐‘ง,๐‘ฆ๐‘–๎€ธโˆ’โŸจ๐‘ง,๐ต๐‘งโŸฉ1/2๎€ธ๎ƒชโˆ‡๎ƒฉ๐‘ ๎“๐‘–=1๐‘ก๐‘–๎€ท๐‘“๎€ท๐‘ง,๐‘ฆ๐‘–๎€ธ๎€ธ++๐ด๐‘ข๐‘๎“๐‘—=1๐œ‡๐‘—๐‘”๐‘—๎ƒชโˆ’โŽ›โŽœโŽœโŽ(๐‘ง)๐‘ ๎“๐‘–=1๐‘ก๐‘–๎€ท๐‘“๎€ท๐‘ง,๐‘ฆ๐‘–๎€ธ+โŸจ๐‘ง,๐ด๐‘งโŸฉ1/2๎€ธ+๎“๐‘—โˆˆ๐ฝ0๐œ‡๐‘—๐‘”๐‘—โŽžโŽŸโŽŸโŽ โˆ‡๎ƒฉ(๐‘ง)๐‘ ๎“๐‘–=1๐‘ก๐‘–๎€ทโ„Ž๎€ท๐‘ง,๐‘ฆ๐‘–๎€ธ๎€ธ๎ƒช๎“โˆ’๐ต๐‘ฃ=0,(4.9)๐‘—โˆˆ๐ฝ๐›ผ๐œ‡๐‘—๐‘”๐‘—๐ฝ(๐‘ง)โ‰ฅ0,๐›ผ=1,2,โ€ฆ,๐‘Ÿ,๐›ผโˆฉ๐ฝ๐›ฝ=๐œ™,for๐›ผโ‰ ๐›ฝ,๐‘Ÿ๎š๐›ผ=0๐ฝ๐›ผ={1,2,โ€ฆ,๐‘}.(4.10)

Theorem 4.2 (weak duality). Let ๐‘ฅ be a feasible solution for (P), and let (๐‘ง,๐œ‡,๐‘ข,๐‘ฃ,๐‘ ,๐‘ก,ฬƒ๐‘ฆ) be a feasible solution for (4.18). Suppose that there exist ๐น,๐œƒ,๐œ™0,๐‘0,๐œŒ0 and ๐œ™๐›ผ,๐‘๐›ผ,๐œŒ๐›ผ, ๐›ผ=1,2,โ€ฆ,๐‘Ÿ such that ๐นโŽ›โŽœโŽœโŽ๎ƒฉ๐‘ฅ,๐‘ง;๐‘ ๎“๐‘–=1๐‘ก๐‘–๎€ทโ„Ž๎€ท๐‘ง,๐‘ฆ๐‘–๎€ธโˆ’โŸจ๐‘ง,๐ต๐‘งโŸฉ1/2๎€ธ๎ƒชโˆ‡โŽ›โŽœโŽœโŽ๐‘ ๎“๐‘–=1๐‘ก๐‘–๎€ท๐‘“๎€ท๐‘ง,๐‘ฆ๐‘–๎€ธ๎€ธ+๎“+๐ด๐‘ข๐‘—โˆˆ๐ฝ0๐œ‡๐‘—๐‘”๐‘—โŽžโŽŸโŽŸโŽ โˆ’โŽ›โŽœโŽœโŽ(๐‘ง)๐‘ ๎“๐‘–=1๐‘ก๐‘–๎€ท๐‘“๎€ท๐‘ง,๐‘ฆ๐‘–๎€ธ+โŸจ๐‘ง,๐ด๐‘งโŸฉ1/2๎€ธ+๎“๐‘—โˆˆ๐ฝ0๐œ‡๐‘—๐‘”๐‘—โŽžโŽŸโŽŸโŽ โˆ‡๎ƒฉ(๐‘ง)๐‘ ๎“๐‘–=1๐‘ก๐‘–๎€ทโ„Ž๎€ท๐‘ง,๐‘ฆ๐‘–๎€ธ๎€ธ๎ƒชโŽžโŽŸโŽŸโŽ โˆ’๐ต๐‘ฃโ‰ฅโˆ’๐œŒ0(โ€–๐œƒ๐‘ฅ,๐‘ง)โ€–2โŸน๐‘0(๐‘ฅ,๐‘ง)๐œ™0โŽ›โŽœโŽœโŽ๎ƒฉ๐‘ ๎“๐‘–=1๐‘ก๐‘–๎€ทโ„Ž๎€ท๐‘ง,๐‘ฆ๐‘–๎€ธโˆ’โŸจ๐‘ง,๐ต๐‘งโŸฉ1/2๎€ธ๎ƒชโŽ›โŽœโŽœโŽ๐‘ ๎“๐‘–=1๐‘ก๐‘–๎€ท๐‘“๎€ท๐‘ฅ,๐‘ฆ๐‘–๎€ธ+โŸจ๐‘ฅ,๐ด๐‘ฅโŸฉ1/2๎€ธ+๎“๐‘—โˆˆ๐ฝ0๐œ‡๐‘—๐‘”๐‘—โŽžโŽŸโŽŸโŽ โˆ’โŽ›โŽœโŽœโŽ(๐‘ฅ)๐‘ ๎“๐‘–=1๐‘ก๐‘–๎€ท๐‘“๎€ท๐‘ง,๐‘ฆ๐‘–๎€ธ+โŸจ๐‘ง,๐ด๐‘งโŸฉ1/2๎€ธ+๎“๐‘—โˆˆ๐ฝ0๐œ‡๐‘—๐‘”๐‘—โŽžโŽŸโŽŸโŽ gร—๎ƒฉ(๐‘ง)๐‘ ๎“๐‘–=1๐‘ก๐‘–๎€ทโ„Ž๎€ท๐‘ฅ,๐‘ฆ๐‘–๎€ธโˆ’โŸจ๐‘ฅ,๐ต๐‘ฅโŸฉ1/2๎€ธ๎ƒช๎ƒชโ‰ฅ0,(4.11)โˆ’๐‘๐›ผ(๐‘ฅ,๐‘ง)๐œ™๐›ผโŽ›โŽœโŽœโŽ๎ƒฉ๐‘ ๎“๐‘–=1๐‘ก๐‘–๎€ทโ„Ž๎€ท๐‘ง,๐‘ฆ๐‘–๎€ธโˆ’โŸจ๐‘ง,๐ต๐‘งโŸฉ1/2๎€ธ๎ƒชโŽ›โŽœโŽœโŽ๎“๐‘—โˆˆ๐ฝ๐›ผ๐œ‡๐‘—๐‘”๐‘—โŽžโŽŸโŽŸโŽ โŽžโŽŸโŽŸโŽ โŽ›โŽœโŽœโŽ(๐‘ง)โ‰ค0โŸน๐น๐‘ฅ,๐‘ง;๐‘ ๎“๐‘–=1๐‘ก๐‘–๎€ทโ„Ž๎€ท๐‘ง,๐‘ฆ๐‘–๎€ธโˆ’โŸจ๐‘ง,๐ต๐‘งโŸฉ1/2๎€ธโŽ›โŽœโŽœโŽ๎“๐‘—โˆˆ๐ฝ๐›ผ๐œ‡๐‘—โˆ‡๐‘”๐‘—โŽžโŽŸโŽŸโŽ โŽžโŽŸโŽŸโŽ (๐‘ง)โ‰คโˆ’๐œŒ๐›ผ(โ€–๐œƒ๐‘ฅ,๐‘ง)โ€–2,๐›ผ=1,2,โ€ฆ,๐‘Ÿ.(4.12) Further, assume that ๐‘Žโ‰ฅ0โŸน๐œ™๐›ผ๐œ™(๐‘Ž)โ‰ฅ0,๐›ผ=1,2,โ€ฆ,๐‘Ÿ,(4.13)0(๐‘๐‘Ž)โ‰ฅ0โŸน๐‘Žโ‰ฅ0,(4.14)0(๐‘ฅ,๐‘ง)>0,๐‘๐›ผ๐œŒ(๐‘ฅ,๐‘ง)โ‰ฅ0,๐›ผ=1,2,โ€ฆ,๐‘Ÿ,(4.15)0+๐‘Ÿ๎“๐›ผ=1๐œŒ๐›ผโ‰ฅ0,(4.16) then sup๐‘ฆโˆˆ๐‘Œ๐‘“(๐‘ฅ,๐‘ฆ)+โŸจ๐‘ฅ,๐ด๐‘ฅโŸฉ1/2โ„Ž(๐‘ฅ,๐‘ฆ)โˆ’โŸจ๐‘ฅ,๐ต๐‘ฅโŸฉ1/2โ‰ฅ๎ƒฉโˆ‘๐‘ ๐‘–=1๐‘ก๐‘–๎€ท๐‘“๎€ท๐‘ง,๐‘ฆ๐‘–๎€ธ+โŸจ๐‘ง,๐ด๐‘งโŸฉ1/2๎€ธ+โˆ‘๐‘—โˆˆ๐ฝ0๐œ‡๐‘—๐‘”๐‘—(๐‘ง)โˆ‘๐‘ ๐‘–=1๐‘ก๐‘–๎€ทโ„Ž๎€ท๐‘ง,๐‘ฆ๐‘–๎€ธโˆ’โŸจ๐‘ง,๐ต๐‘งโŸฉ1/2๎€ธ๎ƒช.(4.17)

Proof. Suppose to contrary that sup๐‘ฆโˆˆ๐‘Œ๐‘“(๐‘ฅ,๐‘ฆ)+โŸจ๐‘ฅ,๐ด๐‘ฅโŸฉ1/2โ„Ž(๐‘ฅ,๐‘ฆ)โˆ’โŸจ๐‘ฅ,๐ต๐‘ฅโŸฉ1/2<๎ƒฉโˆ‘๐‘ ๐‘–=1๐‘ก๐‘–๎€ท๐‘“๎€ท๐‘ง,๐‘ฆ๐‘–๎€ธ+โŸจ๐‘ง,๐ด๐‘งโŸฉ1/2๎€ธ+โˆ‘๐‘—โˆˆ๐ฝ0๐œ‡๐‘—๐‘”๐‘—(๐‘ง)โˆ‘๐‘ ๐‘–=1๐‘ก๐‘–๎€ทโ„Ž๎€ท๐‘ง,๐‘ฆ๐‘–๎€ธโˆ’โŸจ๐‘ง,๐ต๐‘งโŸฉ1/2๎€ธ๎ƒช,(4.18) then, we get ๐‘ ๎“๐‘–=1๐‘ก๐‘–๎€ทโ„Ž๎€ท๐‘ง,๐‘ฆ๐‘–๎€ธโˆ’โŸจ๐‘ง,๐ต๐‘งโŸฉ1/2๎€ธ๎€ท๐‘“(๐‘ฅ,๐‘ฆ)+โŸจ๐‘ฅ,๐ด๐‘ฅโŸฉ1/2๎€ธ<โŽ›โŽœโŽœโŽ๐‘ ๎“๐‘–=1๐‘ก๐‘–๎€ท๐‘“๎€ท๐‘ง,๐‘ฆ๐‘–๎€ธ+โŸจ๐‘ง,๐ด๐‘งโŸฉ1/2๎€ธ+๎“๐‘—โˆˆ๐ฝ0๐œ‡๐‘—๐‘”๐‘—โŽžโŽŸโŽŸโŽ ๎€ท(๐‘ง)โ„Ž(๐‘ฅ,๐‘ฆ)โˆ’โŸจ๐‘ฅ,๐ต๐‘ฅโŸฉ1/2๎€ธ,โˆ€๐‘ฆโˆˆ๐‘Œ.(4.19) Further, this implies ๎ƒฉ๐‘ ๎“๐‘–=1๐‘ก๐‘–๎€ทโ„Ž๎€ท๐‘ง,๐‘ฆ๐‘–๎€ธโˆ’โŸจ๐‘ง,๐ต๐‘งโŸฉ1/2๎€ธ๎ƒช๎ƒฉ๐‘ ๎“๐‘–=1๐‘ก๐‘–๎€ท๐‘“๎€ท๐‘ฅ,๐‘ฆ๐‘–๎€ธ+โŸจ๐‘ฅ,๐ด๐‘ฅโŸฉ1/2๎€ธ๎ƒช<โŽ›โŽœโŽœโŽ๐‘ ๎“๐‘–=1๐‘ก๐‘–๎€ท๐‘“๎€ท๐‘ง,๐‘ฆ๐‘–๎€ธ+โŸจ๐‘ง,๐ด๐‘งโŸฉ1/2๎€ธ+๎“๐‘—โˆˆ๐ฝ0๐œ‡๐‘—๐‘”๐‘—โŽžโŽŸโŽŸโŽ ๎ƒฉ(๐‘ง)๐‘ ๎“๐‘–=1๐‘ก๐‘–๎€ทโ„Ž๎€ท๐‘ฅ,๐‘ฆ๐‘–๎€ธโˆ’โŸจ๐‘ฅ,๐ต๐‘ฅโŸฉ1/2๎€ธ๎ƒช.(4.20) Hence, we have ๎ƒฉ๐‘ ๎“๐‘–=1๐‘ก๐‘–๎€ทโ„Ž๎€ท๐‘ง,๐‘ฆ๐‘–๎€ธโˆ’โŸจ๐‘ง,๐ต๐‘งโŸฉ1/2๎€ธ๎ƒชโŽ›โŽœโŽœโŽ๐‘ ๎“๐‘–=1๐‘ก๐‘–๎€ท๐‘“๎€ท๐‘ฅ,๐‘ฆ๐‘–๎€ธ+โŸจ๐‘ฅ,๐ด๐‘ฅโŸฉ1/2๎€ธ+๎“๐‘—โˆˆ๐ฝ0๐œ‡๐‘—๐‘”๐‘—โŽžโŽŸโŽŸโŽ โˆ’โŽ›โŽœโŽœโŽ(๐‘ฅ)๐‘ ๎“๐‘–=1๐‘ก๐‘–๎€ท๐‘“๎€ท๐‘ง,๐‘ฆ๐‘–๎€ธ+โŸจ๐‘ง,๐ด๐‘งโŸฉ1/2๎€ธ+๎“๐‘—โˆˆ๐ฝ0๐œ‡๐‘—๐‘”๐‘—โŽžโŽŸโŽŸโŽ ๎ƒฉ(๐‘ง)๐‘ ๎“๐‘–=1๐‘ก๐‘–๎€ทโ„Ž๎€ท๐‘ฅ,๐‘ฆ๐‘–๎€ธโˆ’โŸจ๐‘ฅ,๐ต๐‘ฅโŸฉ1/2๎€ธ๎ƒช<๎ƒฉ๐‘ ๎“๐‘–=1๐‘ก๐‘–๎€ทโ„Ž๎€ท๐‘ง,๐‘ฆ๐‘–๎€ธโˆ’โŸจ๐‘ง,๐ต๐‘งโŸฉ1/2๎€ธ๎ƒชโŽ›โŽœโŽœโŽ๎“๐‘—โˆˆ๐ฝ0๐œ‡๐‘—๐‘”๐‘—(โŽžโŽŸโŽŸโŽ .๐‘ฅ)(4.21) Using the fact that (โˆ‘๐‘ ๐‘–=1๐‘ก๐‘–(โ„Ž(๐‘ง,๐‘ฆ๐‘–)โˆ’โŸจ๐‘ง,๐ต๐‘งโŸฉ1/2))>0 and โˆ‘๐‘—โˆˆ๐ฝ0๐œ‡๐‘—๐‘”๐‘—(๐‘ฅ)โ‰ค0 and the last inequality, we have ๎ƒฉ๐‘ ๎“๐‘–=1๐‘ก๐‘–๎€ทโ„Ž๎€ท๐‘ง,๐‘ฆ๐‘–๎€ธโˆ’โŸจ๐‘ง,๐ต๐‘งโŸฉ1/2๎€ธ๎ƒชโŽ›โŽœโŽœโŽ๐‘ ๎“๐‘–=1๐‘ก๐‘–๎€ท๐‘“๎€ท๐‘ฅ,๐‘ฆ๐‘–๎€ธ+โŸจ๐‘ฅ,๐ด๐‘ฅโŸฉ1/2๎€ธ+๎“๐‘—โˆˆ๐ฝ0๐œ‡๐‘—๐‘”๐‘—โŽžโŽŸโŽŸโŽ โˆ’โŽ›โŽœโŽœโŽ(๐‘ฅ)๐‘ ๎“๐‘–=1๐‘ก๐‘–๎€ท๐‘“๎€ท๐‘ง,๐‘ฆ๐‘–๎€ธ+โŸจ๐‘ง,๐ด๐‘งโŸฉ1/2๎€ธ+๎“๐‘—โˆˆ๐ฝ0๐œ‡๐‘—๐‘”๐‘—โŽžโŽŸโŽŸโŽ ๎ƒฉ(๐‘ง)๐‘ ๎“๐‘–=1๐‘ก๐‘–๎€ทโ„Ž๎€ท๐‘ฅ,๐‘ฆ๐‘–๎€ธโˆ’โŸจ๐‘ฅ,๐ต๐‘ฅโŸฉ1/2๎€ธ๎ƒช<0.(4.22) From (4.11),(4.14),(4.15), and (4.22), we get ๐นโŽ›โŽœโŽœโŽ๎ƒฉ๐‘ฅ,๐‘ง;๐‘ ๎“๐‘–=1๐‘ก๐‘–๎€ทโ„Ž๎€ท๐‘ง,๐‘ฆ๐‘–๎€ธโˆ’โŸจ๐‘ง,๐ต๐‘งโŸฉ1/2๎€ธ๎ƒชโˆ‡โŽ›โŽœโŽœโŽ๐‘ ๎“๐‘–=1๐‘ก๐‘–๎€ท๐‘“๎€ท๐‘ง,๐‘ฆ๐‘–๎€ธ๎€ธ+๎“+๐ด๐‘ข๐‘—โˆˆ๐ฝ0๐œ‡๐‘—๐‘”๐‘—โŽžโŽŸโŽŸโŽ โˆ’โŽ›โŽœโŽœโŽ(๐‘ง)๐‘ ๎“๐‘–=1๐‘ก๐‘–๎€ท๐‘“๎€ท๐‘ง,๐‘ฆ๐‘–๎€ธ+โŸจ๐‘ง,๐ด๐‘งโŸฉ1/2๎€ธ+๎“๐‘—โˆˆ๐ฝ0๐œ‡๐‘—๐‘”๐‘—โŽžโŽŸโŽŸโŽ โˆ‡๎ƒฉ(๐‘ง)๐‘ ๎“๐‘–=1๐‘ก๐‘–๎€ทโ„Ž๎€ท๐‘ง,๐‘ฆ๐‘–๎€ธ๎€ธ๎ƒชโŽžโŽŸโŽŸโŽ โˆ’๐ต๐‘ฃ<โˆ’๐œŒ0(โ€–๐œƒ๐‘ฅ,๐‘ง)โ€–2.(4.23) Using (โˆ‘๐‘ ๐‘–=1๐‘ก๐‘–(โ„Ž(๐‘ง,๐‘ฆ๐‘–)โˆ’โŸจ๐‘ง,๐ต๐‘งโŸฉ1/2))>0, (4.10), (4.13), and (4.15), we get โˆ’๐‘๐›ผ(๐‘ฅ,๐‘ง)๐œ™๐›ผโŽ›โŽœโŽœโŽ๐‘ ๎“๐‘–=1๐‘ก๐‘–๎€ทโ„Ž๎€ท๐‘ง,๐‘ฆ๐‘–๎€ธโˆ’โŸจ๐‘ง,๐ต๐‘งโŸฉ1/2๎€ธโŽ›โŽœโŽœโŽ๎“๐‘—โˆˆ๐ฝ๐›ผ๐œ‡๐‘—๐‘”๐‘—โŽžโŽŸโŽŸโŽ โŽžโŽŸโŽŸโŽ (๐‘ง)โ‰ค0,๐›ผ=1,2,โ€ฆ,๐‘Ÿ.(4.24) From (4.12), we have ๐นโŽ›โŽœโŽœโŽ๐‘ฅ,๐‘ง;๐‘ ๎“๐‘–=1๐‘ก๐‘–๎€ทโ„Ž๎€ท๐‘ง,๐‘ฆ๐‘–๎€ธโˆ’โŸจ๐‘ง,๐ต๐‘งโŸฉ1/2๎€ธโŽ›โŽœโŽœโŽ๎“๐‘—โˆˆ๐ฝ๐›ผ๐œ‡๐‘—โˆ‡๐‘”๐‘—โŽžโŽŸโŽŸโŽ โŽžโŽŸโŽŸโŽ (๐‘ง)โ‰คโˆ’๐œŒ๐›ผโ€–๐œƒ(๐‘ฅ,๐‘ง)โ€–2,๐›ผ=1,2,โ€ฆ,๐‘Ÿ.(4.25) On adding (4.23) and (4.25) and making use of sublinearity of ๐น and (4.16), we have ๐น๎ƒฉ๎ƒฉ๐‘ฅ,๐‘ง;๐‘ ๎“๐‘–=1๐‘ก๐‘–๎€ทโ„Ž๎€ท๐‘ง,๐‘ฆ๐‘–๎€ธโˆ’โŸจ๐‘ง,๐ต๐‘งโŸฉ1/2๎€ธ๎ƒชโˆ‡๎ƒฉ๐‘ ๎“๐‘–=1๐‘ก๐‘–๎€ท๐‘“๎€ท๐‘ง,๐‘ฆ๐‘–๎€ธ๎€ธ++๐ด๐‘ข๐‘๎“๐‘—=1๐œ‡๐‘—๐‘”๐‘—๎ƒชโˆ’โŽ›โŽœโŽœโŽ(๐‘ง)๐‘ ๎“๐‘–=1๐‘ก๐‘–๎€ท๐‘“๎€ท๐‘ง,๐‘ฆ๐‘–๎€ธ+โŸจ๐‘ง,๐ด๐‘งโŸฉ1/2๎€ธ+๎“๐‘—โˆˆ๐ฝ0๐œ‡๐‘—๐‘”๐‘—โŽžโŽŸโŽŸโŽ โˆ‡๎ƒฉ(๐‘ง)๐‘ ๎“๐‘–=1๐‘ก๐‘–๎€ทโ„Ž๎€ท๐‘ง,๐‘ฆ๐‘–๎€ธ๎€ธ๎ƒชโŽžโŽŸโŽŸโŽ โˆ’๐ต๐‘ฃ<0,(4.26) which contradicts (4.9). This completes the proof.

Theorem 4.3 (weak duality). Let ๐‘ฅ be a feasible solution for (P) and let (๐‘ง,๐œ‡,๐‘ข,๐‘ฃ,๐‘ ,๐‘ก,ฬƒ๐‘ฆ) be a feasible solution for (4.18). Suppose that there exist ๐น,๐œƒ,๐œ™0,๐‘0,๐œŒ0 and ๐œ™๐›ผ,๐‘๐›ผ,๐œŒ๐›ผ, ๐›ผ=1,2,โ€ฆ,๐‘Ÿ such that ๐‘0(๐‘ฅ,๐‘ง)๐œ™0โŽ›โŽœโŽœโŽ๎ƒฉ๐‘ ๎“๐‘–=1๐‘ก๐‘–๎€ทโ„Ž๎€ท๐‘ง,๐‘ฆ๐‘–๎€ธโˆ’โŸจ๐‘ง,๐ต๐‘งโŸฉ1/2๎€ธ๎ƒชโŽ›โŽœโŽœโŽ๐‘ ๎“๐‘–=1๐‘ก๐‘–๎€ท๐‘“๎€ท๐‘ฅ,๐‘ฆ๐‘–๎€ธ+โŸจ๐‘ฅ,๐ด๐‘ฅโŸฉ1/2๎€ธ+๎“๐‘—โˆˆ๐ฝ0๐œ‡๐‘—๐‘”๐‘—โŽžโŽŸโŽŸโŽ โˆ’โŽ›โŽœโŽœโŽ(๐‘ฅ)๐‘ ๎“๐‘–=1๐‘ก๐‘–๎€ท๐‘“๎€ท๐‘ง,๐‘ฆ๐‘–๎€ธ+โŸจ๐‘ง,๐ด๐‘งโŸฉ1/2๎€ธ+๎“๐‘—โˆˆ๐ฝ0๐œ‡๐‘—๐‘”๐‘—โŽžโŽŸโŽŸโŽ ร—๎ƒฉ(๐‘ง)๐‘ ๎“๐‘–=1๐‘ก๐‘–๎€ทโ„Ž๎€ท๐‘ฅ,๐‘ฆ๐‘–๎€ธโˆ’โŸจ๐‘ฅ,๐ต๐‘ฅโŸฉ1/2๎€ธโŽ›โŽœโŽœโŽ๎ƒฉ๎ƒช๎ƒช<0โŸน๐น๐‘ฅ,๐‘ง;๐‘ ๎“๐‘–=1๐‘ก๐‘–๎€ทโ„Ž๎€ท๐‘ง,๐‘ฆ๐‘–๎€ธโˆ’โŸจ๐‘ง,๐ต๐‘งโŸฉ1/2๎€ธ๎ƒชโˆ‡โŽ›โŽœโŽœโŽ๐‘ ๎“๐‘–=1๐‘ก๐‘–๎€ท๐‘“๎€ท๐‘ง,๐‘ฆ๐‘–๎€ธ๎€ธ+๎“+๐ด๐‘ข๐‘—โˆˆ๐ฝ0๐œ‡๐‘—๐‘”๐‘—โŽžโŽŸโŽŸโŽ โˆ’โŽ›โŽœโŽœโŽ(๐‘ง)๐‘ ๎“๐‘–=1๐‘ก๐‘–๎€ท๐‘“๎€ท๐‘ง,๐‘ฆ๐‘–๎€ธ+โŸจ๐‘ง,๐ด๐‘งโŸฉ1/2๎€ธ+๎“๐‘—โˆˆ๐ฝ0๐œ‡๐‘—๐‘”๐‘—โŽžโŽŸโŽŸโŽ โˆ‡๎ƒฉ(๐‘ง)๐‘ ๎“๐‘–=1๐‘ก๐‘–๎€ทโ„Ž๎€ท๐‘ง,๐‘ฆ๐‘–๎€ธ๎€ธ๎ƒชโŽžโŽŸโŽŸโŽ โˆ’๐ต๐‘ฃโ‰คโˆ’๐œŒ0โ€–๐œƒ(๐‘ฅ,๐‘ง)โ€–2,โˆ’๐‘๐›ผ(๐‘ฅ,๐‘ง)๐œ™๐›ผโŽ›โŽœโŽœโŽ๎ƒฉ๐‘ ๎“๐‘–=1๐‘ก๐‘–๎€ทโ„Ž๎€ท๐‘ง,๐‘ฆ๐‘–๎€ธโˆ’โŸจ๐‘ง,๐ต๐‘งโŸฉ1/2๎€ธ๎ƒชโŽ›โŽœโŽœโŽ๎“๐‘—โˆˆ๐ฝ๐›ผ๐œ‡๐‘—๐‘”๐‘—โŽžโŽŸโŽŸโŽ โŽžโŽŸโŽŸโŽ โŽ›โŽœโŽœโŽ(๐‘ง)โ‰ค0โŸน๐น๐‘ฅ,๐‘ง;๐‘ ๎“๐‘–=1๐‘ก๐‘–๎€ทโ„Ž๎€ท๐‘ง,๐‘ฆ๐‘–๎€ธโˆ’โŸจ๐‘ง,๐ต๐‘งโŸฉ1/2๎€ธโŽ›โŽœโŽœโŽ๎“๐‘—โˆˆ๐ฝ๐›ผ๐œ‡๐‘—โˆ‡๐‘”๐‘—โŽžโŽŸโŽŸโŽ โŽžโŽŸโŽŸโŽ (๐‘ง)<โˆ’๐œŒ๐›ผ(โ€–๐œƒ๐‘ฅ,๐‘ง)โ€–2,๐›ผ=1,2,โ€ฆ,๐‘Ÿ.(4.27) Further, assume that (4.14), (4.15), and (4.16) are satisfied, then sup๐‘ฆโˆˆ๐‘Œ๐‘“(๐‘ฅ,๐‘ฆ)+โŸจ๐‘ฅ,๐ด๐‘ฅโŸฉ1/2โ„Ž(๐‘ฅ,๐‘ฆ)โˆ’โŸจ๐‘ฅ,๐ต๐‘ฅโŸฉ1/2โ‰ฅ๎ƒฉโˆ‘๐‘ ๐‘–=1๐‘ก๐‘–๎€ท๐‘“๎€ท๐‘ง,๐‘ฆ๐‘–๎€ธ+โŸจ๐‘ง,๐ด๐‘งโŸฉ1/2๎€ธ+โˆ‘๐‘—โˆˆ๐ฝ0๐œ‡๐‘—๐‘”๐‘—(๐‘ง)โˆ‘๐‘ ๐‘–=1๐‘ก๐‘–๎€ทโ„Ž๎€ท๐‘ง,๐‘ฆ๐‘–๎€ธโˆ’โŸจ๐‘ง,๐ต๐‘งโŸฉ1/2๎€ธ๎ƒช.(4.28)

Proof. The proof is similar to that of the above theorem and hence omitted.

Theorem 4.4 (strong duality). Assume that ๐‘ฅโˆ— is an optimal solution for (P) and โˆ‡๐‘”๐‘—(๐‘ฅโˆ—), ๐‘—โˆˆ๐ฝ(๐‘ฅโˆ—) are linearly independent. Then there exist (๐‘ โˆ—,๐‘กโˆ—,ฬƒ๐‘ฆโˆ—)โˆˆ๐พ(๐‘ฅโˆ—) and (๐‘ฅโˆ—,๐œ‡โˆ—,๐‘ขโˆ—,๐‘ฃโˆ—)โˆˆ๐ป(๐‘ โˆ—,๐‘กโˆ—,ฬƒ๐‘ฆโˆ—)such that (๐‘ฅโˆ—,๐œ‡โˆ—,๐‘ขโˆ—,๐‘ฃโˆ—,๐‘ โˆ—,๐‘กโˆ—,ฬƒ๐‘ฆโˆ—) is an optimal solution for (4.18). If, in addition, the hypotheses of any of the weak duality (Theorem 4.2 or Theorem 4.3) holds for a feasible point (๐‘ง,๐œ‡,๐‘ข,๐‘ฃ,๐‘ ,๐‘ก,ฬƒ๐‘ฆ), then the problems (P) and (4.18) have the same optimal values.

Proof. By Lemma 4.1, there exist (๐‘ โˆ—,๐‘กโˆ—,ฬƒ๐‘ฆโˆ—)โˆˆ๐พ(๐‘ฅโˆ—) and (๐‘ฅโˆ—,๐œ‡โˆ—,๐‘ขโˆ—,๐‘ฃโˆ—)โˆˆ๐ป(๐‘ โˆ—,๐‘กโˆ—,ฬƒ๐‘ฆโˆ—) such that (๐‘ฅโˆ—,๐œ‡โˆ—,๐‘ขโˆ—,๐‘ฃโˆ—,๐‘ โˆ—,๐‘กโˆ—,ฬƒ๐‘ฆโˆ—) is a feasible for (4.18), optimality of this feasible solution for (4.18) follows from Theorems 4.2 or 4.3 accordingly.

Theorem 4.5 (strict converse duality). Let ๐‘ฅโˆ— and (๐‘ง,๐œ‡,๐‘ข,๐‘ฃ,๐‘ ,๐‘ก,ฬƒ๐‘ฆ) be optimal solutions for (P) and (4.18), respectively. Suppose that โˆ‡๐‘”๐‘—(๐‘ฅโˆ—), ๐‘—โˆˆ๐ฝ(๐‘ฅโˆ—) are linearly independent and there exist ๐น,๐œƒ,๐œ™0,๐‘0,๐œŒ0, and ๐œ™๐›ผ,๐‘๐›ผ,๐œŒ๐›ผ,๐›ผ=1,2,โ€ฆ,๐‘Ÿ such that ๐นโŽ›โŽœโŽœโŽ๐‘ฅโˆ—๎ƒฉ,๐‘ง;๐‘ ๎“๐‘–=1๐‘ก๐‘–๎€ทโ„Ž๎€ท๐‘ง,๐‘ฆ๐‘–๎€ธโˆ’โŸจ๐‘ง,๐ต๐‘งโŸฉ1/2๎€ธ๎ƒชโˆ‡โŽ›โŽœโŽœโŽ๐‘ ๎“๐‘–=1๐‘ก๐‘–๎€ท๐‘“๎€ท๐‘ง,๐‘ฆ๐‘–๎€ธ๎€ธ+๎“+๐ด๐‘ข๐‘—โˆˆ๐ฝ0๐œ‡๐‘—๐‘”๐‘—โŽžโŽŸโŽŸโŽ ร—โŽ›โŽœโŽœโŽ(๐‘ง)๐‘ ๎“๐‘–=1๐‘ก๐‘–๎€ท๐‘“๎€ท๐‘ง,๐‘ฆ๐‘–๎€ธ+โŸจ๐‘ง,๐ด๐‘งโŸฉ1/2๎€ธ+๎“๐‘—โˆˆ๐ฝ0๐œ‡๐‘—๐‘”๐‘—โŽžโŽŸโŽŸโŽ โˆ‡๎ƒฉ(๐‘ง)๐‘ ๎“๐‘–=1๐‘ก๐‘–๎€ทโ„Ž๎€ท๐‘ง,๐‘ฆ๐‘–๎€ธ๎€ธ๎ƒชโŽžโŽŸโŽŸโŽ โˆ’๐ต๐‘ฃโ‰ฅโˆ’๐œŒ0โ€–โ€–๐œƒ๎€ท๐‘ฅโˆ—๎€ธโ€–โ€–,๐‘ง2โŸน๐‘0๎€ท๐‘ฅโˆ—๎€ธ๐œ™,๐‘ง0๎ƒฉ๎ƒฉ๐‘ ๎“๐‘–=1๐‘ก๐‘–๎€ทโ„Ž๎€ท๐‘ง,๐‘ฆ๐‘–๎€ธโˆ’โŸจ๐‘ง,๐ต๐‘งโŸฉ1/2๎€ธ๎ƒชร—โŽ›โŽœโŽœโŽ๐‘ ๎“๐‘–=1๐‘ก๐‘–๎‚€๐‘“๎€ท๐‘ฅโˆ—,๐‘ฆ๐‘–๎€ธ+โŸจ๐‘ฅโˆ—,๐ด๐‘ฅโˆ—โŸฉ1/2๎‚+๎“๐‘—โˆˆ๐ฝ0๐œ‡๐‘—๐‘”๐‘—๎€ท๐‘ฅโˆ—๎€ธโŽžโŽŸโŽŸโŽ โˆ’โŽ›โŽœโŽœโŽ๐‘ ๎“๐‘–=1๐‘ก๐‘–๎€ท๐‘“๎€ท๐‘ง,๐‘ฆ๐‘–๎€ธ+โŸจ๐‘ง,๐ด๐‘งโŸฉ1/2๎€ธ+๎“๐‘—โˆˆ๐ฝ0๐œ‡๐‘—๐‘”๐‘—โŽžโŽŸโŽŸโŽ ๎ƒฉ(๐‘ง)๐‘ ๎“๐‘–=1๐‘ก๐‘–๎‚€โ„Ž๎€ท๐‘ฅโˆ—,๐‘ฆ๐‘–๎€ธโˆ’โŸจ๐‘ฅโˆ—,๐ต๐‘ฅโˆ—โŸฉ1/2๎‚๎ƒช๎ƒชโ‰ฅ0,(4.29)โˆ’๐‘๐›ผ๎€ท๐‘ฅโˆ—๎€ธ๐œ™,๐‘ง๐›ผโŽ›โŽœโŽœโŽ๎ƒฉ๐‘ ๎“๐‘–=1๐‘ก๐‘–๎€ทโ„Ž๎€ท๐‘ง,๐‘ฆ๐‘–๎€ธโˆ’โŸจ๐‘ง,๐ต๐‘งโŸฉ1/2๎€ธ๎ƒชโŽ›โŽœโŽœโŽ๎“๐‘—โˆˆ๐ฝ๐›ผ๐œ‡๐‘—๐‘”๐‘—โŽžโŽŸโŽŸโŽ โŽžโŽŸโŽŸโŽ โŽ›โŽœโŽœโŽ๐‘ฅ(๐‘ง)โ‰ค0โŸน๐นโˆ—,๐‘ง;๐‘ ๎“๐‘–=1๐‘ก๐‘–๎€ทโ„Ž๎€ท๐‘ง,๐‘ฆ๐‘–๎€ธโˆ’โŸจ๐‘ง,๐ต๐‘งโŸฉ1/2๎€ธโŽ›โŽœโŽœโŽ๎“๐‘—โˆˆ๐ฝ๐›ผ๐œ‡๐‘—โˆ‡๐‘”๐‘—(โŽžโŽŸโŽŸโŽ โŽžโŽŸโŽŸโŽ ๐‘ง)โ‰คโˆ’๐œŒ๐›ผโ€–โ€–๐œƒ๎€ท๐‘ฅโˆ—๎€ธโ€–โ€–,๐‘ง2,๐›ผ=1,2,โ€ฆ,๐‘Ÿ.(4.30) Further, assume (4.13), (4.15), (4.16), ๐œ™0(๐‘Ž)โ‰ฅ0โŸน๐‘Ž>0,(4.31) then ๐‘ฅโˆ—=๐‘ง, that is, ๐‘ง is an optimal solution for (P).

Proof. Suppose to contrary that ๐‘ฅโˆ—โ‰ ๐‘ง. From the strong duality Theorem 4.4, we know that sup๐‘ฆโˆˆ๐‘Œ๐‘“๎€ท๐‘ฅโˆ—๎€ธ,๐‘ฆ+โŸจ๐‘ฅโˆ—,๐ด๐‘ฅโˆ—โŸฉ1/2โ„Ž(๐‘ฅโˆ—,๐‘ฆ)โˆ’โŸจ๐‘ฅโˆ—,๐ต๐‘ฅโˆ—โŸฉ1/2=๎ƒฉโˆ‘๐‘ ๐‘–=1๐‘ก๐‘–๎€ท๐‘“๎€ท๐‘ง,๐‘ฆ๐‘–๎€ธ+โŸจ๐‘ง,๐ด๐‘งโŸฉ1/2๎€ธ+โˆ‘๐‘—โˆˆ๐ฝ0๐œ‡๐‘—๐‘”๐‘—(๐‘ง)โˆ‘๐‘ ๐‘–=1๐‘ก๐‘–๎€ทโ„Ž๎€ท๐‘ง,๐‘ฆ๐‘–๎€ธโˆ’โŸจ๐‘ง,๐ต๐‘งโŸฉ1/2๎€ธ๎ƒช.(4.32) Then, we get ๎ƒฉ๐‘ ๎“๐‘–=1๐‘ก๐‘–๎€ทโ„Ž๎€ท๐‘ง,๐‘ฆ๐‘–๎€ธโˆ’โŸจ๐‘ง,๐ต๐‘งโŸฉ1/2๎€ธ๎ƒช๎‚€๐‘“๎€ท๐‘ฅโˆ—๎€ธ,๐‘ฆ+โŸจ๐‘ฅโˆ—,๐ด๐‘ฅโˆ—โŸฉ1/2๎‚โ‰คโŽ›โŽœโŽœโŽ๐‘ ๎“๐‘–=1๐‘ก๐‘–๎€ท๐‘“๎€ท๐‘ง,๐‘ฆ๐‘–๎€ธ+โŸจ๐‘ง,๐ด๐‘งโŸฉ1/2๎€ธ+๎“๐‘—โˆˆ๐ฝ0๐œ‡๐‘—๐‘”๐‘—โŽžโŽŸโŽŸโŽ ๎‚€โ„Ž๎€ท๐‘ฅ(๐‘ง)โˆ—๎€ธ,๐‘ฆโˆ’โŸจ๐‘ฅโˆ—,๐ต๐‘ฅโˆ—โŸฉ1/2๎‚,โˆ€๐‘ฆโˆˆ๐‘Œ.(4.33) Further, this implies ๎ƒฉ๐‘ ๎“๐‘–=1๐‘ก๐‘–๎€ทโ„Ž๎€ท๐‘ง,๐‘ฆ๐‘–๎€ธโˆ’โŸจ๐‘ง,๐ต๐‘งโŸฉ1/2๎€ธ๎ƒช๎ƒฉ๐‘ ๎“๐‘–=1๐‘ก๐‘–๎‚€๐‘“๎€ท๐‘ฅโˆ—,๐‘ฆ๐‘–๎€ธ+โŸจ๐‘ฅโˆ—,๐ด๐‘ฅโˆ—โŸฉ1/2๎‚๎ƒชโ‰คโŽ›โŽœโŽœโŽ๐‘ ๎“๐‘–=1๐‘ก๐‘–๎€ท๐‘“๎€ท๐‘ง,๐‘ฆ๐‘–๎€ธ+โŸจ๐‘ง,๐ด๐‘งโŸฉ1/2๎€ธ+๎“๐‘—โˆˆ๐ฝ0๐œ‡๐‘—๐‘”๐‘—โŽžโŽŸโŽŸโŽ ร—๎ƒฉ(๐‘ง)๐‘ ๎“๐‘–=1๐‘ก๐‘–๎‚€โ„Ž๎€ท๐‘ฅโˆ—,๐‘ฆ๐‘–๎€ธโˆ’โŸจ๐‘ฅโˆ—,๐ต๐‘ฅโˆ—โŸฉ1/2๎‚๎ƒช,โˆ€๐‘ฆโˆˆ๐‘Œ.(4.34) Hence, we have ๎ƒฉ๐‘ ๎“๐‘–=1๐‘ก๐‘–๎€ทโ„Ž๎€ท๐‘ง,๐‘ฆ๐‘–๎€ธโˆ’โŸจ๐‘ง,๐ต๐‘งโŸฉ1/2๎€ธ๎ƒชโŽ›โŽœโŽœโŽ๐‘ ๎“๐‘–=1๐‘ก๐‘–๎‚€๐‘“๎€ท๐‘ฅโˆ—,๐‘ฆ๐‘–๎€ธ+โŸจ๐‘ฅโˆ—,๐ด๐‘ฅโˆ—โŸฉ1/2๎‚+๎“๐‘—โˆˆ๐ฝ0๐œ‡๐‘—๐‘”๐‘—๎€ท๐‘ฅโˆ—๎€ธโŽžโŽŸโŽŸโŽ โˆ’โŽ›โŽœโŽœโŽ๐‘ ๎“๐‘–=1๐‘ก๐‘–๎€ท๐‘“๎€ท๐‘ง,๐‘ฆ๐‘–๎€ธ+โŸจ๐‘ง,๐ด๐‘งโŸฉ1/2๎€ธ+๎“๐‘—โˆˆ๐ฝ0๐œ‡๐‘—๐‘”๐‘—โŽžโŽŸโŽŸโŽ ๎ƒฉ(๐‘ง)๐‘ ๎“๐‘–=1๐‘ก๐‘–๎‚€โ„Ž๎€ท๐‘ฅโˆ—,๐‘ฆ๐‘–๎€ธโˆ’โŸจ๐‘ฅโˆ—,๐ต๐‘ฅโˆ—โŸฉ1/2๎‚๎ƒชโ‰ค๎ƒฉ๐‘ ๎“๐‘–=1๐‘ก๐‘–๎€ทโ„Ž๎€ท๐‘ง,๐‘ฆ๐‘–๎€ธโˆ’โŸจ๐‘ง,๐ต๐‘งโŸฉ1/2๎€ธ๎ƒชโŽ›โŽœโŽœโŽ๎“๐‘—โˆˆ๐ฝ0๐œ‡๐‘—๐‘”๐‘—๎€ท๐‘ฅโˆ—๎€ธโŽžโŽŸโŽŸโŽ .(4.35) Using the fact that (โˆ‘๐‘ ๐‘–=1๐‘ก๐‘–(โ„Ž(๐‘ง,๐‘ฆ๐‘–)โˆ’โŸจ๐‘ง,๐ต๐‘งโŸฉ1/2))>0 and โˆ‘๐‘—โˆˆ๐ฝ0๐œ‡๐‘—๐‘”๐‘—(๐‘ฅโˆ—)โ‰ค0 and the last inequality, we have ๎ƒฉ๐‘ ๎“๐‘–=1๐‘ก๐‘–๎€ทโ„Ž๎€ท๐‘ง,๐‘ฆ๐‘–๎€ธโˆ’โŸจ๐‘ง,๐ต๐‘งโŸฉ1/2๎€ธ๎ƒชโŽ›โŽœโŽœโŽ๐‘ ๎“๐‘–=1๐‘ก๐‘–๎‚€๐‘“๎€ท๐‘ฅโˆ—,๐‘ฆ๐‘–๎€ธ+โŸจ๐‘ฅโˆ—,๐ด๐‘ฅโˆ—โŸฉ1/2๎‚+๎“๐‘—โˆˆ๐ฝ0๐œ‡๐‘—๐‘”๐‘—๎€ท๐‘ฅโˆ—๎€ธโŽžโŽŸโŽŸโŽ โˆ’โŽ›โŽœโŽœโŽ๐‘ ๎“๐‘–=1๐‘ก๐‘–๎€ท๐‘“๎€ท๐‘ง,๐‘ฆ๐‘–๎€ธ+โŸจ๐‘ง,๐ด๐‘งโŸฉ1/2๎€ธ+๎“๐‘—โˆˆ๐ฝ0๐œ‡๐‘—๐‘”๐‘—โŽžโŽŸโŽŸโŽ ๎ƒฉ(๐‘ง)๐‘ ๎“๐‘–=1๐‘ก๐‘–๎‚€โ„Ž๎€ท๐‘ฅโˆ—,๐‘ฆ๐‘–๎€ธโˆ’โŸจ๐‘ฅโˆ—,๐ต๐‘ฅโˆ—โŸฉ1/2๎‚๎ƒชโ‰ค0.(4.36) From (4.15), (4.29), (4.31), and (4.36), we get ๐นโŽ›โŽœโŽœโŽ๐‘ฅโˆ—๎ƒฉ,๐‘ง;๐‘ ๎“๐‘–=1๐‘ก๐‘–๎€ทโ„Ž๎€ท๐‘ง,๐‘ฆ๐‘–๎€ธโˆ’โŸจ๐‘ง,๐ต๐‘งโŸฉ1/2๎€ธ๎ƒชโˆ‡โŽ›โŽœโŽœโŽ๐‘ ๎“๐‘–=1๐‘ก๐‘–๎€ท๐‘“๎€ท๐‘ง,๐‘ฆ๐‘–๎€ธ๎€ธ+๎“+๐ด๐‘ข๐‘—โˆˆ๐ฝ0๐œ‡๐‘—๐‘”๐‘—โŽžโŽŸโŽŸโŽ โˆ’โŽ›โŽœโŽœโŽ(๐‘ง)๐‘ ๎“๐‘–=1๐‘ก๐‘–๎€ท๐‘“๎€ท๐‘ง,๐‘ฆ๐‘–๎€ธ+โŸจ๐‘ง,๐ด๐‘งโŸฉ1/2๎€ธ+๎“๐‘—โˆˆ๐ฝ0๐œ‡๐‘—๐‘”๐‘—โŽžโŽŸโŽŸโŽ โˆ‡๎ƒฉ(๐‘ง)๐‘ ๎“๐‘–=1๐‘ก๐‘–๎€ทโ„Ž๎€ท๐‘ง,๐‘ฆ๐‘–๎€ธ๎€ธ๎ƒชโŽžโŽŸโŽŸโŽ โˆ’๐ต๐‘ฃ<โˆ’๐œŒ0โ€–โ€–๐œƒ๎€ท๐‘ฅโˆ—๎€ธโ€–โ€–,๐‘ง2.(4.37) Using (โˆ‘๐‘ ๐‘–=1๐‘ก๐‘–(โ„Ž(๐‘ง,๐‘ฆ๐‘–)โˆ’โŸจ๐‘ง,๐ต๐‘งโŸฉ1/2))>0, (4.10), (4.13), and (4.15), we get โˆ’๐‘๐›ผ๎€ท๐‘ฅโˆ—๎€ธ๐œ™,๐‘ง๐›ผโŽ›โŽœโŽœโŽ๐‘ ๎“๐‘–=1๐‘ก๐‘–๎€ทโ„Ž๎€ท๐‘ง,๐‘ฆ๐‘–๎€ธโˆ’โŸจ๐‘ง,๐ต๐‘งโŸฉ1/2๎€ธโŽ›โŽœโŽœโŽ๎“๐‘—โˆˆ๐ฝ๐›ผ๐œ‡๐‘—๐‘”๐‘—โŽžโŽŸโŽŸโŽ โŽžโŽŸโŽŸโŽ (๐‘ง)โ‰ค0,๐›ผ=1,2,โ€ฆ,๐‘Ÿ.(4.38) From (4.30), we have ๐นโŽ›โŽœโŽœโŽ๐‘ฅโˆ—,๐‘ง;๐‘ ๎“๐‘–=1๐‘ก๐‘–๎€ทโ„Ž๎€ท๐‘ง,๐‘ฆ๐‘–๎€ธโˆ’โŸจ๐‘ง,๐ต๐‘งโŸฉ1/2๎€ธโŽ›โŽœโŽœโŽ๎“๐‘—โˆˆ๐ฝ๐›ผ๐œ‡๐‘—โˆ‡๐‘”๐‘—โŽžโŽŸโŽŸโŽ โŽžโŽŸโŽŸโŽ (๐‘ง)โ‰คโˆ’๐œŒ๐›ผโ€–โ€–๐œƒ๎€ท๐‘ฅโˆ—๎€ธโ€–โ€–,๐‘ง2,๐›ผ=1,2,โ€ฆ,๐‘Ÿ.(4.39) On adding (4.37) and (4.39) and making use of sublinearity of ๐น and (4.16), we have ๐น๎ƒฉ๐‘ฅโˆ—๎ƒฉ