International Scholarly Research Notices

International Scholarly Research Notices / 2011 / Article

Research Article | Open Access

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

Anurag Jayswal, "Optimality and Duality for Nondifferentiable Minimax Fractional Programming with Generalized Convexity", International Scholarly Research Notices, vol. 2011, Article ID 491941, 19 pages, 2011. 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 (𝐹,𝛼,𝜌,𝑑) 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 𝛼-univexity. Yuan et al. [14] introduced the concept of generalized (𝐢,𝛼,𝜌,𝑑)-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 (𝐢,𝛼,𝜌,𝑑)-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 𝑅𝑛 denote the 𝑛-dimensional Euclidean space and let 𝑅𝑛+ be its nonnegative orthant.

In this paper, we consider the following nondifferentiable minimax fractional programming problem:minπ‘₯βˆˆπ‘…π‘›supπ‘¦βˆˆπ‘Œπ‘“(π‘₯,𝑦)+⟨π‘₯,𝐴π‘₯⟩1/2β„Ž(π‘₯,𝑦)βˆ’βŸ¨π‘₯,𝐡π‘₯⟩1/2subjectto𝑔(π‘₯)≀0,(P) where 𝑓,β„ŽβˆΆπ‘…π‘›Γ—π‘…π‘šβ†’π‘… and π‘”βˆΆπ‘…π‘›β†’π‘…π‘ are continuous differentiable functions, π‘Œ is a compact subset of π‘…π‘š, and 𝐴 and 𝐡 are 𝑛×𝑛 positive semidefinite matrices. The problem (P) is nondifferentiable programming problem if either 𝐴 or 𝐡 is nonzero. If 𝐴 and 𝐡 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 ℑ𝑃 = {π‘₯βˆˆπ‘…π‘›βˆΆπ‘”(π‘₯)≀0} be the set of all feasible solutions of (P). For each (π‘₯,𝑦)βˆˆπ‘…π‘›Γ—π‘…π‘š, we defineπœ™(π‘₯,𝑦)=𝑓(π‘₯,𝑦)+⟨π‘₯,𝐴π‘₯⟩1/2β„Ž(π‘₯,𝑦)βˆ’βŸ¨π‘₯,𝐡π‘₯⟩1/2.(2.1) Assume that for each (π‘₯,𝑦)βˆˆπ‘…π‘›Γ—π‘Œ, 𝑓(π‘₯,𝑦)+⟨π‘₯,𝐴π‘₯⟩1/2β‰₯0, and β„Ž(π‘₯,𝑦)βˆ’βŸ¨π‘₯,𝐡π‘₯⟩1/2>0.

Denoteξƒ―π‘Œ(π‘₯)=π‘“ξ€·π‘¦βˆˆπ‘ŒβˆΆπ‘₯,𝑦+⟨π‘₯,𝐴π‘₯⟩1/2β„Žξ€·π‘₯,π‘¦ξ€Έβˆ’βŸ¨π‘₯,𝐡π‘₯⟩1/2=supπ‘§βˆˆπ‘Œπ‘“(π‘₯,𝑧)+⟨π‘₯,𝐴π‘₯⟩1/2β„Ž(π‘₯,𝑧)βˆ’βŸ¨π‘₯,𝐡π‘₯⟩1/2ξƒ°,𝐽={1,2,…,𝑝},𝐽(π‘₯)=π‘—βˆˆπ½βˆΆπ‘”π‘—ξ€Ύ,ξ‚»(π‘₯)=0𝐾(π‘₯)=(𝑠,𝑑,̃𝑦)βˆˆπ‘Γ—π‘…π‘ +Γ—π‘…π‘šπ‘ ξ€·π‘‘βˆΆ1≦𝑠≦𝑛+1,𝑑=1,𝑑2,…,π‘‘π‘ ξ€Έβˆˆπ‘…π‘ +with𝑠𝑖=1𝑑𝑖=1,̃𝑦=𝑦1,𝑦2…,𝑦𝑠,π‘¦π‘–βˆˆξƒ°.π‘Œ(π‘₯),𝑖=1,2,…,𝑠(2.2)

Since 𝑓 and β„Ž are continuously differentiable and π‘Œ is a compact subset of π‘…π‘š, it follows that for each π‘₯βˆ—βˆˆβ„‘π‘ƒ, π‘Œ(π‘₯βˆ—)β‰ πœ™. Thus, for any π‘¦π‘–βˆˆπ‘Œ(π‘₯βˆ—), we have a positive constant π‘£βˆ—=πœ™(π‘₯βˆ—,𝑦𝑖).

Definition 2.1. A functional πΉβˆΆπ‘‹Γ—π‘‹Γ—π‘…π‘›β†’π‘…(whereπ‘‹βŠ†π‘…π‘›) is said to be sublinear in its third argument, if for all (π‘₯,π‘₯0)βˆˆπ‘‹Γ—π‘‹, 𝐹π‘₯,π‘₯0;π‘Ž1+π‘Ž2≀𝐹π‘₯,π‘₯0;π‘Ž1ξ€Έξ€·+𝐹π‘₯,π‘₯0;π‘Ž2ξ€Έ,βˆ€π‘Ž1,π‘Ž2βˆˆπ‘…π‘›,𝐹π‘₯,π‘₯0ξ€Έξ€·;π›Όπ‘Ž=𝛼𝐹π‘₯,π‘₯0ξ€Έ;π‘Ž,βˆ€π›Όβˆˆπ‘…,𝛼β‰₯0,βˆ€π‘Žβˆˆπ‘…π‘›.(2.3) The following result from Lai and Lee [10] is needed in the sequel.

Lemma 2.2. Let π‘₯βˆ— be an optimal solution for (P) satisfying ⟨π‘₯βˆ—,𝐴π‘₯βˆ—βŸ©>0,⟨π‘₯βˆ—,𝐡π‘₯βˆ—βŸ©>0 and let βˆ‡π‘”π‘—(π‘₯βˆ—), π‘—βˆˆπ½(π‘₯βˆ—) be linearly independent, then there exist (𝑠,π‘‘βˆ—,̃𝑦)∈𝐾(π‘₯βˆ—), π‘£βˆ—βˆˆπ‘…+, 𝑒,π‘£βˆˆπ‘…π‘› and πœ‡βˆ—βˆˆπ‘…π‘+ such that 𝑠𝑖=1π‘‘βˆ—π‘–ξ€·ξ€·π‘₯βˆ‡π‘“βˆ—,𝑦𝑖+π΄π‘’βˆ’π‘£βˆ—ξ€·ξ€·π‘₯βˆ‡β„Žβˆ—,𝑦𝑖+βˆ’π΅π‘£ξ€Έξ€Έπ‘ξ“π‘—=1πœ‡βˆ—π‘—βˆ‡π‘”π‘—ξ€·π‘₯βˆ—ξ€Έ=0,(2.4)𝑓π‘₯βˆ—,𝑦𝑖+⟨π‘₯βˆ—,𝐴π‘₯βˆ—βŸ©1/2βˆ’π‘£βˆ—ξ‚€β„Žξ€·π‘₯βˆ—,π‘¦π‘–ξ€Έβˆ’βŸ¨π‘₯βˆ—,𝐡π‘₯βˆ—βŸ©1/2=0,𝑖=1,2,…,𝑠,(2.5)𝑝𝑗=1πœ‡βˆ—π‘—π‘”π‘—ξ€·π‘₯βˆ—ξ€Έ=0,(2.6)π‘‘βˆ—π‘–βˆˆπ‘…π‘ +,𝑠𝑖=1π‘‘βˆ—π‘–=1,𝑦𝑖π‘₯βˆˆπ‘Œβˆ—ξ€Έ,𝑖=1,2,…,𝑠,(2.7)βŸ¨π‘’,π΄π‘’βŸ©β‰€1,βŸ¨π‘£,π΅π‘£βŸ©β‰€1,⟨π‘₯βˆ—,π΄π‘’βŸ©=⟨π‘₯βˆ—,𝐴π‘₯βˆ—βŸ©1/2,⟨π‘₯βˆ—,π΅π‘£βŸ©=⟨π‘₯βˆ—,𝐡π‘₯βˆ—βŸ©1/2.(2.8)

It should be noted that both the matrices 𝐴 and 𝐡 are positive definite at the solution π‘₯βˆ— in the above Lemma. If one of ⟨𝐴π‘₯βˆ—,π‘₯βˆ—βŸ© and ⟨𝐡π‘₯βˆ—,π‘₯βˆ—βŸ© is zero, or both 𝐴 and 𝐡 are singular at π‘₯βˆ—, then for (𝑠,π‘‘βˆ—,̃𝑦)∈𝐾(π‘₯βˆ—), we can take 𝑍̃𝑦(π‘₯βˆ—) defined in Lai and Lee [10] by𝑍̃𝑦π‘₯βˆ—ξ€Έ=ξ€½π‘§βˆˆπ‘…π‘›βˆΆξ«βˆ‡π‘”π‘—ξ€·π‘₯βˆ—ξ€Έξ¬ξ€·π‘₯,𝑧≀0,π‘—βˆˆπ½βˆ—ξ€Έξ€Ύwithanyoneofthefollowing(𝑖)-(𝑖𝑖𝑖)holds(i)⟨𝐴π‘₯βˆ—,π‘₯βˆ—βŸ©>0,⟨𝐡π‘₯βˆ—,π‘₯βˆ—βŸΉξ„”βŸ©=0𝑠𝑖=1π‘‘βˆ—π‘–ξ€·π‘₯βˆ‡π‘“βˆ—,𝑦𝑖+𝐴π‘₯βˆ—βŸ¨π΄π‘₯βˆ—,π‘₯βˆ—βŸ©1/2βˆ’π‘£βˆ—ξ€·π‘₯βˆ‡β„Žβˆ—,𝑦𝑖+𝑣,π‘§ξ‚¬ξ‚€βˆ—2𝐡𝑧,𝑧1/2<0,(ii)⟨𝐴π‘₯βˆ—,π‘₯βˆ—βŸ©=0,⟨𝐡π‘₯βˆ—,π‘₯βˆ—βŸΉξ„”βŸ©>0𝑠𝑖=1π‘‘βˆ—π‘–ξƒ©ξ€·π‘₯βˆ‡π‘“βˆ—,π‘¦π‘–ξ€Έβˆ’π‘£βˆ—ξƒ©ξ€·π‘₯βˆ‡β„Žβˆ—,π‘¦π‘–ξ€Έβˆ’π΅π‘₯βˆ—βŸ¨π΅π‘₯βˆ—,π‘₯βˆ—βŸ©1/2ξ„•ξƒͺξƒͺ,𝑧+βŸ¨π΅π‘§,π‘§βŸ©1/2<0,(iii)⟨𝐴π‘₯βˆ—,π‘₯βˆ—βŸ©=0,⟨𝐡π‘₯βˆ—,π‘₯βˆ—βŸΉξ„”βŸ©=0𝑠𝑖=1π‘‘βˆ—π‘–ξ€·ξ€·π‘₯βˆ‡π‘“βˆ—,π‘¦π‘–ξ€Έβˆ’π‘£βˆ—ξ€·π‘₯βˆ‡β„Žβˆ—,𝑦𝑖+𝑣,π‘§ξ«ξ€·βˆ—π΅ξ€Έξ¬π‘§,𝑧1/2+βŸ¨π΅π‘§,π‘§βŸ©1/2<0.(2.9) If we take the condition 𝑍̃𝑦(π‘₯βˆ—)=π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 πΉβˆΆπ‘‹Γ—π‘‹Γ—π‘…π‘›β†’π‘… be sublinear functional, πœ™0,πœ™1βˆΆπ‘…β†’π‘…, πœƒβˆΆπ‘…π‘›Γ—π‘…π‘›β†’π‘…π‘›, and 𝑏0,𝑏1βˆΆπ‘‹Γ—π‘‹β†’π‘…+. Let 𝜌0, 𝜌1 be real numbers.

Theorem 3.1. Let π‘₯βˆ—βˆˆβ„‘π‘ƒ be a feasible solution for (P), and there exist π‘£βˆ—βˆˆπ‘…+,(𝑠,π‘‘βˆ—,̃𝑦)∈𝐾(π‘₯βˆ—), 𝑒,π‘£βˆˆπ‘…π‘›, and πœ‡βˆ—βˆˆπ‘…π‘+ satisfying (2.4)–(2.8). Suppose that there exist 𝐹,πœƒ,πœ™0,𝑏0,𝜌0 and πœ™1,𝑏1,𝜌1 such that 𝐹π‘₯,π‘₯βˆ—;𝑠𝑖=1π‘‘βˆ—π‘–ξ€·π‘₯ξ€·ξ€·βˆ‡π‘“βˆ—,𝑦𝑖+π΄π‘’βˆ’π‘£βˆ—ξ€·ξ€·π‘₯βˆ‡β„Žβˆ—,𝑦𝑖ξƒͺβˆ’π΅π‘£ξ€Έξ€Έβ‰₯βˆ’πœŒ0β€–β€–πœƒξ€·π‘₯,π‘₯βˆ—ξ€Έβ€–β€–2βŸΉπ‘0ξ€·π‘₯,π‘₯βˆ—ξ€Έπœ™0𝑠𝑖=1π‘‘βˆ—π‘–ξ€·π‘“ξ€·π‘₯,𝑦𝑖+⟨π‘₯,π΄π‘’βŸ©βˆ’π‘£βˆ—ξ€·β„Žξ€·π‘₯,π‘¦π‘–ξ€Έβˆ’βˆ’βŸ¨π‘₯,π΅π‘£βŸ©ξ€Έξ€Έπ‘ ξ“π‘–=1π‘‘βˆ—π‘–ξ€·π‘“ξ€·π‘₯βˆ—,𝑦𝑖+⟨π‘₯βˆ—,π΄π‘’βŸ©βˆ’π‘£βˆ—ξ€·β„Žξ€·π‘₯βˆ—,π‘¦π‘–ξ€Έβˆ’βŸ¨π‘₯βˆ—ξƒ­,π΅π‘£βŸ©ξ€Έξ€Έβ‰₯0,(3.1)βˆ’π‘1ξ€·π‘₯,π‘₯βˆ—ξ€Έπœ™1𝑝𝑗=1πœ‡βˆ—π‘—π‘”π‘—ξ€·π‘₯βˆ—ξ€Έξƒͺ≀0⟹𝐹π‘₯,π‘₯βˆ—;𝑝𝑗=1πœ‡βˆ—π‘—βˆ‡π‘”π‘—ξ€·π‘₯βˆ—ξ€Έξƒͺβ‰€βˆ’πœŒ1β€–β€–πœƒξ€·π‘₯,π‘₯βˆ—ξ€Έβ€–β€–2.(3.2) Further, assume that π‘Žβ‰₯0βŸΉπœ™1πœ™(π‘Ž)β‰₯0,(3.3)0(π‘π‘Ž)β‰₯0βŸΉπ‘Žβ‰₯0,(3.4)0ξ€·π‘₯,π‘₯βˆ—ξ€Έβ‰₯0,𝑏1ξ€·π‘₯,π‘₯βˆ—ξ€ΈπœŒ>0,(3.5)0+𝜌1β‰₯0,(3.6) then π‘₯βˆ— is an optimal solution of (P).

Proof. Suppose to the contrary that π‘₯βˆ— is not an optimal solution of (P), then there exists π‘₯βˆˆβ„‘π‘ƒ such that supπ‘¦βˆˆπ‘Œπ‘“(π‘₯,𝑦)+⟨π‘₯,𝐴π‘₯⟩1/2β„Ž(π‘₯,𝑦)βˆ’βŸ¨π‘₯,𝐡π‘₯⟩1/2<supπ‘¦βˆˆπ‘Œπ‘“ξ€·π‘₯βˆ—ξ€Έ,𝑦+⟨π‘₯βˆ—,𝐴π‘₯βˆ—βŸ©1/2β„Ž(π‘₯βˆ—,𝑦)βˆ’βŸ¨π‘₯βˆ—,𝐡π‘₯βˆ—βŸ©1/2.(3.7) We note that supπ‘¦βˆˆπ‘Œπ‘“ξ€·π‘₯βˆ—ξ€Έ,𝑦+⟨π‘₯βˆ—,𝐴π‘₯βˆ—βŸ©1/2β„Ž(π‘₯βˆ—,𝑦)βˆ’βŸ¨π‘₯βˆ—,𝐡π‘₯βˆ—βŸ©1/2=𝑓π‘₯βˆ—,𝑦𝑖+⟨π‘₯βˆ—,𝐴π‘₯βˆ—βŸ©1/2β„Žξ€·π‘₯βˆ—,π‘¦π‘–ξ€Έβˆ’βŸ¨π‘₯βˆ—,𝐡π‘₯βˆ—βŸ©1/2=π‘£βˆ—,(3.8) for π‘¦π‘–βˆˆπ‘Œ(π‘₯βˆ—), 𝑖=1,2,…,𝑠, 𝑓π‘₯,𝑦𝑖+⟨π‘₯,𝐴π‘₯⟩1/2β„Žξ€·π‘₯,π‘¦π‘–ξ€Έβˆ’βŸ¨π‘₯,𝐡π‘₯⟩1/2≀supπ‘¦βˆˆπ‘Œπ‘“(π‘₯,𝑦)+⟨π‘₯,𝐴π‘₯⟩1/2β„Ž(π‘₯,𝑦)βˆ’βŸ¨π‘₯,𝐡π‘₯⟩1/2.(3.9) Thus, we have 𝑓π‘₯,𝑦𝑖+⟨π‘₯,𝐴π‘₯⟩1/2β„Žξ€·π‘₯,π‘¦π‘–ξ€Έβˆ’βŸ¨π‘₯,𝐡π‘₯⟩1/2<π‘£βˆ—for𝑖=1,2,…,𝑠.(3.10) It follows that 𝑓π‘₯,𝑦𝑖+⟨π‘₯,𝐴π‘₯⟩1/2βˆ’π‘£βˆ—ξ€·β„Žξ€·π‘₯,π‘¦π‘–ξ€Έβˆ’βŸ¨π‘₯,𝐡π‘₯⟩1/2ξ€Έ<0,for𝑖=1,2,…,𝑠.(3.11) From (2.5), (2.7), (2.8), and (3.11), we get 𝑠𝑖=1π‘‘βˆ—π‘–ξ€·π‘“ξ€·π‘₯,𝑦𝑖+⟨π‘₯,π΄π‘’βŸ©βˆ’π‘£βˆ—ξ€·β„Žξ€·π‘₯,𝑦𝑖<βˆ’βŸ¨π‘₯,π΅π‘£βŸ©ξ€Έξ€Έπ‘ ξ“π‘–=1π‘‘βˆ—π‘–ξ€·π‘“ξ€·π‘₯βˆ—,𝑦𝑖+⟨π‘₯βˆ—,π΄π‘’βŸ©βˆ’π‘£βˆ—ξ€·β„Žξ€·π‘₯βˆ—,π‘¦π‘–ξ€Έβˆ’βŸ¨π‘₯βˆ—.,π΅π‘£βŸ©ξ€Έξ€Έ(3.12) On the other hand, from (2.6), (3.3), and (3.5), we have βˆ’π‘1ξ€·π‘₯,π‘₯βˆ—ξ€Έπœ™1𝑝𝑗=1πœ‡βˆ—π‘—π‘”π‘—ξ€·π‘₯βˆ—ξ€Έξƒͺ≀0.(3.13) It follows from (3.2) that 𝐹π‘₯,π‘₯βˆ—;𝑝𝑗=1πœ‡βˆ—π‘—βˆ‡π‘”π‘—ξ€·π‘₯βˆ—ξ€Έξƒͺβ‰€βˆ’πœŒ1β€–β€–πœƒξ€·π‘₯,π‘₯βˆ—ξ€Έβ€–β€–2.(3.14) From (2.4), the sublinearity of 𝐹, and (3.6), we get 𝐹π‘₯,π‘₯βˆ—;𝑠𝑖=1π‘‘βˆ—π‘–ξ€·ξ€·π‘₯βˆ‡π‘“βˆ—,𝑦𝑖+π΄π‘’βˆ’π‘£βˆ—ξ€·ξ€·π‘₯βˆ‡β„Žβˆ—,𝑦𝑖ξƒͺβˆ’π΅π‘£ξ€Έξ€Έβ‰₯βˆ’πœŒ0β€–β€–πœƒξ€·π‘₯,π‘₯βˆ—ξ€Έβ€–β€–2.(3.15) Then by (3.1), we have 𝑏0ξ€·π‘₯,π‘₯βˆ—ξ€Έπœ™0𝑠𝑖=1π‘‘βˆ—π‘–ξ€·π‘“ξ€·π‘₯,𝑦𝑖+⟨π‘₯,π΄π‘’βŸ©βˆ’π‘£βˆ—ξ€·β„Žξ€·π‘₯,π‘¦π‘–ξ€Έβˆ’βˆ’βŸ¨π‘₯,π΅π‘£βŸ©ξ€Έξ€Έπ‘ ξ“π‘–=1π‘‘βˆ—π‘–ξ€·π‘“ξ€·π‘₯βˆ—,𝑦𝑖+⟨π‘₯βˆ—,π΄π‘’βŸ©βˆ’π‘£βˆ—ξ€·β„Žξ€·π‘₯βˆ—,π‘¦π‘–ξ€Έβˆ’βŸ¨π‘₯βˆ—ξƒ­,π΅π‘£βŸ©ξ€Έξ€Έβ‰₯0.(3.16) From (3.4), (3.5), and the above inequality, we obtain 𝑠𝑖=1π‘‘βˆ—π‘–ξ€·π‘“ξ€·π‘₯,𝑦𝑖+⟨π‘₯,π΄π‘’βŸ©βˆ’π‘£βˆ—ξ€·β„Žξ€·π‘₯,π‘¦π‘–ξ€Έβˆ’βˆ’βŸ¨π‘₯,π΅π‘£βŸ©ξ€Έξ€Έπ‘ ξ“π‘–=1π‘‘βˆ—π‘–ξ€·π‘“ξ€·π‘₯βˆ—,𝑦𝑖+⟨π‘₯βˆ—,π΄π‘’βŸ©βˆ’π‘£βˆ—ξ€·β„Žξ€·π‘₯βˆ—,π‘¦π‘–ξ€Έβˆ’βŸ¨π‘₯βˆ—,π΅π‘£βŸ©ξ€Έξ€Έβ‰₯0,(3.17) which contradicts (3.12). Therefore, π‘₯βˆ— is an optimal solution for (P). This completes the proof.

Remark 3.2. If both 𝐴 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 π‘₯βˆ—βˆˆβ„‘π‘ƒ be a feasible solution for (P), and there exist π‘£βˆ—βˆˆπ‘…+,(𝑠,π‘‘βˆ—,̃𝑦)∈𝐾(π‘₯βˆ—), 𝑒,π‘£βˆˆπ‘…π‘›, and πœ‡βˆ—βˆˆπ‘…π‘+ satisfying (2.4)–(2.8). Suppose that there exist 𝐹,πœƒ,πœ™0,𝑏0,𝜌0 and πœ™1,𝑏1,𝜌1 such that 𝐹π‘₯,π‘₯βˆ—;𝑠𝑖=1π‘‘βˆ—π‘–ξ€·ξ€·π‘₯βˆ‡π‘“βˆ—,𝑦𝑖+π΄π‘’βˆ’π‘£βˆ—ξ€·ξ€·π‘₯βˆ‡β„Žβˆ—,𝑦𝑖ξƒͺβˆ’π΅π‘£ξ€Έξ€Έβ‰₯βˆ’πœŒ0β€–β€–πœƒξ€·π‘₯,π‘₯βˆ—ξ€Έβ€–β€–2βŸΉπ‘0ξ€·π‘₯,π‘₯βˆ—ξ€Έπœ™0𝑠𝑖=1π‘‘βˆ—π‘–ξ€·π‘“ξ€·π‘₯,𝑦𝑖+⟨π‘₯,π΄π‘’βŸ©βˆ’π‘£βˆ—ξ€·β„Žξ€·π‘₯,π‘¦π‘–ξ€Έβˆ’βˆ’βŸ¨π‘₯,π΅π‘£βŸ©ξ€Έξ€Έπ‘ ξ“π‘–=1π‘‘βˆ—π‘–ξ€·π‘“ξ€·π‘₯βˆ—,𝑦𝑖+⟨π‘₯βˆ—,π΄π‘’βŸ©βˆ’π‘£βˆ—ξ€·β„Žξ€·π‘₯βˆ—,π‘¦π‘–ξ€Έβˆ’βŸ¨π‘₯βˆ—ξƒ­,π΅π‘£βŸ©ξ€Έξ€Έ>0,(3.18) or equivalently, 𝑏0ξ€·π‘₯,π‘₯βˆ—ξ€Έπœ™0𝑠𝑖=1π‘‘βˆ—π‘–ξ€·π‘“ξ€·π‘₯,𝑦𝑖+⟨π‘₯,π΄π‘’βŸ©βˆ’π‘£βˆ—ξ€·β„Žξ€·π‘₯,π‘¦π‘–ξ€Έβˆ’βˆ’βŸ¨π‘₯,π΅π‘£βŸ©ξ€Έξ€Έπ‘ ξ“π‘–=1π‘‘βˆ—π‘–ξ€·π‘“ξ€·π‘₯βˆ—,𝑦𝑖+⟨π‘₯βˆ—,π΄π‘’βŸ©βˆ’π‘£βˆ—ξ€·β„Žξ€·π‘₯βˆ—,π‘¦π‘–ξ€Έβˆ’βŸ¨π‘₯βˆ—ξƒ­ξƒ©,π΅π‘£βŸ©ξ€Έξ€Έβ‰€0⟹𝐹π‘₯,π‘₯βˆ—;𝑠𝑖=1π‘‘βˆ—π‘–ξ€·ξ€·π‘₯βˆ‡π‘“βˆ—,𝑦𝑖+π΄π‘’βˆ’π‘£βˆ—ξ€·ξ€·π‘₯βˆ‡β„Žβˆ—,𝑦𝑖ξƒͺβˆ’π΅π‘£ξ€Έξ€Έ<βˆ’πœŒ0β€–β€–πœƒξ€·π‘₯,π‘₯βˆ—ξ€Έβ€–β€–2,(3.19)βˆ’π‘1ξ€·π‘₯,π‘₯βˆ—ξ€Έπœ™1𝑝𝑗=1πœ‡βˆ—π‘—π‘”π‘—ξ€·π‘₯βˆ—ξ€Έξƒͺ≀0⟹𝐹π‘₯,π‘₯βˆ—;𝑝𝑗=1πœ‡βˆ—π‘—βˆ‡π‘”π‘—ξ€·π‘₯βˆ—ξ€Έξƒͺβ‰€βˆ’πœŒ1β€–β€–πœƒξ€·π‘₯,π‘₯βˆ—ξ€Έβ€–β€–2.(3.20) Further, assume that (3.3), (3.5), (3.6), and π‘Žβ‰€0βŸΉπœ™0(π‘Ž)≀0,(3.21) are satisfied, then π‘₯βˆ— is an optimal solution of (P).

Proof. Suppose to the contrary that π‘₯βˆ— is not an optimal solution of (P). Following the proof of Theorem 3.1, we get 𝑠𝑖=1π‘‘βˆ—π‘–ξ€·π‘“ξ€·π‘₯,𝑦𝑖+⟨π‘₯,π΄π‘’βŸ©βˆ’π‘£βˆ—ξ€·β„Žξ€·π‘₯,𝑦𝑖<βˆ’βŸ¨π‘₯,π΅π‘£βŸ©ξ€Έξ€Έπ‘ ξ“π‘–=1π‘‘βˆ—π‘–ξ€·π‘“ξ€·π‘₯βˆ—,𝑦𝑖+⟨π‘₯βˆ—,π΄π‘’βŸ©βˆ’π‘£βˆ—ξ€·β„Žξ€·π‘₯βˆ—,π‘¦π‘–ξ€Έβˆ’βŸ¨π‘₯βˆ—.,π΅π‘£βŸ©ξ€Έξ€Έ(3.22) Using (3.5), (3.19), (3.21), and (3.22), we have 𝐹π‘₯,π‘₯βˆ—;𝑠𝑖=1π‘‘βˆ—π‘–ξ€·ξ€·π‘₯βˆ‡π‘“βˆ—,𝑦𝑖+π΄π‘’βˆ’π‘£βˆ—ξ€·ξ€·π‘₯βˆ‡β„Žβˆ—,𝑦𝑖ξƒͺβˆ’π΅π‘£ξ€Έξ€Έ<βˆ’πœŒ0β€–β€–πœƒξ€·π‘₯,π‘₯βˆ—ξ€Έβ€–β€–2.(3.23) By (2.6), (3.3), (3.5), and (3.20), it follows that 𝐹π‘₯,π‘₯βˆ—;𝑝𝑗=1πœ‡βˆ—π‘—βˆ‡π‘”π‘—ξ€·π‘₯βˆ—ξ€Έξƒͺβ‰€βˆ’πœŒ1β€–β€–πœƒξ€·π‘₯,π‘₯βˆ—ξ€Έβ€–β€–2.(3.24) On adding (3.23) and (3.24), and making use of the sublinearity of 𝐹 and (3.6), we have 𝐹π‘₯,π‘₯βˆ—;𝑠𝑖=1π‘‘βˆ—π‘–ξ€·ξ€·π‘₯βˆ‡π‘“βˆ—,𝑦𝑖+π΄π‘’βˆ’π‘£βˆ—ξ€·ξ€·π‘₯βˆ‡β„Žβˆ—,𝑦𝑖ξƒͺξƒ©βˆ’π΅π‘£ξ€Έξ€Έ+𝐹π‘₯,π‘₯βˆ—;𝑝𝑗=1πœ‡βˆ—π‘—βˆ‡π‘”π‘—ξ€·π‘₯βˆ—ξ€Έξƒͺξ€·πœŒβ‰€βˆ’0+𝜌1ξ€Έβ€–β€–πœƒξ€·π‘₯,π‘₯βˆ—ξ€Έβ€–β€–2<0⟹𝐹π‘₯,π‘₯βˆ—;𝑠𝑖=1π‘‘βˆ—π‘–ξƒ©ξ€·π‘₯βˆ‡π‘“βˆ—,𝑦𝑖+π΄π‘’βˆ’π‘£βˆ—ξ€·ξ€·π‘₯βˆ‡β„Žβˆ—,𝑦𝑖+βˆ’π΅π‘£π‘ξ“π‘—=1πœ‡βˆ—π‘—βˆ‡π‘”π‘—ξ€·π‘₯βˆ—ξ€Έξƒͺξƒͺ<0.(3.25) On the other hand, (2.4) implies 𝐹π‘₯,π‘₯βˆ—;𝑠𝑖=1π‘‘βˆ—π‘–ξƒ©ξ€·π‘₯βˆ‡π‘“βˆ—,𝑦𝑖+π΄π‘’βˆ’π‘£βˆ—ξ€·ξ€·π‘₯βˆ‡β„Žβˆ—,𝑦𝑖+βˆ’π΅π‘£π‘ξ“π‘—=1πœ‡βˆ—π‘—βˆ‡π‘”π‘—ξ€·π‘₯βˆ—ξ€Έξƒͺξƒͺ=0.(3.26) Hence we have a contradiction to inequality (3.25). Therefore, π‘₯βˆ— is an optimal solution for (P). This completes the proof.

Theorem 3.4. Let π‘₯βˆ—βˆˆβ„‘π‘ƒ be a feasible solution for (P) and there existπ‘£βˆ—βˆˆπ‘…+,(𝑠,π‘‘βˆ—,̃𝑦)∈𝐾(π‘₯βˆ—), 𝑒,π‘£βˆˆπ‘…π‘›, and πœ‡βˆ—βˆˆπ‘…π‘+ satisfying (2.4)–(2.8). Suppose that there exist 𝐹,πœƒ,πœ™0,𝑏0,𝜌0 and πœ™1,𝑏1,𝜌1 such that 𝑏0ξ€·π‘₯,π‘₯βˆ—ξ€Έπœ™0𝑠𝑖=1π‘‘βˆ—π‘–ξ€·π‘“ξ€·π‘₯,𝑦𝑖+⟨π‘₯,π΄π‘’βŸ©βˆ’π‘£βˆ—ξ€·β„Žξ€·π‘₯,π‘¦π‘–ξ€Έβˆ’βˆ’βŸ¨π‘₯,π΅π‘£βŸ©ξ€Έξ€Έπ‘ ξ“π‘–=1π‘‘βˆ—π‘–ξ€·π‘“ξ€·π‘₯βˆ—,𝑦𝑖+⟨π‘₯βˆ—,π΄π‘’βŸ©βˆ’π‘£βˆ—ξ€·β„Žξ€·π‘₯βˆ—,π‘¦π‘–ξ€Έβˆ’βŸ¨π‘₯βˆ—ξƒ­ξƒ©,π΅π‘£βŸ©ξ€Έξ€Έβ‰€0⟹𝐹π‘₯,π‘₯βˆ—;𝑠𝑖=1π‘‘βˆ—π‘–ξ€·ξ€·π‘₯βˆ‡π‘“βˆ—,𝑦𝑖+π΄π‘’βˆ’π‘£βˆ—ξ€·ξ€·π‘₯βˆ‡β„Žβˆ—,𝑦𝑖ξƒͺβˆ’π΅π‘£ξ€Έξ€Έβ‰€βˆ’πœŒ0β€–β€–πœƒξ€·π‘₯,π‘₯βˆ—ξ€Έβ€–β€–2,𝐹π‘₯,π‘₯βˆ—;𝑝𝑗=1πœ‡βˆ—π‘—βˆ‡π‘”π‘—ξ€·π‘₯βˆ—ξ€Έξƒͺβ‰₯βˆ’πœŒ1β€–β€–πœƒξ€·π‘₯,π‘₯βˆ—ξ€Έβ€–β€–2βŸΉβˆ’π‘1ξ€·π‘₯,π‘₯βˆ—ξ€Έπœ™1𝑝𝑗=1πœ‡βˆ—π‘—π‘”π‘—ξ€·π‘₯βˆ—ξ€Έξƒͺ>0.(3.27) Further, assume that (3.3), (3.5), (3.6), and (3.21) are satisfied, then π‘₯βˆ— 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 𝐴 and 𝐡 are zero matrices, then Theorems 3.3 and 3.4 above reduce to Theorems 3.3 given in Yang and Hou [8].
(ii) If 𝐹(π‘₯,𝑒;π‘Ž)=βŸ¨πœ‚(π‘₯,𝑒),π‘ŽβŸ© where πœ‚ is a function from 𝑋×𝑋→𝑅𝑛, βˆ’πœ™1(βˆ‘π‘π‘—=1πœ‡βˆ—π‘—π‘”π‘—(π‘₯βˆ—))=πœ™1(βˆ‘π‘π‘—=1πœ‡βˆ—π‘—π‘”π‘—(π‘₯)βˆ’πœ‡βˆ—π‘—π‘”π‘—(π‘₯βˆ—)), and 𝜌0=𝜌1=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,…,𝑝} into several parts. Let 𝐽𝛼(0β‰€π›Όβ‰€π‘Ÿ) be a partition of {1,2,…𝑝}, that is, π½π›Όβˆ©π½π›½=πœ™,for𝛼≠𝛽,π‘Ÿξšπ›Ό=0𝐽𝛼={1,2,…,𝑝}.(4.1)

We note that for (P)-optimal π‘₯βˆ—, (2.6) impliesξ“π‘—βˆˆπ½π›Όπœ‡βˆ—π‘—π‘”π‘—ξ€·π‘₯βˆ—ξ€Έ=0,𝛼=0,1,…,π‘Ÿ.(4.2) We now recast the necessary condition in Lemma 2.2 in the following form.

Lemma 4.1. Let π‘₯βˆ— be an optimal solution for (P) satisfying ⟨π‘₯βˆ—,𝐴π‘₯βˆ—βŸ©>0, ⟨π‘₯βˆ—,𝐡π‘₯βˆ—βŸ©>0 and let βˆ‡π‘”π‘—(π‘₯βˆ—), π‘—βˆˆπ½(π‘₯βˆ—) be linearly independent, then there exist (𝑠,π‘‘βˆ—,̃𝑦)∈𝐾(π‘₯βˆ—), 𝑒,π‘£βˆˆπ‘…π‘› and πœ‡βˆ—βˆˆπ‘…π‘+ such that 𝑠𝑖=1π‘‘βˆ—π‘–ξ‚€β„Žξ€·π‘₯βˆ—,π‘¦π‘–ξ€Έβˆ’βŸ¨π‘₯βˆ—,𝐡π‘₯βˆ—βŸ©1/2ξƒͺβˆ‡ξƒ©π‘ ξ“π‘–=1π‘‘βˆ—π‘–ξ€·π‘“ξ€·π‘₯βˆ—,𝑦𝑖++𝐴𝑒𝑝𝑗=1πœ‡βˆ—π‘—π‘”π‘—ξ€·π‘₯βˆ—ξ€Έξƒͺβˆ’βŽ›βŽœβŽœβŽπ‘ ξ“π‘–=1π‘‘βˆ—π‘–ξ‚€π‘“ξ€·π‘₯βˆ—,𝑦𝑖+⟨π‘₯βˆ—,𝐴π‘₯βˆ—βŸ©1/2+ξ“π‘—βˆˆπ½0πœ‡βˆ—π‘—π‘”π‘—ξ€·π‘₯βˆ—ξ€ΈβŽžβŽŸβŽŸβŽ βˆ‡ξƒ©π‘ ξ“π‘–=1π‘‘βˆ—π‘–ξ€·β„Žξ€·π‘₯βˆ—,𝑦𝑖ξƒͺξ“βˆ’π΅π‘£=0,(4.3)π‘—βˆˆπ½π›Όπœ‡βˆ—π‘—π‘”π‘—ξ€·π‘₯βˆ—ξ€Έπœ‡=0,𝛼=1,2,…,π‘Ÿ,(4.4)βˆ—βˆˆπ‘…π‘+π‘‘βˆ—π‘–β‰₯0,𝑠𝑖=1π‘‘βˆ—π‘–=1,𝑦𝑖π‘₯βˆˆπ‘Œβˆ—ξ€Έ,𝑖=1,2,…,𝑠,(4.5) where 𝐽𝛼(0β‰€π›Όβ‰€π‘Ÿ) is a partition of {1,2,…𝑝}.

Proof. It suffices to establish (4.3). From (2.4) and (2.5), ξƒ©βˆ‡π‘ ξ“π‘–=1π‘‘βˆ—π‘–ξ€·π‘“ξ€·π‘₯βˆ—,𝑦𝑖ξƒͺβˆ’π‘“ξ€·π‘₯+π΄π‘’βˆ—,𝑦𝑖+⟨π‘₯βˆ—,𝐴π‘₯βˆ—βŸ©1/2β„Žξ€·π‘₯βˆ—,π‘¦π‘–ξ€Έβˆ’βŸ¨π‘₯βˆ—,𝐡π‘₯βˆ—βŸ©1/2ξƒ©βˆ‡π‘ ξ“π‘–=1π‘‘βˆ—π‘–ξ€·β„Žξ€·π‘₯βˆ—,𝑦𝑖ξƒͺβˆ’π΅π‘£+βˆ‡π‘ξ“π‘—=1πœ‡βˆ—π‘—π‘”π‘—ξ€·π‘₯βˆ—ξ€Έ=0,𝑖=1,2,…,𝑠.(4.6) Multiply the respective equation above by π‘‘βˆ—π‘–(β„Ž(π‘₯βˆ—,𝑦𝑖)βˆ’βŸ¨π‘₯βˆ—,𝐡π‘₯βˆ—βŸ©1/2), 𝑖=1,2,…,𝑠 and add them together, we have 𝑠𝑖=1π‘‘βˆ—π‘–ξ‚€β„Žξ€·π‘₯βˆ—,π‘¦π‘–ξ€Έβˆ’βŸ¨π‘₯βˆ—,𝐡π‘₯βˆ—βŸ©1/2ξƒͺβˆ‡ξƒ©π‘ ξ“π‘–=1π‘‘βˆ—π‘–ξ€·π‘“ξ€·π‘₯βˆ—,𝑦𝑖++𝐴𝑒𝑝𝑗=1πœ‡βˆ—π‘—π‘”π‘—ξ€·π‘₯βˆ—ξ€Έξƒͺβˆ’ξƒ©π‘ ξ“π‘–=1π‘‘βˆ—π‘–ξ‚€π‘“ξ€·π‘₯βˆ—,𝑦𝑖+⟨π‘₯βˆ—,𝐴π‘₯βˆ—βŸ©1/2ξƒͺβˆ‡ξƒ©π‘ ξ“π‘–=1π‘‘βˆ—π‘–ξ€·β„Žξ€·π‘₯βˆ—,𝑦𝑖ξƒͺβˆ’π΅π‘£=0.(4.7) The above equation together with (2.6) implies that 𝑠𝑖=1π‘‘βˆ—π‘–ξ‚€β„Žξ€·π‘₯βˆ—,π‘¦π‘–ξ€Έβˆ’βŸ¨π‘₯βˆ—,𝐡π‘₯βˆ—βŸ©1/2ξƒͺβˆ‡ξƒ©π‘ ξ“π‘–=1π‘‘βˆ—π‘–ξ€·π‘“ξ€·π‘₯βˆ—,𝑦𝑖++𝐴𝑒𝑝𝑗=1πœ‡βˆ—π‘—π‘”π‘—ξ€·π‘₯βˆ—ξ€Έξƒͺβˆ’βŽ›βŽœβŽœβŽπ‘ ξ“π‘–=1π‘‘βˆ—π‘–ξ‚€π‘“ξ€·π‘₯βˆ—,𝑦𝑖+⟨π‘₯βˆ—,𝐴π‘₯βˆ—βŸ©1/2+ξ“π‘—βˆˆπ½0πœ‡βˆ—π‘—π‘”π‘—ξ€·π‘₯βˆ—ξ€ΈβŽžβŽŸβŽŸβŽ βˆ‡ξƒ©π‘ ξ“π‘–=1π‘‘βˆ—π‘–ξ€·β„Žξ€·π‘₯βˆ—,𝑦𝑖ξƒͺβˆ’π΅π‘£=0.(4.8) Hence, the lemma is established.

Our dual model is as follows: max𝑠,𝑑,Μƒπ‘¦ξ€ΈβˆˆπΎ(𝑧)supξ€·(𝑧,πœ‡,𝑒,𝑣)βˆˆπ»π‘ ,𝑑,Μƒπ‘¦ξ€Έξƒ©βˆ‘π‘ π‘–=1𝑑𝑖𝑓𝑧,𝑦𝑖+βŸ¨π‘§,π΄π‘§βŸ©1/2+βˆ‘π‘—βˆˆπ½0πœ‡π‘—π‘”π‘—(𝑧)βˆ‘π‘ π‘–=1π‘‘π‘–ξ€·β„Žξ€·π‘§,π‘¦π‘–ξ€Έβˆ’βŸ¨π‘§,π΄π‘§βŸ©1/2ξ€Έξƒͺ,(D) where 𝐻(𝑠,𝑑,̃𝑦) denotes the set of all (𝑧,πœ‡,𝑒,𝑣)βˆˆπ‘…π‘›Γ—π‘…π‘›+×𝑅𝑛×𝑅𝑛 satisfying𝑠𝑖=1π‘‘π‘–ξ€·β„Žξ€·π‘§,π‘¦π‘–ξ€Έβˆ’βŸ¨π‘§,π΅π‘§βŸ©1/2ξ€Έξƒͺβˆ‡ξƒ©π‘ ξ“π‘–=1𝑑𝑖𝑓𝑧,𝑦𝑖++𝐴𝑒𝑝𝑗=1πœ‡π‘—π‘”π‘—ξƒͺβˆ’βŽ›βŽœβŽœβŽ(𝑧)𝑠𝑖=1𝑑𝑖𝑓𝑧,𝑦𝑖+βŸ¨π‘§,π΄π‘§βŸ©1/2ξ€Έ+ξ“π‘—βˆˆπ½0πœ‡π‘—π‘”π‘—βŽžβŽŸβŽŸβŽ βˆ‡ξƒ©(𝑧)𝑠𝑖=1π‘‘π‘–ξ€·β„Žξ€·π‘§,𝑦𝑖ξƒͺξ“βˆ’π΅π‘£=0,(4.9)π‘—βˆˆπ½π›Όπœ‡π‘—π‘”π‘—π½(𝑧)β‰₯0,𝛼=1,2,…,π‘Ÿ,π›Όβˆ©π½π›½=πœ™,for𝛼≠𝛽,π‘Ÿξšπ›Ό=0𝐽𝛼={1,2,…,𝑝}.(4.10)

Theorem 4.2 (weak duality). Let π‘₯ be a feasible solution for (P), and let (𝑧,πœ‡,𝑒,𝑣,𝑠,𝑑,̃𝑦) be a feasible solution for (4.18). Suppose that there exist 𝐹,πœƒ,πœ™0,𝑏0,𝜌0 and πœ™π›Ό,𝑏𝛼,πœŒπ›Ό, 𝛼=1,2,…,π‘Ÿ such that πΉβŽ›βŽœβŽœβŽξƒ©π‘₯,𝑧;𝑠𝑖=1π‘‘π‘–ξ€·β„Žξ€·π‘§,π‘¦π‘–ξ€Έβˆ’βŸ¨π‘§,π΅π‘§βŸ©1/2ξ€Έξƒͺβˆ‡βŽ›βŽœβŽœβŽπ‘ ξ“π‘–=1𝑑𝑖𝑓𝑧,𝑦𝑖++π΄π‘’π‘—βˆˆπ½0πœ‡π‘—π‘”π‘—βŽžβŽŸβŽŸβŽ βˆ’βŽ›βŽœβŽœβŽ(𝑧)𝑠𝑖=1𝑑𝑖𝑓𝑧,𝑦𝑖+βŸ¨π‘§,π΄π‘§βŸ©1/2ξ€Έ+ξ“π‘—βˆˆπ½0πœ‡π‘—π‘”π‘—βŽžβŽŸβŽŸβŽ βˆ‡ξƒ©(𝑧)𝑠𝑖=1π‘‘π‘–ξ€·β„Žξ€·π‘§,𝑦𝑖ξƒͺβŽžβŽŸβŽŸβŽ βˆ’π΅π‘£β‰₯βˆ’πœŒ0(β€–πœƒπ‘₯,𝑧)β€–2βŸΉπ‘0(π‘₯,𝑧)πœ™0βŽ›βŽœβŽœβŽξƒ©π‘ ξ“π‘–=1π‘‘π‘–ξ€·β„Žξ€·π‘§,π‘¦π‘–ξ€Έβˆ’βŸ¨π‘§,π΅π‘§βŸ©1/2ξ€ΈξƒͺβŽ›βŽœβŽœβŽπ‘ ξ“π‘–=1𝑑𝑖𝑓π‘₯,𝑦𝑖+⟨π‘₯,𝐴π‘₯⟩1/2ξ€Έ+ξ“π‘—βˆˆπ½0πœ‡π‘—π‘”π‘—βŽžβŽŸβŽŸβŽ βˆ’βŽ›βŽœβŽœβŽ(π‘₯)𝑠𝑖=1𝑑𝑖𝑓𝑧,𝑦𝑖+βŸ¨π‘§,π΄π‘§βŸ©1/2ξ€Έ+ξ“π‘—βˆˆπ½0πœ‡π‘—π‘”π‘—βŽžβŽŸβŽŸβŽ g×(𝑧)𝑠𝑖=1π‘‘π‘–ξ€·β„Žξ€·π‘₯,π‘¦π‘–ξ€Έβˆ’βŸ¨π‘₯,𝐡π‘₯⟩1/2ξ€Έξƒͺξƒͺβ‰₯0,(4.11)βˆ’π‘π›Ό(π‘₯,𝑧)πœ™π›ΌβŽ›βŽœβŽœβŽξƒ©π‘ ξ“π‘–=1π‘‘π‘–ξ€·β„Žξ€·π‘§,π‘¦π‘–ξ€Έβˆ’βŸ¨π‘§,π΅π‘§βŸ©1/2ξ€ΈξƒͺβŽ›βŽœβŽœβŽξ“π‘—βˆˆπ½π›Όπœ‡π‘—π‘”π‘—βŽžβŽŸβŽŸβŽ βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽ(𝑧)≀0⟹𝐹π‘₯,𝑧;𝑠𝑖=1π‘‘π‘–ξ€·β„Žξ€·π‘§,π‘¦π‘–ξ€Έβˆ’βŸ¨π‘§,π΅π‘§βŸ©1/2ξ€ΈβŽ›βŽœβŽœβŽξ“π‘—βˆˆπ½π›Όπœ‡π‘—βˆ‡π‘”π‘—βŽžβŽŸβŽŸβŽ βŽžβŽŸβŽŸβŽ (𝑧)β‰€βˆ’πœŒπ›Ό(β€–πœƒπ‘₯,𝑧)β€–2,𝛼=1,2,…,π‘Ÿ.(4.12) Further, assume that π‘Žβ‰₯0βŸΉπœ™π›Όπœ™(π‘Ž)β‰₯0,𝛼=1,2,…,π‘Ÿ,(4.13)0(π‘π‘Ž)β‰₯0βŸΉπ‘Žβ‰₯0,(4.14)0(π‘₯,𝑧)>0,π‘π›ΌπœŒ(π‘₯,𝑧)β‰₯0,𝛼=1,2,…,π‘Ÿ,(4.15)0+π‘Ÿξ“π›Ό=1πœŒπ›Όβ‰₯0,(4.16) then supπ‘¦βˆˆπ‘Œπ‘“(π‘₯,𝑦)+⟨π‘₯,𝐴π‘₯⟩1/2β„Ž(π‘₯,𝑦)βˆ’βŸ¨π‘₯,𝐡π‘₯⟩1/2β‰₯ξƒ©βˆ‘π‘ π‘–=1𝑑𝑖𝑓𝑧,𝑦𝑖+βŸ¨π‘§,π΄π‘§βŸ©1/2ξ€Έ+βˆ‘π‘—βˆˆπ½0πœ‡π‘—π‘”π‘—(𝑧)βˆ‘π‘ π‘–=1π‘‘π‘–ξ€·β„Žξ€·π‘§,π‘¦π‘–ξ€Έβˆ’βŸ¨π‘§,π΅π‘§βŸ©1/2ξ€Έξƒͺ.(4.17)

Proof. Suppose to contrary that supπ‘¦βˆˆπ‘Œπ‘“(π‘₯,𝑦)+⟨π‘₯,𝐴π‘₯⟩1/2β„Ž(π‘₯,𝑦)βˆ’βŸ¨π‘₯,𝐡π‘₯⟩1/2<ξƒ©βˆ‘π‘ π‘–=1𝑑𝑖𝑓𝑧,𝑦𝑖+βŸ¨π‘§,π΄π‘§βŸ©1/2ξ€Έ+βˆ‘π‘—βˆˆπ½0πœ‡π‘—π‘”π‘—(𝑧)βˆ‘π‘ π‘–=1π‘‘π‘–ξ€·β„Žξ€·π‘§,π‘¦π‘–ξ€Έβˆ’βŸ¨π‘§,π΅π‘§βŸ©1/2ξ€Έξƒͺ,(4.18) then, we get 𝑠𝑖=1π‘‘π‘–ξ€·β„Žξ€·π‘§,π‘¦π‘–ξ€Έβˆ’βŸ¨π‘§,π΅π‘§βŸ©1/2𝑓(π‘₯,𝑦)+⟨π‘₯,𝐴π‘₯⟩1/2ξ€Έ<βŽ›βŽœβŽœβŽπ‘ ξ“π‘–=1𝑑𝑖𝑓𝑧,𝑦𝑖+βŸ¨π‘§,π΄π‘§βŸ©1/2ξ€Έ+ξ“π‘—βˆˆπ½0πœ‡π‘—π‘”π‘—βŽžβŽŸβŽŸβŽ ξ€·(𝑧)β„Ž(π‘₯,𝑦)βˆ’βŸ¨π‘₯,𝐡π‘₯⟩1/2ξ€Έ,βˆ€π‘¦βˆˆπ‘Œ.(4.19) Further, this implies 𝑠𝑖=1π‘‘π‘–ξ€·β„Žξ€·π‘§,π‘¦π‘–ξ€Έβˆ’βŸ¨π‘§,π΅π‘§βŸ©1/2ξ€Έξƒͺ𝑠𝑖=1𝑑𝑖𝑓π‘₯,𝑦𝑖+⟨π‘₯,𝐴π‘₯⟩1/2ξ€Έξƒͺ<βŽ›βŽœβŽœβŽπ‘ ξ“π‘–=1𝑑𝑖𝑓𝑧,𝑦𝑖+βŸ¨π‘§,π΄π‘§βŸ©1/2ξ€Έ+ξ“π‘—βˆˆπ½0πœ‡π‘—π‘”π‘—βŽžβŽŸβŽŸβŽ ξƒ©(𝑧)𝑠𝑖=1π‘‘π‘–ξ€·β„Žξ€·π‘₯,π‘¦π‘–ξ€Έβˆ’βŸ¨π‘₯,𝐡π‘₯⟩1/2ξ€Έξƒͺ.(4.20) Hence, we have 𝑠𝑖=1π‘‘π‘–ξ€·β„Žξ€·π‘§,π‘¦π‘–ξ€Έβˆ’βŸ¨π‘§,π΅π‘§βŸ©1/2ξ€ΈξƒͺβŽ›βŽœβŽœβŽπ‘ ξ“π‘–=1𝑑𝑖𝑓π‘₯,𝑦𝑖+⟨π‘₯,𝐴π‘₯⟩1/2ξ€Έ+ξ“π‘—βˆˆπ½0πœ‡π‘—π‘”π‘—βŽžβŽŸβŽŸβŽ βˆ’βŽ›βŽœβŽœβŽ(π‘₯)𝑠𝑖=1𝑑𝑖𝑓𝑧,𝑦𝑖+βŸ¨π‘§,π΄π‘§βŸ©1/2ξ€Έ+ξ“π‘—βˆˆπ½0πœ‡π‘—π‘”π‘—βŽžβŽŸβŽŸβŽ ξƒ©(𝑧)𝑠𝑖=1π‘‘π‘–ξ€·β„Žξ€·π‘₯,π‘¦π‘–ξ€Έβˆ’βŸ¨π‘₯,𝐡π‘₯⟩1/2ξ€Έξƒͺ<𝑠𝑖=1π‘‘π‘–ξ€·β„Žξ€·π‘§,π‘¦π‘–ξ€Έβˆ’βŸ¨π‘§,π΅π‘§βŸ©1/2ξ€ΈξƒͺβŽ›βŽœβŽœβŽξ“π‘—βˆˆπ½0πœ‡π‘—π‘”π‘—(⎞⎟⎟⎠.π‘₯)(4.21) Using the fact that (βˆ‘π‘ π‘–=1𝑑𝑖(β„Ž(𝑧,𝑦𝑖)βˆ’βŸ¨π‘§,π΅π‘§βŸ©1/2))>0 and βˆ‘π‘—βˆˆπ½0πœ‡π‘—π‘”π‘—(π‘₯)≀0 and the last inequality, we have 𝑠𝑖=1π‘‘π‘–ξ€·β„Žξ€·π‘§,π‘¦π‘–ξ€Έβˆ’βŸ¨π‘§,π΅π‘§βŸ©1/2ξ€ΈξƒͺβŽ›βŽœβŽœβŽπ‘ ξ“π‘–=1𝑑𝑖𝑓π‘₯,𝑦𝑖+⟨π‘₯,𝐴π‘₯⟩1/2ξ€Έ+ξ“π‘—βˆˆπ½0πœ‡π‘—π‘”π‘—βŽžβŽŸβŽŸβŽ βˆ’βŽ›βŽœβŽœβŽ(π‘₯)𝑠𝑖=1𝑑𝑖𝑓𝑧,𝑦𝑖+βŸ¨π‘§,π΄π‘§βŸ©1/2ξ€Έ+ξ“π‘—βˆˆπ½0πœ‡π‘—π‘”π‘—βŽžβŽŸβŽŸβŽ ξƒ©(𝑧)𝑠𝑖=1π‘‘π‘–ξ€·β„Žξ€·π‘₯,π‘¦π‘–ξ€Έβˆ’βŸ¨π‘₯,𝐡π‘₯⟩1/2ξ€Έξƒͺ<0.(4.22) From (4.11),(4.14),(4.15), and (4.22), we get πΉβŽ›βŽœβŽœβŽξƒ©π‘₯,𝑧;𝑠𝑖=1π‘‘π‘–ξ€·β„Žξ€·π‘§,π‘¦π‘–ξ€Έβˆ’βŸ¨π‘§,π΅π‘§βŸ©1/2ξ€Έξƒͺβˆ‡βŽ›βŽœβŽœβŽπ‘ ξ“π‘–=1𝑑𝑖𝑓𝑧,𝑦𝑖++π΄π‘’π‘—βˆˆπ½0πœ‡π‘—π‘”π‘—βŽžβŽŸβŽŸβŽ βˆ’βŽ›βŽœβŽœβŽ(𝑧)𝑠𝑖=1𝑑𝑖𝑓𝑧,𝑦𝑖+βŸ¨π‘§,π΄π‘§βŸ©1/2ξ€Έ+ξ“π‘—βˆˆπ½0πœ‡π‘—π‘”π‘—βŽžβŽŸβŽŸβŽ βˆ‡ξƒ©(𝑧)𝑠𝑖=1π‘‘π‘–ξ€·β„Žξ€·π‘§,𝑦𝑖ξƒͺβŽžβŽŸβŽŸβŽ βˆ’π΅π‘£<βˆ’πœŒ0(β€–πœƒπ‘₯,𝑧)β€–2.(4.23) Using (βˆ‘π‘ π‘–=1𝑑𝑖(β„Ž(𝑧,𝑦𝑖)βˆ’βŸ¨π‘§,π΅π‘§βŸ©1/2))>0, (4.10), (4.13), and (4.15), we get βˆ’π‘π›Ό(π‘₯,𝑧)πœ™π›ΌβŽ›βŽœβŽœβŽπ‘ ξ“π‘–=1π‘‘π‘–ξ€·β„Žξ€·π‘§,π‘¦π‘–ξ€Έβˆ’βŸ¨π‘§,π΅π‘§βŸ©1/2ξ€ΈβŽ›βŽœβŽœβŽξ“π‘—βˆˆπ½π›Όπœ‡π‘—π‘”π‘—βŽžβŽŸβŽŸβŽ βŽžβŽŸβŽŸβŽ (𝑧)≀0,𝛼=1,2,…,π‘Ÿ.(4.24) From (4.12), we have πΉβŽ›βŽœβŽœβŽπ‘₯,𝑧;𝑠𝑖=1π‘‘π‘–ξ€·β„Žξ€·π‘§,π‘¦π‘–ξ€Έβˆ’βŸ¨π‘§,π΅π‘§βŸ©1/2ξ€ΈβŽ›βŽœβŽœβŽξ“π‘—βˆˆπ½π›Όπœ‡π‘—βˆ‡π‘”π‘—βŽžβŽŸβŽŸβŽ βŽžβŽŸβŽŸβŽ (𝑧)β‰€βˆ’πœŒπ›Όβ€–πœƒ(π‘₯,𝑧)β€–2,𝛼=1,2,…,π‘Ÿ.(4.25) On adding (4.23) and (4.25) and making use of sublinearity of 𝐹 and (4.16), we have 𝐹π‘₯,𝑧;𝑠𝑖=1π‘‘π‘–ξ€·β„Žξ€·π‘§,π‘¦π‘–ξ€Έβˆ’βŸ¨π‘§,π΅π‘§βŸ©1/2ξ€Έξƒͺβˆ‡ξƒ©π‘ ξ“π‘–=1𝑑𝑖𝑓𝑧,𝑦𝑖++𝐴𝑒𝑝𝑗=1πœ‡π‘—π‘”π‘—ξƒͺβˆ’βŽ›βŽœβŽœβŽ(𝑧)𝑠𝑖=1𝑑𝑖𝑓𝑧,𝑦𝑖+βŸ¨π‘§,π΄π‘§βŸ©1/2ξ€Έ+ξ“π‘—βˆˆπ½0πœ‡π‘—π‘”π‘—βŽžβŽŸβŽŸβŽ βˆ‡ξƒ©(𝑧)𝑠𝑖=1π‘‘π‘–ξ€·β„Žξ€·π‘§,𝑦𝑖ξƒͺβŽžβŽŸβŽŸβŽ βˆ’π΅π‘£<0,(4.26) which contradicts (4.9). This completes the proof.

Theorem 4.3 (weak duality). Let π‘₯ be a feasible solution for (P) and let (𝑧,πœ‡,𝑒,𝑣,𝑠,𝑑,̃𝑦) be a feasible solution for (4.18). Suppose that there exist 𝐹,πœƒ,πœ™0,𝑏0,𝜌0 and πœ™π›Ό,𝑏𝛼,πœŒπ›Ό, 𝛼=1,2,…,π‘Ÿ such that 𝑏0(π‘₯,𝑧)πœ™0βŽ›βŽœβŽœβŽξƒ©π‘ ξ“π‘–=1π‘‘π‘–ξ€·β„Žξ€·π‘§,π‘¦π‘–ξ€Έβˆ’βŸ¨π‘§,π΅π‘§βŸ©1/2ξ€ΈξƒͺβŽ›βŽœβŽœβŽπ‘ ξ“π‘–=1𝑑𝑖𝑓π‘₯,𝑦𝑖+⟨π‘₯,𝐴π‘₯⟩1/2ξ€Έ+ξ“π‘—βˆˆπ½0πœ‡π‘—π‘”π‘—βŽžβŽŸβŽŸβŽ βˆ’βŽ›βŽœβŽœβŽ(π‘₯)𝑠𝑖=1𝑑𝑖𝑓𝑧,𝑦𝑖+βŸ¨π‘§,π΄π‘§βŸ©1/2ξ€Έ+ξ“π‘—βˆˆπ½0πœ‡π‘—π‘”π‘—βŽžβŽŸβŽŸβŽ Γ—ξƒ©(𝑧)𝑠𝑖=1π‘‘π‘–ξ€·β„Žξ€·π‘₯,π‘¦π‘–ξ€Έβˆ’βŸ¨π‘₯,𝐡π‘₯⟩1/2ξ€ΈβŽ›βŽœβŽœβŽξƒ©ξƒͺξƒͺ<0⟹𝐹π‘₯,𝑧;𝑠𝑖=1π‘‘π‘–ξ€·β„Žξ€·π‘§,π‘¦π‘–ξ€Έβˆ’βŸ¨π‘§,π΅π‘§βŸ©1/2ξ€Έξƒͺβˆ‡βŽ›βŽœβŽœβŽπ‘ ξ“π‘–=1𝑑𝑖𝑓𝑧,𝑦𝑖++π΄π‘’π‘—βˆˆπ½0πœ‡π‘—π‘”π‘—βŽžβŽŸβŽŸβŽ βˆ’βŽ›βŽœβŽœβŽ(𝑧)𝑠𝑖=1𝑑𝑖𝑓𝑧,𝑦𝑖+βŸ¨π‘§,π΄π‘§βŸ©1/2ξ€Έ+ξ“π‘—βˆˆπ½0πœ‡π‘—π‘”π‘—βŽžβŽŸβŽŸβŽ βˆ‡ξƒ©(𝑧)𝑠𝑖=1π‘‘π‘–ξ€·β„Žξ€·π‘§,𝑦𝑖ξƒͺβŽžβŽŸβŽŸβŽ βˆ’π΅π‘£β‰€βˆ’πœŒ0β€–πœƒ(π‘₯,𝑧)β€–2,βˆ’π‘π›Ό(π‘₯,𝑧)πœ™π›ΌβŽ›βŽœβŽœβŽξƒ©π‘ ξ“π‘–=1π‘‘π‘–ξ€·β„Žξ€·π‘§,π‘¦π‘–ξ€Έβˆ’βŸ¨π‘§,π΅π‘§βŸ©1/2ξ€ΈξƒͺβŽ›βŽœβŽœβŽξ“π‘—βˆˆπ½π›Όπœ‡π‘—π‘”π‘—βŽžβŽŸβŽŸβŽ βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽ(𝑧)≀0⟹𝐹π‘₯,𝑧;𝑠𝑖=1π‘‘π‘–ξ€·β„Žξ€·π‘§,π‘¦π‘–ξ€Έβˆ’βŸ¨π‘§,π΅π‘§βŸ©1/2ξ€ΈβŽ›βŽœβŽœβŽξ“π‘—βˆˆπ½π›Όπœ‡π‘—βˆ‡π‘”π‘—βŽžβŽŸβŽŸβŽ βŽžβŽŸβŽŸβŽ (𝑧)<βˆ’πœŒπ›Ό(β€–πœƒπ‘₯,𝑧)β€–2,𝛼=1,2,…,π‘Ÿ.(4.27) Further, assume that (4.14), (4.15), and (4.16) are satisfied, then supπ‘¦βˆˆπ‘Œπ‘“(π‘₯,𝑦)+⟨π‘₯,𝐴π‘₯⟩1/2β„Ž(π‘₯,𝑦)βˆ’βŸ¨π‘₯,𝐡π‘₯⟩1/2β‰₯ξƒ©βˆ‘π‘ π‘–=1𝑑𝑖𝑓𝑧,𝑦𝑖+βŸ¨π‘§,π΄π‘§βŸ©1/2ξ€Έ+βˆ‘π‘—βˆˆπ½0πœ‡π‘—π‘”π‘—(𝑧)βˆ‘π‘ π‘–=1π‘‘π‘–ξ€·β„Žξ€·π‘§,π‘¦π‘–ξ€Έβˆ’βŸ¨π‘§,π΅π‘§βŸ©1/2ξ€Έξƒͺ.(4.28)

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

Theorem 4.4 (strong duality). Assume that π‘₯βˆ— is an optimal solution for (P) and βˆ‡π‘”π‘—(π‘₯βˆ—), π‘—βˆˆπ½(π‘₯βˆ—) are linearly independent. Then there exist (π‘ βˆ—,π‘‘βˆ—,Μƒπ‘¦βˆ—)∈𝐾(π‘₯βˆ—) and (π‘₯βˆ—,πœ‡βˆ—,π‘’βˆ—,π‘£βˆ—)∈𝐻(π‘ βˆ—,π‘‘βˆ—,Μƒπ‘¦βˆ—)such that (π‘₯βˆ—,πœ‡βˆ—,π‘’βˆ—,π‘£βˆ—,π‘ βˆ—,π‘‘βˆ—,Μƒπ‘¦βˆ—) 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 (𝑧,πœ‡,𝑒,𝑣,𝑠,𝑑,̃𝑦), then the problems (P) and (4.18) have the same optimal values.

Proof. By Lemma 4.1, there exist (π‘ βˆ—,π‘‘βˆ—,Μƒπ‘¦βˆ—)∈𝐾(π‘₯βˆ—) and (π‘₯βˆ—,πœ‡βˆ—,π‘’βˆ—,π‘£βˆ—)∈𝐻(π‘ βˆ—,π‘‘βˆ—,Μƒπ‘¦βˆ—) such that (π‘₯βˆ—,πœ‡βˆ—,π‘’βˆ—,π‘£βˆ—,π‘ βˆ—,π‘‘βˆ—,Μƒπ‘¦βˆ—) 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 π‘₯βˆ— and (𝑧,πœ‡,𝑒,𝑣,𝑠,𝑑,̃𝑦) be optimal solutions for (P) and (4.18), respectively. Suppose that βˆ‡π‘”π‘—(π‘₯βˆ—), π‘—βˆˆπ½(π‘₯βˆ—) are linearly independent and there exist 𝐹,πœƒ,πœ™0,𝑏0,𝜌0, and πœ™π›Ό,𝑏𝛼,πœŒπ›Ό,𝛼=1,2,…,π‘Ÿ such that πΉβŽ›βŽœβŽœβŽπ‘₯βˆ—ξƒ©,𝑧;𝑠𝑖=1π‘‘π‘–ξ€·β„Žξ€·π‘§,π‘¦π‘–ξ€Έβˆ’βŸ¨π‘§,π΅π‘§βŸ©1/2ξ€Έξƒͺβˆ‡βŽ›βŽœβŽœβŽπ‘ ξ“π‘–=1𝑑𝑖𝑓𝑧,𝑦𝑖++π΄π‘’π‘—βˆˆπ½0πœ‡π‘—π‘”π‘—βŽžβŽŸβŽŸβŽ Γ—βŽ›βŽœβŽœβŽ(𝑧)𝑠𝑖=1𝑑𝑖𝑓𝑧,𝑦𝑖+βŸ¨π‘§,π΄π‘§βŸ©1/2ξ€Έ+ξ“π‘—βˆˆπ½0πœ‡π‘—π‘”π‘—βŽžβŽŸβŽŸβŽ βˆ‡ξƒ©(𝑧)𝑠𝑖=1π‘‘π‘–ξ€·β„Žξ€·π‘§,𝑦𝑖ξƒͺβŽžβŽŸβŽŸβŽ βˆ’π΅π‘£β‰₯βˆ’πœŒ0β€–β€–πœƒξ€·π‘₯βˆ—ξ€Έβ€–β€–,𝑧2βŸΉπ‘0ξ€·π‘₯βˆ—ξ€Έπœ™,𝑧0𝑠𝑖=1π‘‘π‘–ξ€·β„Žξ€·π‘§,π‘¦π‘–ξ€Έβˆ’βŸ¨π‘§,π΅π‘§βŸ©1/2ξ€ΈξƒͺΓ—βŽ›βŽœβŽœβŽπ‘ ξ“π‘–=1𝑑𝑖𝑓π‘₯βˆ—,𝑦𝑖+⟨π‘₯βˆ—,𝐴π‘₯βˆ—βŸ©1/2+ξ“π‘—βˆˆπ½0πœ‡π‘—π‘”π‘—ξ€·π‘₯βˆ—ξ€ΈβŽžβŽŸβŽŸβŽ βˆ’βŽ›βŽœβŽœβŽπ‘ ξ“π‘–=1𝑑𝑖𝑓𝑧,𝑦𝑖+βŸ¨π‘§,π΄π‘§βŸ©1/2ξ€Έ+ξ“π‘—βˆˆπ½0πœ‡π‘—π‘”π‘—βŽžβŽŸβŽŸβŽ ξƒ©(𝑧)𝑠𝑖=1π‘‘π‘–ξ‚€β„Žξ€·π‘₯βˆ—,π‘¦π‘–ξ€Έβˆ’βŸ¨π‘₯βˆ—,𝐡π‘₯βˆ—βŸ©1/2ξƒͺξƒͺβ‰₯0,(4.29)βˆ’π‘π›Όξ€·π‘₯βˆ—ξ€Έπœ™,π‘§π›ΌβŽ›βŽœβŽœβŽξƒ©π‘ ξ“π‘–=1π‘‘π‘–ξ€·β„Žξ€·π‘§,π‘¦π‘–ξ€Έβˆ’βŸ¨π‘§,π΅π‘§βŸ©1/2ξ€ΈξƒͺβŽ›βŽœβŽœβŽξ“π‘—βˆˆπ½π›Όπœ‡π‘—π‘”π‘—βŽžβŽŸβŽŸβŽ βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘₯(𝑧)≀0βŸΉπΉβˆ—,𝑧;𝑠𝑖=1π‘‘π‘–ξ€·β„Žξ€·π‘§,π‘¦π‘–ξ€Έβˆ’βŸ¨π‘§,π΅π‘§βŸ©1/2ξ€ΈβŽ›βŽœβŽœβŽξ“π‘—βˆˆπ½π›Όπœ‡π‘—βˆ‡π‘”π‘—(βŽžβŽŸβŽŸβŽ βŽžβŽŸβŽŸβŽ π‘§)β‰€βˆ’πœŒπ›Όβ€–β€–πœƒξ€·π‘₯βˆ—ξ€Έβ€–β€–,𝑧2,𝛼=1,2,…,π‘Ÿ.(4.30) Further, assume (4.13), (4.15), (4.16), πœ™0(π‘Ž)β‰₯0βŸΉπ‘Ž>0,(4.31) then π‘₯βˆ—=𝑧, that is, 𝑧 is an optimal solution for (P).

Proof. Suppose to contrary that π‘₯βˆ—β‰ π‘§. From the strong duality Theorem 4.4, we know that supπ‘¦βˆˆπ‘Œπ‘“ξ€·π‘₯βˆ—ξ€Έ,𝑦+⟨π‘₯βˆ—,𝐴π‘₯βˆ—βŸ©1/2β„Ž(π‘₯βˆ—,𝑦)βˆ’βŸ¨π‘₯βˆ—,𝐡π‘₯βˆ—βŸ©1/2=ξƒ©βˆ‘π‘ π‘–=1𝑑𝑖𝑓𝑧,𝑦𝑖+βŸ¨π‘§,π΄π‘§βŸ©1/2ξ€Έ+βˆ‘π‘—βˆˆπ½0πœ‡π‘—π‘”π‘—(𝑧)βˆ‘π‘ π‘–=1π‘‘π‘–ξ€·β„Žξ€·π‘§,π‘¦π‘–ξ€Έβˆ’βŸ¨π‘§,π΅π‘§βŸ©1/2ξ€Έξƒͺ.(4.32) Then, we get 𝑠𝑖=1π‘‘π‘–ξ€·β„Žξ€·π‘§,π‘¦π‘–ξ€Έβˆ’βŸ¨π‘§,π΅π‘§βŸ©1/2ξ€Έξƒͺ𝑓π‘₯βˆ—ξ€Έ,𝑦+⟨π‘₯βˆ—,𝐴π‘₯βˆ—βŸ©1/2ξ‚β‰€βŽ›βŽœβŽœβŽπ‘ ξ“π‘–=1𝑑𝑖𝑓𝑧,𝑦𝑖+βŸ¨π‘§,π΄π‘§βŸ©1/2ξ€Έ+ξ“π‘—βˆˆπ½0πœ‡π‘—π‘”π‘—βŽžβŽŸβŽŸβŽ ξ‚€β„Žξ€·π‘₯(𝑧)βˆ—ξ€Έ,π‘¦βˆ’βŸ¨π‘₯βˆ—,𝐡π‘₯βˆ—βŸ©1/2,βˆ€π‘¦βˆˆπ‘Œ.(4.33) Further, this implies 𝑠𝑖=1π‘‘π‘–ξ€·β„Žξ€·π‘§,π‘¦π‘–ξ€Έβˆ’βŸ¨π‘§,π΅π‘§βŸ©1/2ξ€Έξƒͺ𝑠𝑖=1𝑑𝑖𝑓π‘₯βˆ—,𝑦𝑖+⟨π‘₯βˆ—,𝐴π‘₯βˆ—βŸ©1/2ξƒͺβ‰€βŽ›βŽœβŽœβŽπ‘ ξ“π‘–=1𝑑𝑖𝑓𝑧,𝑦𝑖+βŸ¨π‘§,π΄π‘§βŸ©1/2ξ€Έ+ξ“π‘—βˆˆπ½0πœ‡π‘—π‘”π‘—βŽžβŽŸβŽŸβŽ Γ—ξƒ©(𝑧)𝑠𝑖=1π‘‘π‘–ξ‚€β„Žξ€·π‘₯βˆ—,π‘¦π‘–ξ€Έβˆ’βŸ¨π‘₯βˆ—,𝐡π‘₯βˆ—βŸ©1/2ξƒͺ,βˆ€π‘¦βˆˆπ‘Œ.(4.34) Hence, we have 𝑠𝑖=1π‘‘π‘–ξ€·β„Žξ€·π‘§,π‘¦π‘–ξ€Έβˆ’βŸ¨π‘§,π΅π‘§βŸ©1/2ξ€ΈξƒͺβŽ›βŽœβŽœβŽπ‘ ξ“π‘–=1𝑑𝑖𝑓π‘₯βˆ—,𝑦𝑖+⟨π‘₯βˆ—,𝐴π‘₯βˆ—βŸ©1/2+ξ“π‘—βˆˆπ½0πœ‡π‘—π‘”π‘—ξ€·π‘₯βˆ—ξ€ΈβŽžβŽŸβŽŸβŽ βˆ’βŽ›βŽœβŽœβŽπ‘ ξ“π‘–=1𝑑𝑖𝑓𝑧,𝑦𝑖+βŸ¨π‘§,π΄π‘§βŸ©1/2ξ€Έ+ξ“π‘—βˆˆπ½0πœ‡π‘—π‘”π‘—βŽžβŽŸβŽŸβŽ ξƒ©(𝑧)𝑠𝑖=1π‘‘π‘–ξ‚€β„Žξ€·π‘₯βˆ—,π‘¦π‘–ξ€Έβˆ’βŸ¨π‘₯βˆ—,𝐡π‘₯βˆ—βŸ©1/2ξƒͺ≀𝑠𝑖=1π‘‘π‘–ξ€·β„Žξ€·π‘§,π‘¦π‘–ξ€Έβˆ’βŸ¨π‘§,π΅π‘§βŸ©1/2ξ€ΈξƒͺβŽ›βŽœβŽœβŽξ“π‘—βˆˆπ½0πœ‡π‘—π‘”π‘—ξ€·π‘₯βˆ—ξ€ΈβŽžβŽŸβŽŸβŽ .(4.35) Using the fact that (βˆ‘π‘ π‘–=1𝑑𝑖(β„Ž(𝑧,𝑦𝑖)βˆ’βŸ¨π‘§,π΅π‘§βŸ©1/2))>0 and βˆ‘π‘—βˆˆπ½0πœ‡π‘—π‘”π‘—(π‘₯βˆ—)≀0 and the last inequality, we have 𝑠𝑖=1π‘‘π‘–ξ€·β„Žξ€·π‘§,π‘¦π‘–ξ€Έβˆ’βŸ¨π‘§,π΅π‘§βŸ©1/2ξ€ΈξƒͺβŽ›βŽœβŽœβŽπ‘ ξ“π‘–=1𝑑𝑖𝑓π‘₯βˆ—,𝑦𝑖+⟨π‘₯βˆ—,𝐴π‘₯βˆ—βŸ©1/2+ξ“π‘—βˆˆπ½0πœ‡π‘—π‘”π‘—ξ€·π‘₯βˆ—ξ€ΈβŽžβŽŸβŽŸβŽ βˆ’βŽ›βŽœβŽœβŽπ‘ ξ“π‘–=1𝑑𝑖𝑓𝑧,𝑦𝑖+βŸ¨π‘§,π΄π‘§βŸ©1/2ξ€Έ+ξ“π‘—βˆˆπ½0πœ‡π‘—π‘”π‘—βŽžβŽŸβŽŸβŽ ξƒ©(𝑧)𝑠𝑖=1π‘‘π‘–ξ‚€β„Žξ€·π‘₯βˆ—,π‘¦π‘–ξ€Έβˆ’βŸ¨π‘₯βˆ—,𝐡π‘₯βˆ—βŸ©1/2ξƒͺ≀0.(4.36) From (4.15), (4.29), (4.31), and (4.36), we get πΉβŽ›βŽœβŽœβŽπ‘₯βˆ—ξƒ©,𝑧;𝑠𝑖=1π‘‘π‘–ξ€·β„Žξ€·π‘§,π‘¦π‘–ξ€Έβˆ’βŸ¨π‘§,π΅π‘§βŸ©1/2ξ€Έξƒͺβˆ‡βŽ›βŽœβŽœβŽπ‘ ξ“π‘–=1𝑑𝑖𝑓𝑧,𝑦𝑖++π΄π‘’π‘—βˆˆπ½0πœ‡π‘—π‘”π‘—βŽžβŽŸβŽŸβŽ βˆ’βŽ›βŽœβŽœβŽ(𝑧)𝑠𝑖=1𝑑𝑖𝑓𝑧,𝑦𝑖+βŸ¨π‘§,π΄π‘§βŸ©1/2ξ€Έ