Table of Contents
ISRN Mathematical Analysis
VolumeΒ 2011, Article IDΒ 786978, 22 pages
http://dx.doi.org/10.5402/2011/786978
Research Article

Sufficient Conditions and Duality Theorems for Nondifferentiable Minimax Fractional Programming

Center of General Education, Chung Jen College of Nursing, Health Science and Management, Dalin 62241, Taiwan

Received 20 October 2010; Accepted 28 November 2010

Academic Editors: R.Β Barrio and S.Β Zhang

Copyright Β© 2011 Shun-Chin Ho. 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 consider nondifferentiable minimax fractional programming problems involving 𝐡-(𝑝, π‘Ÿ)-invex functions with respect to πœ‚ and 𝑏. Sufficient optimality conditions and duality results for a class of nondifferentiable minimax fractional programming problems are obtained undr 𝐡-(𝑝, π‘Ÿ)-invexity assumption on objective and constraint functions. Parametric duality, Mond-Weir duality, and Wolfe duality problems may be formulated, and duality results are derived under 𝐡-(𝑝, π‘Ÿ)-invex functions.

1. Introduction

Convexity plays an important role in many aspects of mathematical programming including sufficient optimality conditions and duality theorems. In general, we use the invex function to replace convexity on sufficient optimality conditions and duality theorems (see, e.g., [1–6]).

Many authors investigated the optimality conditions and duality theorems for minimax (fractional) programming problems. For details, one can consult [1–14]. In particular, Lai et al. [10] have established the theorems of necessary and sufficient optimality conditions for nondifferentiable minimax fractional problem under the conditions of convexity. In [11], Lai and Lee employed the optimality conditions to construct two parameter-free dual models of nondifferentiable minimax fractional programming problem which involve pseudoconvex and quasiconvex functions, and derived weak and strong duality theorems. In the formulation of the dual models in [11] optimality conditions given in [10] are used. Mishra et al. [4] derived a Kuhn-Tucker-type sufficient optimality condition for an optimal solution to the nondifferentiable minimax fractional programming problem and established weak, strong, and converse duality theorems for the problem and its three different forms of dual problems under generalized univexity. Mishra et al. [5, 13] considered the nondifferentiable minimax fractional programming problem and obtain optimality and duality results under generalized 𝛼-invexity [5] and generalized 𝛼-unvexity [13]. Recently, Antczak [15] defined a new class of functions, named 𝐡-(𝑝,π‘Ÿ)-invex, which is an extension of invex function. In [1], parametric and nonparametric sufficient optimality conditions and several parametric and parameter-free duality models for the generalized fractional minimax programs are obtained under 𝐡-(𝑝,π‘Ÿ)-invexity assumption on objective and constraint functions.

In this paper, we are inspired to extend the result of Lai et al. [10] to 𝐡-(𝑝,π‘Ÿ)-invexity and organize this paper as follows. In Section 2 we introduce some basic results. We establish sufficient optimality conditions for nondifferentiable minimax fractional programming problem under 𝐡-(𝑝,π‘Ÿ)-invex with respect to the same function πœ‚ and with respect to, not necessarily, the same function 𝑏 in Section 3. Employing these results, we construct three dual problems in Sections 4–6. Here we investigate weak, strong, and strict converse duality theorems under the framework of 𝐡-(𝑝,π‘Ÿ)-invex with respect to the same function πœ‚ and with respect to, not necessarily, the same function 𝑏.

2. Some Notations and Preliminary Results

Let ℝ𝑛 be the 𝑛-dimensional Euclidean space and ℝ𝑛+ its nonnegative orthant. Throughout the paper, let 𝑋 be a nonempty open set of β„π•Ÿ.

The following definition can be found in [15].

Definition 2.1 (see [15]). Let 𝑝 and π‘Ÿ be any real numbers. The differentiable function π‘“βˆΆπ‘‹β†’β„ is said to be (strictly) 𝐡-(𝑝,π‘Ÿ)-invex with respect to πœ‚ and 𝑏 at π‘’βˆˆπ‘‹ on a nonempty set π‘‹βŠ‚β„π‘› if, there exist a function πœ‚βˆΆπ‘‹Γ—π‘‹β†’β„π‘› and a function π‘βˆΆπ‘‹Γ—π‘‹β†’β„+⧡{0} such that, for all π‘₯βˆˆπ‘‹, the inequalities 1π‘Ÿξ€·π‘’π‘(π‘₯,𝑒)π‘Ÿ(𝑓(π‘₯)βˆ’π‘“(𝑒))ξ€Έβ‰₯ξƒ―1βˆ’1π‘ξ€·π‘’βˆ‡π‘“(𝑒)π‘πœ‚(π‘₯,𝑒)ξ€Έξƒ―1βˆ’πŸ(>ifπ‘₯≠𝑒)for𝑝≠0,π‘Ÿβ‰ 0,βˆ‡π‘“(𝑒)πœ‚(π‘₯,𝑒)(>ifπ‘₯≠𝑒)for𝑝=0,π‘Ÿβ‰ 0,𝑏(π‘₯,𝑒)(𝑓(π‘₯)βˆ’π‘“(𝑒))β‰₯π‘ξ€·π‘’βˆ‡π‘“(𝑒)π‘πœ‚(π‘₯,𝑒)ξ€Έ(βˆ’πŸ(>ifπ‘₯≠𝑒)for𝑝≠0,π‘Ÿ=0,βˆ‡π‘“(𝑒)πœ‚(π‘₯,𝑒)>ifπ‘₯≠𝑒)for𝑝=0,π‘Ÿ=0(2.1) hold.

𝑓 is said to be 𝐡-(𝑝,π‘Ÿ)-invex (strictly 𝐡-(𝑝,π‘Ÿ)-invex) with respect to πœ‚ and 𝑏 on 𝑋 if it is 𝐡-(𝑝,π‘Ÿ)-invex with respect to the same πœ‚ and 𝑏 at each π‘’βˆˆπ‘‹.

It should be pointed out that exponentials appearing on the right-hand sides of the above inequalities are understood to be taken componentwise and 𝟏=(1,1,…,1)βˆˆβ„π‘›.

We consider the following nondifferentiable minimax fractional programming problem: minimize𝐹(π‘₯)=supπ‘¦βˆˆπ‘Œξ€·π‘₯𝑓(π‘₯,𝑦)+βŠ€ξ€Έπ΅π‘₯1/2ξ€·π‘₯β„Ž(π‘₯,𝑦)βˆ’βŠ€ξ€Έπ·π‘₯1/2(subjectto𝑔(π‘₯)≀0,P) where π‘Œ is a compact subset of β„π‘š, 𝑓(β‹…,β‹…)βˆΆβ„π‘›Γ—β„π‘šβ†¦β„, β„Ž(β‹…,β‹…)βˆΆβ„π‘›Γ—β„π‘šβ†¦β„, and 𝑔(β‹…)βˆΆβ„π‘›β†¦β„π‘ are 𝐢1-functions, 𝐡 and 𝐷 are 𝑛×𝑛 positive semidefinite matrices, 𝑓(π‘₯,𝑦)+(π‘₯⊀𝐡π‘₯)1/2β‰₯0, and β„Ž(π‘₯,𝑦)βˆ’(π‘₯⊀𝐷π‘₯)1/2>0 for each (π‘₯,𝑦) in π‘‹βˆ˜Γ—π‘Œ, where π‘‹βˆ˜ is the set of feasible solutions of problem (P); that is, π‘‹βˆ˜={π‘₯βˆˆπ‘‹βˆΆπ‘”(π‘₯)≀0}. This is a nondifferentiable programming problem if either 𝐡 or 𝐷 is nonzero. If 𝐡 and 𝐷 are null matrices, then problem (P) is a minimax fractional programming problem.

For each (π‘₯,𝑦)βˆˆβ„π‘›Γ—β„π‘š define ξ€·π‘₯πœ™(π‘₯,𝑦)=𝑓(π‘₯,𝑦)+βŠ€ξ€Έπ΅π‘₯1/2ξ€·π‘₯β„Ž(π‘₯,𝑦)βˆ’βŠ€ξ€Έπ·π‘₯1/2.(2.2) We let𝐽={1,2,…,𝑝},𝐽(π‘₯)=π‘—βˆˆπ½βˆ£π‘”π‘—ξ€Ύ,ξ‚»(π‘₯)=0π‘Œ(π‘₯)=π‘¦βˆˆπ‘Œβˆ£πœ™(π‘₯,𝑦)=supπ‘§βˆˆπ‘Œξ‚Ό,ξƒ―πœ™(π‘₯,𝑧)𝐾(π‘₯)=(𝑠,𝑑,𝑦)βˆˆβ„•Γ—β„π‘ +Γ—β„π‘šπ‘ ξ€·π‘‘βˆ£1≀𝑠≀𝑛+1,𝑑=1,𝑑2,…,π‘‘π‘ ξ€Έβˆˆβ„π‘ +with𝑠𝑖=1𝑑𝑖𝑦=1,𝑦=1,𝑦2,…,𝑦𝑠with𝑦𝑖.βˆˆπ‘Œ(π‘₯),𝑖=1,2,…,𝑠(2.3)

Because 𝑓 and β„Ž are contionuous differentiable and π‘Œ is compact subset of β„π‘š, we see that for each π‘₯0βˆˆπ‘‹βˆ˜, π‘Œ(π‘₯0)β‰ βˆ…, and for any π‘¦π‘–βˆˆπ‘Œ(π‘₯0), we have a postive constant π‘˜0ξ€·π‘₯=πœ™0,𝑦𝑖=𝑓π‘₯0,𝑦𝑖+ξ€·π‘₯⊀0𝐡π‘₯0ξ€Έ1/2β„Žξ€·π‘₯0,π‘¦π‘–ξ€Έβˆ’ξ€·π‘₯⊀0𝐷π‘₯0ξ€Έ1/2.(2.4)

We will use the generalized Schwarz inequality π‘₯βŠ€ξ€·π‘₯π΅π‘£β‰€βŠ€ξ€Έπ΅π‘₯1/2ξ€·π‘£βŠ€ξ€Έπ΅π‘£1/2,forπ‘₯,π‘£βˆˆβ„π‘›;(2.5) the equality holds when 𝐡π‘₯=πœ†π΅π‘£, for some πœ†β‰₯0.

Hence if π‘£βŠ€π΅π‘£β‰€1, we have π‘₯βŠ€ξ€·π‘₯π΅π‘£β‰€βŠ€ξ€Έπ΅π‘₯1/2.(2.6)

In [10] Lai et al. derived the following necessary conditions for optimality (P).

Theorem 2.2 ((necessary conditions) see, [10]). Let π‘₯0 be a (𝑃)-optimal solution and satisfying π‘₯⊀0𝐡π‘₯0>0, π‘₯⊀0𝐷π‘₯0>0, and βˆ‡π‘”π‘—(π‘₯0),π‘—βˆˆπ½(π‘₯0) is linearly independent. Then there exist (𝑠,π‘‘βˆ—,𝑦)∈𝐾(π‘₯0), π‘˜0βˆˆβ„+, 𝑀,π‘£βˆˆβ„π‘›, and πœ‡βˆ—βˆˆβ„π‘+ such that 𝑠𝑖=1π‘‘βˆ—π‘–ξ€½ξ€·π‘₯βˆ‡π‘“0,𝑦𝑖+π΅π‘€βˆ’π‘˜0ξ€·ξ€·π‘₯βˆ‡β„Ž0,π‘¦π‘–ξ€Έβˆ’π·π‘£ξ€Έξ€Ύ+βˆ‡π‘ξ“π‘—=1πœ‡βˆ—π‘—π‘”π‘—ξ€·π‘₯0𝑓π‘₯=0,(2.7)0,𝑦𝑖+ξ€·π‘₯⊀0𝐡π‘₯0ξ€Έ1/2βˆ’π‘˜0ξ‚€β„Žξ€·π‘₯0,π‘¦π‘–ξ€Έβˆ’ξ€·π‘₯⊀0𝐷π‘₯0ξ€Έ1/2=0,𝑖=1,2,…,𝑠,(2.8)𝑝𝑗=1πœ‡βˆ—π‘—π‘”π‘—ξ€·π‘₯0𝑑=0,(2.9)βˆ—π‘–β‰₯0,𝑠𝑖=1π‘‘βˆ—π‘–π‘€=1,(2.10)βŠ€π΅π‘€β‰€1,π‘£βŠ€π‘₯𝐷𝑣≀1,⊀0ξ€·π‘₯𝐡𝑀=⊀0𝐡π‘₯0ξ€Έ1/2,π‘₯⊀0ξ€·π‘₯𝐷𝑣=⊀0𝐡π‘₯0ξ€Έ1/2.(2.11)

It should be noted that both the matrices 𝐡 and 𝐷 are positive definite at the solution π‘₯0 in the above theorem. If one of π‘₯⊀0𝐡π‘₯0 and π‘₯⊀0𝐷π‘₯0 is zero, or both 𝐡 and 𝐷 are singular at π‘₯0, then, for (𝑠,π‘‘βˆ—,𝑦)∈𝐾(π‘₯0), we define a set 𝑍𝑦(π‘₯0) by 𝑍𝑦π‘₯0ξ€Έ=ξ€½π‘§βˆˆβ„π‘›βˆ£π‘§βŠ€βˆ‡π‘”π‘—ξ€·π‘₯0ξ€Έξ€·π‘₯≀0,π‘—βˆˆπ½0ξ€Έξ€Ύ.,withanyoneofthenextconditions(i)-(iii)holds(2.12)

Here conditions (i)–(iii) are given as follows:(i)if π‘₯⊀0𝐡π‘₯0>0 and π‘₯⊀0𝐷π‘₯0=0, then π‘§βŠ€ξƒ©π‘ ξ“π‘–=1π‘‘βˆ—π‘–ξƒ©ξ€·π‘₯βˆ‡π‘“0,𝑦𝑖+𝐡π‘₯0ξ€·π‘₯⊀0𝐡π‘₯0ξ€Έ1/2βˆ’π‘˜0ξ€·π‘₯βˆ‡β„Ž0,𝑦𝑖+𝑧ξƒͺξƒͺβŠ€ξ€·π‘˜20𝐷𝑧1/2<0,(2.13)(ii)if π‘₯⊀0𝐡π‘₯0=0 and π‘₯⊀0𝐷π‘₯0>0, then π‘§βŠ€ξƒ©π‘ ξ“π‘–=1π‘‘βˆ—π‘–ξƒ©ξ€·π‘₯βˆ‡π‘“0,π‘¦π‘–ξ€Έβˆ’π‘˜0π‘₯βˆ‡β„Ž0,π‘¦π‘–ξ€Έβˆ’π·π‘₯0ξ€·π‘₯⊀0𝐷π‘₯0ξ€Έ1/2+𝑧ξƒͺξƒͺξƒͺβŠ€ξ€Έπ΅π‘§1/2<0,(2.14)(iii)if π‘₯⊀0𝐡π‘₯0=0 and π‘₯⊀0𝐷π‘₯0=0, then π‘§βŠ€ξƒ©π‘ ξ“π‘–=1π‘‘βˆ—π‘–ξ€·ξ€·π‘₯βˆ‡π‘“0,π‘¦π‘–ξ€Έβˆ’π‘˜0ξ€·π‘₯βˆ‡β„Ž0,𝑦𝑖ξƒͺ+ξ€·π‘§ξ€Έξ€ΈβŠ€ξ€·π‘˜0𝐷𝑧1/2+ξ€·π‘§βŠ€ξ€Έπ΅π‘§1/2<0.(2.15)

If we take condition 𝑍𝑦(π‘₯0)=βˆ… in Theorem 2.2, then the result of Theorem 2.2 still holds.

3. Optimality Conditions

In this section we derive sufficient conditions for optimality of (P) under the assumpition of a particular form of generalized 𝐡-(𝑝,π‘Ÿ)-invexity. All theorems in this work will be proved only in the case when 𝑝≠0, π‘Ÿβ‰ 0 (other cases can be dealt with by similarity since the only difference is arised from the form of the inequality defining the class of the 𝐡-(𝑝,π‘Ÿ)-invex functions with respect to πœ‚ and 𝑏 for given 𝑝 and π‘Ÿ). The proofs of the other cases are easier than this one.

We would establish the sufficient conditions under the 𝐡-(𝑝,π‘Ÿ)-invex function.

Theorem 3.1 (sufficient optimality conditions). Let π‘₯0βˆˆπ‘‹βˆ˜ be a feasible solution of (P). There exist a positive interger 𝑠, 1≀𝑠≀𝑛+1, π‘‘βˆ—βˆˆβ„π‘ , π‘¦π‘–βˆˆπ‘Œ(π‘₯0)(𝑖=1,2,…,𝑠), π‘˜0βˆˆβ„+, (𝑀,𝑣)βˆˆβ„π‘›Γ—β„π‘›, and πœ‡βˆ—βˆˆβ„π‘+ to satisfy relations (2.7)~(2.11). Furthermore suppose that any one of conditions (a) and (b) holds: (a)βˆ‘Ξ₯(β‹…)=𝑠𝑖=1π‘‘βˆ—π‘–((𝑓(β‹…,𝑦𝑖)+(β‹…)βŠ€π΅π‘€)βˆ’π‘˜0(β„Ž(β‹…,𝑦𝑖)βˆ’(β‹…)βŠ€π·π‘£)) is 𝐡-(𝑝,π‘Ÿ)-invex with respect to πœ‚ and 𝑏 at π‘₯0, and βˆ‘π‘π‘—=1πœ‡βˆ—π‘—π‘”π‘—(β‹…) is 𝐡-(𝑝,π‘Ÿ)-invex with respect to the same function πœ‚ and another function 𝑏1 at π‘₯0 on π‘‹βˆ˜, not necessarily, equal to 𝑏, (b)Ξ₯1βˆ‘(β‹…)=𝑠𝑖=1π‘‘βˆ—π‘–((𝑓(β‹…,𝑦𝑖)+(β‹…)βŠ€π΅π‘€)βˆ’π‘˜0(β„Ž(β‹…,𝑦𝑖)βˆ’(β‹…)βŠ€βˆ‘π·π‘£))+𝑝𝑗=1πœ‡βˆ—π‘—π‘”π‘—(β‹…) is 𝐡-(𝑝,π‘Ÿ)-invex with respect to πœ‚ and 𝑏 at π‘₯0 on π‘‹βˆ˜,then π‘₯0 is an optimal solution of (P).

Proof. Suppose that π‘₯0 is not an optimal solution of (P). Then there exists a (𝑃)-feasible solution π‘₯1 such that supπ‘¦βˆˆπ‘Œπ‘“ξ€·π‘₯1ξ€Έ+ξ€·π‘₯,π‘¦βŠ€1𝐡π‘₯1ξ€Έ1/2β„Žξ€·π‘₯1ξ€Έβˆ’ξ€·π‘₯,π‘¦βŠ€1𝐷π‘₯1ξ€Έ1/2<supπ‘¦βˆˆπ‘Œπ‘“ξ€·π‘₯0ξ€Έ+ξ€·π‘₯,π‘¦βŠ€0𝐡π‘₯0ξ€Έ1/2β„Žξ€·π‘₯0ξ€Έβˆ’ξ€·π‘₯,π‘¦βŠ€0𝐷π‘₯0ξ€Έ1/2.(3.1) We note that supπ‘¦βˆˆπ‘Œπ‘“ξ€·π‘₯0ξ€Έ+ξ€·π‘₯,π‘¦βŠ€0𝐡π‘₯0ξ€Έ1/2β„Žξ€·π‘₯0ξ€Έβˆ’ξ€·π‘₯,π‘¦βŠ€0𝐷π‘₯0ξ€Έ1/2=𝑓π‘₯0,𝑦𝑖+ξ€·π‘₯⊀0𝐡π‘₯0ξ€Έ1/2β„Žξ€·π‘₯0,π‘¦π‘–ξ€Έβˆ’ξ€·π‘₯⊀0𝐷π‘₯0ξ€Έ1/2=π‘˜0,(3.2) for π‘¦π‘–βˆˆπ‘Œ(π‘₯0), 𝑖=1,2,…,𝑠, and 𝑓π‘₯1,𝑦𝑖+ξ€·π‘₯⊀1𝐡π‘₯1ξ€Έ1/2β„Žξ€·π‘₯1,π‘¦π‘–ξ€Έβˆ’ξ€·π‘₯⊀1𝐷π‘₯1ξ€Έ1/2≀supπ‘¦βˆˆπ‘Œπ‘“ξ€·π‘₯1ξ€Έ+ξ€·π‘₯,π‘¦βŠ€1𝐡π‘₯1ξ€Έ1/2β„Žξ€·π‘₯1ξ€Έβˆ’ξ€·π‘₯,π‘¦βŠ€1𝐷π‘₯1ξ€Έ1/2.(3.3) Then, we obtain 𝑓π‘₯1,𝑦𝑖+ξ€·π‘₯⊀1𝐡π‘₯1ξ€Έ1/2β„Žξ€·π‘₯1,π‘¦π‘–ξ€Έβˆ’ξ€·π‘₯⊀1𝐷π‘₯1ξ€Έ1/2<π‘˜0,for𝑖=1,2,…,𝑠.(3.4) It follows that 𝑓π‘₯1,𝑦𝑖+ξ€·π‘₯⊀1𝐡π‘₯1ξ€Έ1/2βˆ’π‘˜0ξ‚€β„Žξ€·π‘₯1,π‘¦π‘–ξ€Έβˆ’ξ€·π‘₯⊀1𝐷π‘₯1ξ€Έ1/2<0,for𝑖=1,2,…,𝑠.(3.5) From relations (2.6), (2.11), (2.8), (2.10), and (3.5), we have Ξ₯ξ€·π‘₯1ξ€Έ=𝑠𝑖=1π‘‘βˆ—π‘–ξ€·π‘“ξ€·π‘₯1,𝑦𝑖+ξ€·π‘₯⊀1ξ€Έπ΅π‘€βˆ’π‘˜0ξ€·β„Žξ€·π‘₯1,π‘¦π‘–ξ€Έβˆ’ξ€·π‘₯⊀1≀𝐷𝑣𝑠𝑖=1π‘‘βˆ—π‘–ξ‚€π‘“ξ€·π‘₯1,𝑦𝑖+ξ€·π‘₯⊀1𝐡π‘₯1ξ€Έ1/2βˆ’π‘˜0ξ‚€β„Žξ€·π‘₯1,π‘¦π‘–ξ€Έβˆ’ξ€·π‘₯⊀1𝐷π‘₯1ξ€Έ1/2=<0𝑠𝑖=1π‘‘βˆ—π‘–ξ‚€π‘“ξ€·π‘₯0,𝑦𝑖+ξ€·π‘₯⊀0𝐡π‘₯0ξ€Έ1/2βˆ’π‘˜0ξ‚€β„Žξ€·π‘₯0,π‘¦π‘–ξ€Έβˆ’ξ€·π‘₯⊀0𝐷π‘₯0ξ€Έ1/2=𝑠𝑖=1π‘‘βˆ—π‘–ξ€·π‘“ξ€·π‘₯0,𝑦𝑖+ξ€·π‘₯⊀0ξ€Έπ΅π‘€βˆ’π‘˜0ξ€·β„Žξ€·π‘₯0,π‘¦π‘–ξ€Έβˆ’ξ€·π‘₯⊀0ξ€·π‘₯𝐷𝑣=Ξ₯0ξ€Έ.(3.6) That is, Ξ₯ξ€·π‘₯1ξ€Έξ€·π‘₯<Ξ₯0ξ€Έ.(3.7) From relations (P) and (2.9), we obtain 𝑝𝑗=1πœ‡βˆ—π‘—π‘”π‘—ξ€·π‘₯1≀0=𝑝𝑗=1πœ‡βˆ—π‘—π‘”π‘—ξ€·π‘₯0ξ€Έ.(3.8) If hypothesis (a) holds, from the 𝐡-(𝑝,π‘Ÿ)-invexity with respect to πœ‚ and 𝑏1 at π‘₯0 of βˆ‘π‘π‘—=1πœ‡βˆ—π‘—π‘”π‘—(β‹…), we have 1π‘Ÿπ‘1ξ€·π‘₯1,π‘₯0ξ€Έξ‚€π‘’βˆ‘π‘Ÿ(𝑝𝑗=1πœ‡βˆ—π‘—π‘”π‘—(π‘₯1βˆ‘)βˆ’π‘π‘—=1πœ‡βˆ—π‘—π‘”π‘—(π‘₯0))β‰₯1βˆ’1π‘βˆ‡π‘ξ“π‘—=1πœ‡βˆ—π‘—π‘”π‘—ξ€·π‘₯0π‘’ξ€Έξ€·π‘πœ‚(π‘₯1,π‘₯0)ξ€Έ.βˆ’πŸ(3.9) From the inequalities (3.8) and (3.9), we get 1π‘βˆ‡ξƒ©π‘ξ“π‘—=1πœ‡βˆ—π‘—π‘”π‘—ξ€·π‘₯0ξ€Έξƒͺξ€·π‘’π‘πœ‚(π‘₯1,π‘₯0)ξ€Έβˆ’πŸβ‰€0.(3.10) Now, multiplying equality (2.7) by (1/𝑝)(π‘’π‘πœ‚(π‘₯1,π‘₯0)βˆ’πŸ), we know 1π‘ξ€·π‘’π‘πœ‚(π‘₯1,π‘₯0)ξ€Έξƒ©βˆ’πŸπ‘ ξ“π‘–=1π‘‘βˆ—π‘–ξ€½ξ€·π‘₯βˆ‡π‘“0,𝑦𝑖+π΅π‘€βˆ’π‘˜0ξ€·ξ€·π‘₯βˆ‡β„Ž0,π‘¦π‘–ξ€Έβˆ’π·π‘£ξ€Έξ€Ύ+βˆ‡π‘ξ“π‘—=1πœ‡βˆ—π‘—π‘”jξ€·π‘₯0ξ€Έξƒͺ=0.(3.11) From relations (3.10) and (3.11), we have 1π‘ξ€·π‘’π‘πœ‚(π‘₯1,π‘₯0)ξ€Έξƒ©βˆ’πŸπ‘ ξ“π‘–=1π‘‘βˆ—π‘–ξ€½ξ€·π‘₯βˆ‡π‘“0,𝑦𝑖+π΅π‘€βˆ’π‘˜0ξ€·ξ€·π‘₯βˆ‡β„Ž0,𝑦𝑖ξƒͺβˆ’π·π‘£ξ€Έξ€Ύβ‰₯0.(3.12) From the 𝐡-(𝑝,π‘Ÿ)-invexity with respect to the same function πœ‚ and the function 𝑏 at π‘₯0 of Ξ₯(β‹…), 1π‘Ÿξ€·π‘’π‘(π‘₯,𝑒)π‘Ÿ(Ξ₯(π‘₯1)βˆ’Ξ₯(π‘₯0))ξ€Έβ‰₯1βˆ’1𝑝π‘₯βˆ‡Ξ₯0π‘’ξ€Έξ€·π‘πœ‚(π‘₯1,π‘₯0)ξ€Έ.βˆ’πŸ(3.13) From inequality (3.12) and the above inequality, we obtain Ξ₯ξ€·π‘₯1ξ€Έξ€·π‘₯β‰₯Ξ₯0ξ€Έ,(3.14) which contradicts (3.7), and proves that π‘₯0 is an optimal solution to (P).
If hypothesis (b) holds, from the 𝐡-(𝑝,π‘Ÿ)-invexity with respect to πœ‚ and 𝑏 at π‘₯0 of Ξ₯1(β‹…), then 1π‘Ÿξ€·π‘’π‘(π‘₯,𝑒)π‘Ÿ(Ξ₯1(π‘₯1)βˆ’Ξ₯1(π‘₯0))ξ€Έβ‰₯1βˆ’1π‘βˆ‡Ξ₯1ξ€·π‘₯0π‘’ξ€Έξ€·π‘πœ‚(π‘₯1,π‘₯0)ξ€Έβˆ’πŸ.(3.15) The above inequality along with (2.7) yields Ξ₯1ξ€·π‘₯1ξ€Έβ‰₯Ξ₯1ξ€·π‘₯0ξ€Έ,(3.16) which contradicts (3.7). Hence, the proof is completed.

4. Parametric Dual-Type Model

We use the optimality conditions of the preceding section and show that the following formation is a dual (D) to the minimax problem (P):max(𝑠,𝑑,𝑦)∈𝐾(𝑧)sup(𝑧,πœ‡,π‘˜,𝑀,𝑣)∈𝐻1(𝑠,𝑑,𝑦)(π‘˜,D) where 𝐻1(𝑠,𝑑,𝑦) denotes the set of (𝑧,πœ‡,π‘˜,𝑀,𝑣)βˆˆβ„π‘›Γ—β„π‘+×ℝ+×ℝ𝑛×ℝ𝑛 satisfying 𝑠𝑖=1π‘‘π‘–ξ€½ξ€·βˆ‡π‘“π‘§,𝑦𝑖+π΅π‘€βˆ’π‘˜βˆ‡β„Žπ‘§,π‘¦π‘–ξ€Έβˆ’π·π‘£ξ€Έξ€Ύ+βˆ‡π‘ξ“π‘—=1πœ‡π‘—π‘”π‘—(𝑧)=0,(4.1)𝑠𝑖=1𝑑𝑖𝑓𝑧,y𝑖+π‘§βŠ€ξ€·β„Žξ€·π΅π‘€βˆ’π‘˜π‘§,π‘¦π‘–ξ€Έβˆ’π‘§βŠ€π·π‘£ξ€Έξ€Ύβ‰₯0,(4.2)𝑝𝑗=1πœ‡π‘—π‘”π‘—π‘€(𝑧)β‰₯0,(4.3)βŠ€π΅π‘€β‰€1,π‘£βŠ€π·π‘£β‰€1,(𝑠,𝑑,𝑦)∈𝐾(𝑧).(4.4) If for a triplet (𝑠,𝑑,𝑦)∈𝐾(𝑧) the set 𝐻1(𝑠,𝑑,𝑦) is empty, then we define the supremum over it to be βˆ’βˆž.

Let Ξ“ denote the set of all feasible points of (D). Moreover, we denote π‘π‘Ÿπ‘‹Ξ“={π‘§βˆˆπ‘‹βˆ£(𝑧,πœ‡,𝑠,𝑑,𝑦,𝑀,𝑣,π‘˜)βˆˆΞ“}.

We can derive the following weak duality theorem between (P) and (D).

Theorem 4.1 (weak duality). Let π‘₯ and (𝑧,πœ‡,𝑠,𝑑,𝑦,𝑀,𝑣,π‘˜) be (𝑃)-feasible and (𝐷)-feasible, respectively. Suppose that any one of the following conditions (π‘Ž) and (𝑏) holds: (a)Ξ₯2βˆ‘(β‹…)=𝑠𝑖=1𝑑𝑖((𝑓(β‹…,𝑦𝑖)+(β‹…)βŠ€π΅π‘€)βˆ’π‘˜(β„Ž(β‹…,𝑦𝑖)βˆ’(β‹…)βŠ€π·π‘£)) is 𝐡-(𝑝,π‘Ÿ)-invex with respect to πœ‚ and 𝑏 at 𝑧 and βˆ‘π‘π‘—=1πœ‡π‘—π‘”π‘—(β‹…) is 𝐡-(𝑝,π‘Ÿ)-invex with respect to the same function πœ‚ and another function 𝑏2 at 𝑧 on π‘‹βˆ˜βˆͺπ‘π‘Ÿπ‘‹Ξ“, not necessarily, equal to 𝑏, (b)Ξ₯3βˆ‘(β‹…)=𝑠𝑖=1𝑑𝑖((𝑓(β‹…,𝑦𝑖)+(β‹…)βŠ€π΅π‘€)βˆ’π‘˜(β„Ž(β‹…,𝑦𝑖)βˆ’(β‹…)βŠ€βˆ‘π·π‘£))+𝑝𝑗=1πœ‡π‘—π‘”π‘—(β‹…) is (𝑝,π‘Ÿ)-invex with respect to πœ‚ and 𝑏 at 𝑧 on π‘‹βˆ˜βˆͺπ‘π‘Ÿπ‘‹Ξ“. Then supπ‘¦βˆˆπ‘Œξ€·π‘₯𝑓(π‘₯,𝑦)+βŠ€ξ€Έπ΅π‘₯1/2ξ€·π‘₯β„Ž(π‘₯,𝑦)βˆ’βŠ€ξ€Έπ·π‘₯1/2β‰₯π‘˜.(4.5)

Proof. Suppose on the contrary that supπ‘¦βˆˆπ‘Œξ€·π‘₯𝑓(π‘₯,𝑦)+βŠ€ξ€Έπ΅π‘₯1/2ξ€·π‘₯β„Ž(π‘₯,𝑦)βˆ’βŠ€ξ€Έπ·π‘₯1/2<π‘˜.(4.6) Then, we have an inequality ξ€·π‘₯𝑓(π‘₯,𝑦)+βŠ€ξ€Έπ΅π‘₯1/2ξ‚€ξ€·π‘₯βˆ’π‘˜β„Ž(π‘₯,𝑦)βˆ’βŠ€ξ€Έπ·π‘₯1/2<0βˆ€π‘¦βˆˆπ‘Œ.(4.7) It follows that for 𝑑𝑖β‰₯0, 𝑖=1,2,…,𝑠 with βˆ‘π‘ π‘–=1𝑑𝑖=1, we have 𝑑𝑖𝑓π‘₯,𝑦𝑖+ξ€·π‘₯βŠ€ξ€Έπ΅π‘₯1/2ξ‚€β„Žξ€·βˆ’π‘˜π‘₯,π‘¦π‘–ξ€Έβˆ’ξ€·π‘₯βŠ€ξ€Έπ·π‘₯1/2≀0,(4.8) with at least one strict inequality because 𝑑=(𝑑1,𝑑2,…,𝑑𝑠)β‰ 0. From relations (2.6), (4.4), (4.8), and (4.2), we obtain Ξ₯2(π‘₯)=𝑠𝑖=1𝑑𝑖𝑓π‘₯,𝑦𝑖+π‘₯βŠ€ξ€·β„Žξ€·π΅π‘€βˆ’π‘˜π‘₯,π‘¦π‘–ξ€Έβˆ’π‘₯βŠ€β‰€π·π‘£ξ€Έξ€Έπ‘ ξ“π‘–=1𝑑𝑖𝑓π‘₯,𝑦𝑖+ξ€·π‘₯βŠ€ξ€Έπ΅π‘₯1/2ξ‚€β„Žξ€·βˆ’π‘˜π‘₯,π‘¦π‘–ξ€Έβˆ’ξ€·π‘₯βŠ€ξ€Έπ·π‘₯1/2≀<0𝑠𝑖=1𝑑𝑖𝑓𝑧,𝑦𝑖+π‘§βŠ€ξ€·β„Žξ€·π΅π‘€βˆ’π‘˜π‘§,π‘¦π‘–ξ€Έβˆ’π‘§βŠ€π·π‘£ξ€Έξ€Έ=Ξ₯2(𝑧).(4.9) That is, Ξ₯2(π‘₯)<Ξ₯2(𝑧).(4.10) From relations (P) and (4.3), we have 𝑝𝑗=1πœ‡π‘—π‘”π‘—(π‘₯)≀0≀𝑝𝑗=1πœ‡π‘—π‘”π‘—(𝑧),(4.11) If hypothesis (a) holds, from 𝐡-(𝑝,π‘Ÿ)-invexity with respect to πœ‚ and 𝑏 at 𝑧 of Ξ₯2(β‹…), we get 1π‘Ÿξ€·e𝑏(π‘₯,𝑧)π‘Ÿ(Ξ₯2(π‘₯)βˆ’Ξ₯2(𝑧))ξ€Έβ‰₯1βˆ’1π‘βˆ‡Ξ₯2𝑒(𝑧)π‘πœ‚(π‘₯,𝑧)ξ€Έ.βˆ’πŸ(4.12) From the above inequality together with relation (4.10), we have 1π‘βˆ‡Ξ₯2𝑒(𝑧)π‘πœ‚(π‘₯,𝑧)ξ€Έβˆ’πŸ<0.(4.13) Multiplying (4.1) by (1/𝑝)(π‘’π‘πœ‚(π‘₯,𝑧)βˆ’πŸ), we obtain 1π‘ξ€·π‘’π‘πœ‚(π‘₯,𝑧)ξ€Έξƒ©βˆ’πŸπ‘ ξ“π‘–=1π‘‘π‘–ξ€½ξ€·βˆ‡π‘“π‘§,𝑦𝑖+π΅π‘€βˆ’π‘˜βˆ‡β„Žπ‘§,π‘¦π‘–ξ€Έβˆ’π·π‘£ξ€Έξ€Ύ+βˆ‡π‘ξ“π‘—=1πœ‡π‘—π‘”π‘—ξƒͺ(𝑧)=0.(4.14) From the above equality and inequality (4.13), we get 1π‘ξ€·π‘’π‘πœ‚(π‘₯,𝑧)ξ€Έβˆ’πŸπ‘ξ“π‘—=1πœ‡π‘—βˆ‡π‘”π‘—(𝑧)>0.(4.15) Using the 𝐡-(𝑝,π‘Ÿ)-invexity of βˆ‘π‘π‘—=1πœ‡π‘—π‘”π‘—(β‹…) with respect to the same function πœ‚ and the function 𝑏2 at 𝑧 and inequality (4.15), we get 𝑝𝑗=1πœ‡π‘—π‘”π‘—(π‘₯)>𝑝𝑗=1πœ‡π‘—π‘”π‘—(𝑧).(4.16) which contradicts (4.11) and proves that supπ‘¦βˆˆπ‘Œ((𝑓(π‘₯,𝑦)+(π‘₯⊀𝐡π‘₯)1/2)/(β„Ž(π‘₯,𝑦)βˆ’(π‘₯⊀𝐷x)1/2))β‰₯π‘˜.
If hypothesis (b) holds, from the 𝐡-(𝑝,π‘Ÿ)-invexity with respect to πœ‚ and 𝑏 at 𝑧 of Ξ₯3(β‹…), then 1π‘Ÿξ€·π‘’π‘(π‘₯,𝑧)π‘Ÿ(Ξ₯3(π‘₯)βˆ’Ξ₯3(𝑧))ξ€Έβ‰₯1βˆ’1π‘βˆ‡Ξ₯3𝑒(𝑧)π‘πœ‚(π‘₯,𝑧)ξ€Έ.βˆ’πŸ(4.17) By the above inequality and equality (4.1), we have Ξ₯3(π‘₯)β‰₯Ξ₯3(𝑧).(4.18) From relations (4.10) and (4.11), we obtain Ξ₯3(π‘₯)<Ξ₯3(𝑧),(4.19) which contradicts inequality (4.18). Thus, the proof is complete.

Theorem 4.2 (strong duality). Let π‘₯βˆ— be an optimal solution of (P), and let π‘₯βˆ— satisfy a constraint qualification for (P). Then there exist (π‘ βˆ—,π‘‘βˆ—,π‘¦βˆ—)∈𝐾(π‘₯βˆ—) and (π‘₯βˆ—,πœ‡βˆ—,π‘˜βˆ—,π‘€βˆ—,π‘£βˆ—)∈𝐻1(π‘ βˆ—,π‘‘βˆ—,π‘¦βˆ—) such that (π‘₯βˆ—,πœ‡βˆ—,π‘ βˆ—,π‘‘βˆ—,π‘¦βˆ—,π‘˜βˆ—,π‘€βˆ—,π‘£βˆ—) is a feasible solution of (D). If in addition the hypothesis of Theorem 4.1 holds, then (π‘₯βˆ—,πœ‡βˆ—,π‘ βˆ—,π‘‘βˆ—,π‘¦βˆ—,π‘˜βˆ—,π‘€βˆ—,π‘£βˆ—) is an optimal solution of (D) and the two problems (P) and (D) have the same optimal value.

Proof. By Theorem 2.2, there exist (π‘ βˆ—,π‘‘βˆ—,π‘¦βˆ—)∈𝐾(π‘₯βˆ—) and (π‘₯βˆ—,πœ‡βˆ—,π‘˜βˆ—,π‘€βˆ—,π‘£βˆ—)∈𝐻1(π‘ βˆ—,π‘‘βˆ—,π‘¦βˆ—) such that (π‘₯βˆ—,πœ‡βˆ—,π‘ βˆ—,π‘‘βˆ—,π‘¦βˆ—,π‘˜βˆ—,π‘€βˆ—,π‘£βˆ—) is feasible for (D), and π‘˜βˆ—=𝑓π‘₯βˆ—,π‘¦βˆ—π‘–ξ€Έ+ξ‚€π‘₯βˆ—βŠ€π΅π‘₯βˆ—ξ‚1/2β„Žξ€·π‘₯βˆ—,π‘¦βˆ—π‘–ξ€Έβˆ’ξ€·π‘₯βˆ—βŠ€π·π‘₯βˆ—ξ€Έ1/2.(4.20) The optimality of this feasible solution for (D) follows from Theorem 4.1.

Theorem 4.3 (strict converse duality). Let π‘₯ and (π‘§βˆ—,πœ‡βˆ—,π‘ βˆ—,π‘‘βˆ—,π‘¦βˆ—,π‘˜βˆ—,π‘€βˆ—π‘£βˆ—) be optimal solutions of (P) and (D), respectively, and assume that the hypothesis of Theorem 4.2 is fulfilled. Suppose that any one of the following conditions (a) and (b) holds: (a)Ξ₯4βˆ‘(β‹…)=π‘ βˆ—π‘–=1π‘‘βˆ—π‘–{𝑓(β‹…,π‘¦βˆ—π‘–)+(β‹…)βŠ€π΅π‘€βˆ—βˆ’π‘˜βˆ—(β„Ž(β‹…,π‘¦βˆ—π‘–)βˆ’(β‹…)βŠ€π·π‘£βˆ—)} is strictly 𝐡-(𝑝,π‘Ÿ)-invex with respect to πœ‚ and 𝑏 at π‘§βˆ— and βˆ‘π‘π‘—=1πœ‡βˆ—π‘—π‘”π‘—(β‹…) is (𝑝,π‘Ÿ)-invex with respect to the same function πœ‚ and another function 𝑏4 at π‘§βˆ— on π‘‹βˆ˜βˆͺπ‘π‘Ÿπ‘‹Ξ“, not necessarily, equal to 𝑏, (b)Ξ₯5βˆ‘(β‹…)=π‘ βˆ—π‘–=1π‘‘βˆ—π‘–{𝑓(β‹…,π‘¦βˆ—π‘–)+(β‹…)βŠ€π΅π‘€βˆ—βˆ’π‘˜βˆ—(β„Ž(β‹…,π‘¦βˆ—π‘–)βˆ’(β‹…)βŠ€π·π‘£βˆ—)}+βˆ‘π‘π‘—=1πœ‡βˆ—π‘—π‘”π‘—(β‹…) is strictly (𝑝,π‘Ÿ)-invex with respect to πœ‚ and 𝑏 at π‘§βˆ— on π‘‹βˆ˜βˆͺπ‘π‘Ÿπ‘‹Ξ“.Then π‘₯=π‘§βˆ—, that is, π‘§βˆ— solves (P) and supπ‘¦βˆˆπ‘Œ((𝑓(π‘§βˆ—,𝑦)+((π‘§βˆ—)βŠ€π΅π‘§βˆ—)1/2)/(β„Ž(π‘§βˆ—,𝑦)βˆ’((π‘§βˆ—)βŠ€π·π‘§βˆ—)1/2))=π‘˜βˆ—.

Proof. We shall assume that π‘₯β‰ π‘§βˆ— and reach a contradiction. From Theorem 4.2, we know that there exist (𝑠,𝑑,𝑦)∈𝐾(π‘₯) and (π‘₯,πœ‡,π‘˜,𝑀,𝑣)∈𝐻1(𝑠,𝑑,𝑦) such that (π‘₯,πœ‡,𝑠,𝑑,𝑦,π‘˜,𝑀,𝑣) is an optimal solution for (D) with the optimal value supπ‘¦βˆˆπ‘Œπ‘“ξ€·ξ€Έ+ξ‚€π‘₯,𝑦π‘₯⊀𝐡π‘₯1/2β„Žξ€·ξ€Έβˆ’ξ‚€π‘₯,𝑦π‘₯⊀𝐷π‘₯1/2=π‘˜.(4.21) Now like the proof of Theorem 4.1 by π‘₯ replacing by π‘₯ and (𝑧,πœ‡,𝑠,𝑑,𝑦,𝑀,𝑣,π‘˜) by (π‘§βˆ—,πœ‡βˆ—,π‘ βˆ—,π‘‘βˆ—,π‘¦βˆ—,π‘˜βˆ—,π‘€βˆ—,π‘£βˆ—), we obtain supπ‘¦βˆˆπ‘Œπ‘“ξ€·ξ€Έ+ξ‚€π‘₯,𝑦π‘₯⊀𝐡π‘₯1/2β„Žξ€·ξ€Έβˆ’ξ‚€π‘₯,𝑦π‘₯⊀𝐷π‘₯1/2>π‘˜βˆ—.(4.22) The above inequality contradicts supπ‘¦βˆˆπ‘Œπ‘“ξ€·ξ€Έ+ξ‚€π‘₯,𝑦π‘₯⊀𝐡π‘₯1/2β„Žξ€·ξ€Έβˆ’ξ‚€π‘₯,𝑦π‘₯⊀𝐷π‘₯1/2=π‘˜βˆ—=π‘˜.(4.23) Therefore, we conclude that π‘₯=π‘§βˆ—. Here, the proof of the theorem is complete.

Remark 4.4. In Theorem 4.3, if βˆ‘π‘π‘—=1πœ‡βˆ—π‘—π‘”π‘—(β‹…) is a strictly 𝐡-(𝑝,π‘Ÿ)-invex function with respect to πœ‚ and 𝑏4 and Ξ₯4(β‹…) is a 𝐡-(𝑝,π‘Ÿ)-invex function with respect to the same function πœ‚ and the function 𝑏, not necessarily, equal to 𝑏4, then Theorem 4.3 also holds.

5. Mond-Weir Dual-Type Model

In this section, we formulate the Mond-Weir-type dual model to the problem (𝑃) as follows:max(𝑠,𝑑,𝑦)∈𝐾(𝑧)sup(𝑧,πœ‡,𝑀,𝑣)∈𝐻2(𝑠,𝑑,𝑦)(β„±(𝑧),MWD) where 𝐻2(𝑠,𝑑,𝑦) denotes the set of (𝑧,πœ‡,𝑀,𝑣)βˆˆβ„π‘›Γ—β„π‘+×ℝ𝑛×ℝ𝑛 satisfying𝑠𝑖=1π‘‘π‘–β„Žξ€·ξ‚†ξ‚€π‘§,π‘¦π‘–ξ€Έβˆ’ξ€·π‘§βŠ€ξ€Έπ·π‘§1/2ξ‚ξ€·ξ€·βˆ‡π‘“π‘§,π‘¦π‘–ξ€Έξ€Έβˆ’ξ‚€π‘“ξ€·+𝐡𝑀𝑧,𝑦𝑖+ξ€·π‘§βŠ€ξ€Έπ΅π‘§1/2ξ‚ξ€·ξ€·βˆ‡β„Žπ‘§,𝑦𝑖+βˆ’π·π‘£π‘ξ“π‘—=1πœ‡π‘—βˆ‡π‘”π‘—(𝑧)=0,(5.1)𝑝𝑗=1πœ‡π‘—π‘”π‘—π‘€(𝑧)β‰₯0,(5.2)βŠ€π΅π‘€β‰€1,π‘£βŠ€ξ€·π‘§π·π‘£β‰€1,𝑧𝐡𝑀=βŠ€ξ€Έπ΅π‘§1/2𝑧𝑧𝐷𝑣=βŠ€ξ€Έπ΅π‘§1/2,(5.3) where β„±(𝑧)=supπ‘¦βˆˆπ‘Œξ€·π‘§π‘“(𝑧,𝑦)+βŠ€ξ€Έπ΅π‘§1/2ξ€·π‘§β„Ž(𝑧,𝑦)βˆ’βŠ€ξ€Έπ·π‘§1/2=supπ‘¦βˆˆπ‘Œπœ™(𝑧,𝑦).(5.4)

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

Let Γ denote the set of all feasible points of (MWD). Moreover, we denote π‘π‘Ÿπ‘‹ξ‚ξ‚Ξ“={π‘§βˆˆπ‘‹βˆ£(𝑧,πœ‡,𝑠,𝑑,𝑦,𝑀,𝑣)βˆˆΞ“}.

We establish the weak, strong, and strict converse duality theorems for (MWD) with respect to the primal problem (P).

Theorem 5.1 (weak duality). Let π‘₯ and (𝑧,πœ‡,s,𝑑,𝑦,𝑀,𝑣) be (𝑃)-feasible and (π‘€π‘Šπ·)-feasible, respectively. Suppose that any one of the following conditions (a) and (b) holds: (a)Ξ₯6βˆ‘(β‹…)=𝑠𝑖=1𝑑𝑖{(β„Ž(𝑧,𝑦𝑖)βˆ’π‘§βŠ€π·π‘£)(𝑓(β‹…,𝑦𝑖)+(β‹…)βŠ€π΅π‘€)βˆ’(𝑓(𝑧,𝑦𝑖)+π‘§βŠ€π΅π‘€)(β„Ž(β‹…,𝑦𝑖)βˆ’(β‹…)βŠ€π·π‘£)} is 𝐡-(𝑝,π‘Ÿ)-invex, respect to πœ‚ and 𝑏 at 𝑧 and βˆ‘π‘π‘—=1πœ‡π‘—π‘”π‘—(β‹…) is 𝐡-(𝑝,π‘Ÿ)-invex with respect to the same function πœ‚ and another function 𝑏6 at 𝑧 on π‘‹βˆ˜βˆͺΓ, not necessarily, equal to 𝑏, (b)Ξ₯7=βˆ‘π‘ π‘–=1𝑑𝑖{(β„Ž(𝑧,𝑦𝑖)βˆ’π‘§βŠ€π·π‘£)(𝑓(β‹…,𝑦𝑖)+(β‹…)βŠ€π΅π‘€)βˆ’(𝑓(𝑧,𝑦𝑖)+π‘§βŠ€π΅π‘€)(β„Ž(β‹…,𝑦𝑖)βˆ’(β‹…)βŠ€π·π‘£)}+βˆ‘π‘π‘—=1πœ‡π‘—π‘”π‘—(β‹…) is 𝐡-(𝑝,π‘Ÿ)-invex with respect to πœ‚ and 𝑏 at 𝑧 on π‘‹βˆ˜βˆͺΓ. Then supπ‘¦βˆˆπ‘Œξ€·π‘₯𝑓(π‘₯,𝑦)+βŠ€ξ€Έπ΅π‘₯1/2ξ€·π‘₯β„Ž(π‘₯,𝑦)βˆ’βŠ€ξ€Έπ·π‘₯1/2β‰₯β„±(𝑧).(5.5)

Proof. On the contrary, if possible, suppose that for each π‘₯βˆˆπ‘‹βˆ˜, supπ‘¦βˆˆπ‘Œξ€·π‘₯𝑓(π‘₯,𝑦)+βŠ€ξ€Έπ΅π‘₯1/2ξ€·π‘₯β„Ž(π‘₯,𝑦)βˆ’βŠ€ξ€Έπ·π‘₯1/2<𝐹(𝑧).(5.6) From the above inequality and π‘¦π‘–βˆˆπ‘Œ(𝑧), 𝑖=1,2,…,𝑠, we obtain 𝑓π‘₯,𝑦𝑖+ξ€·π‘₯βŠ€ξ€Έπ΅π‘₯1/2β„Žξ€·π‘₯,π‘¦π‘–ξ€Έβˆ’ξ€·π‘₯βŠ€ξ€Έπ·π‘₯1/2𝑓<𝐹(𝑧)=𝑧,𝑦𝑖+ξ€·π‘§βŠ€ξ€Έπ΅π‘§1/2β„Žξ€·π‘§,π‘¦π‘–ξ€Έβˆ’ξ€·π‘§βŠ€ξ€Έπ·π‘§1/2,𝑖=1,2,…,𝑠.(5.7) By the above inequality, we know that 𝑓π‘₯,𝑦𝑖+ξ€·π‘₯βŠ€ξ€Έπ΅π‘₯1/2β„Žξ€·ξ‚ξ‚€π‘§,π‘¦π‘–ξ€Έβˆ’ξ€·π‘§βŠ€ξ€Έπ·π‘§1/2ξ‚βˆ’ξ‚€β„Žξ€·π‘₯,π‘¦π‘–ξ€Έβˆ’ξ€·π‘₯βŠ€ξ€Έπ·π‘₯1/2𝑓𝑧,𝑦𝑖+ξ€·π‘§βŠ€ξ€Έπ΅π‘§1/2<0,(5.8) for all 𝑖=1,2,…,𝑠 and π‘¦π‘–βˆˆπ‘Œ(𝑧).
Multiplying the above inequality by π‘‘π‘–βˆˆβ„π‘ + with βˆ‘π‘ π‘–=1𝑑𝑖=1, we have 𝑠𝑖=1𝑑𝑖𝑓π‘₯,𝑦𝑖+ξ€·π‘₯βŠ€ξ€Έπ΅π‘₯1/2β„Žξ€·ξ‚ξ‚€π‘§,π‘¦π‘–ξ€Έβˆ’ξ€·π‘§βŠ€ξ€Έπ·π‘§1/2ξ‚βˆ’ξ‚€β„Žξ€·π‘₯,π‘¦π‘–ξ€Έβˆ’ξ€·π‘₯βŠ€ξ€Έπ·π‘₯1/2𝑓𝑧,𝑦𝑖+ξ€·π‘§βŠ€ξ€Έπ΅π‘§1/2<0.(5.9) From relations (2.6), (5.3), and (5.9), we get Ξ₯6(π‘₯)=𝑠𝑖=1𝑑𝑖𝑓π‘₯,𝑦𝑖+π‘₯βŠ€β„Žξ€·π΅π‘€ξ€Έξ€·π‘§,π‘¦π‘–ξ€Έβˆ’π‘§βŠ€ξ€Έβˆ’ξ€·β„Žξ€·π·π‘£π‘₯,π‘¦π‘–ξ€Έβˆ’π‘₯βŠ€π‘“ξ€·π·π‘£ξ€Έξ€·π‘§,𝑦𝑖+π‘§βŠ€β‰€π΅π‘€ξ€Έξ€Ύπ‘ ξ“π‘–=1𝑑𝑖𝑓π‘₯,𝑦𝑖+ξ€·π‘₯βŠ€ξ€Έπ΅π‘₯1/2β„Žξ€·ξ‚ξ‚€π‘§,π‘¦π‘–ξ€Έβˆ’ξ€·π‘§βŠ€ξ€Έπ·π‘§1/2ξ‚βˆ’ξ‚€β„Žξ€·π‘₯,π‘¦π‘–ξ€Έβˆ’ξ€·π‘₯βŠ€ξ€Έπ·π‘₯1/2𝑓𝑧,𝑦𝑖+ξ€·π‘§βŠ€ξ€Έπ΅π‘§1/2<0=Ξ₯6(𝑧).(5.10) By relations (P) and (5.2), we have 𝑝𝑗=1πœ‡π‘—π‘”π‘—(π‘₯)≀0≀𝑝𝑗=1πœ‡π‘—π‘”π‘—(𝑧).(5.11) Now, if condition (a) holds, from 𝐡-(𝑝,π‘Ÿ)-invexity with respect to πœ‚ and 𝑏6 at 𝑧 of βˆ‘π‘π‘—=1πœ‡π‘—π‘”π‘—(β‹…), we get 1π‘Ÿπ‘1ξ€·π‘₯1,π‘₯0ξ€Έξ‚€π‘’βˆ‘π‘Ÿ(𝑝𝑗=1πœ‡βˆ—π‘—π‘”π‘—βˆ‘(π‘₯)βˆ’π‘π‘—=1πœ‡βˆ—π‘—π‘”π‘—(𝑧))β‰₯1βˆ’1π‘βˆ‡π‘ξ“π‘—=1πœ‡βˆ—π‘—π‘”π‘—ξ€·π‘’(𝑧)π‘πœ‚(π‘₯,𝑧)ξ€Έ.βˆ’πŸ(5.12) From the above inequality and inequality (5.11), we obtain 1π‘βˆ‡π‘ξ“π‘—=1ξ€·πœ‡π‘—π‘”π‘—π‘’(𝑧)ξ€Έξ€·π‘πœ‚(π‘₯,𝑧)ξ€Έβˆ’πŸβ‰€0.(5.13) Multiplying (5.1) by (1/𝑝)(π‘’π‘πœ‚(π‘₯,𝑧)βˆ’πŸ), we have 1π‘ξ€·π‘’π‘πœ‚(π‘₯,𝑧)ξ€Έξƒ―βˆ’πŸπ‘ ξ“π‘–=1π‘‘π‘–β„Žξ€·ξ‚†ξ‚€π‘§,π‘¦π‘–ξ€Έβˆ’ξ€·π‘§βŠ€ξ€Έπ·π‘§1/2ξ‚ξ€·ξ€·βˆ‡π‘“π‘§,π‘¦π‘–ξ€Έξ€Έβˆ’ξ‚€π‘“ξ€·+𝐡𝑀𝑧,𝑦𝑖+ξ€·π‘§βŠ€ξ€Έπ΅π‘§1/2ξ‚ξ€·ξ€·βˆ‡β„Žπ‘§,𝑦𝑖+βˆ’π·π‘£π‘ξ“π‘—=1πœ‡π‘—βˆ‡π‘”π‘—ξƒ°(𝑧)=0.(5.14) By the above equality and inequality (5.13), we obtain 1π‘ξ€·π‘’π‘πœ‚(π‘₯,𝑧)ξ€Έξƒ―βˆ’πŸπ‘ ξ“π‘–=1π‘‘π‘–β„Žξ€·ξ‚†ξ‚€π‘§,π‘¦π‘–ξ€Έβˆ’ξ€·π‘§βŠ€ξ€Έπ·π‘§1/2ξ‚ξ€·ξ€·βˆ‡π‘“π‘§,π‘¦π‘–ξ€Έξ€Έβˆ’ξ‚€π‘“ξ€·+𝐡𝑀𝑧,𝑦𝑖+ξ€·π‘§βŠ€ξ€Έπ΅π‘§1/2ξ‚ξ€·ξ€·βˆ‡β„Žπ‘§,π‘¦π‘–ξ€Έξ€Έξ‚‡ξƒ°βˆ’π·π‘£β‰₯0.(5.15) Using the 𝐡-(𝑝,π‘Ÿ)-invexity with respect to the same function πœ‚ and the function 𝑏 at 𝑧 of Ξ₯6(β‹…) and the above inequality, we have Ξ₯6(π‘₯)β‰₯Ξ₯6(𝑧),(5.16) which contradicts (5.10) and proves that supπ‘¦βˆˆπ‘Œ((𝑓(π‘₯,𝑦)+(π‘₯⊀𝐡π‘₯)1/2)/(β„Ž(π‘₯,𝑦)βˆ’(π‘₯⊀𝐷π‘₯)1/2))β‰₯β„±(𝑧).
If hypothesis (b) holds, from the 𝐡-(𝑝,π‘Ÿ)-invexity with respect to πœ‚ and 𝑏 at 𝑧 of Ξ₯7(β‹…) and the equality (5.1), then Ξ₯7(π‘₯)β‰₯Ξ₯7(𝑧).(5.17) From relations (5.10) and (5.11), we have 𝑠𝑖=1π‘‘π‘–β„Žξ€·ξ€½ξ€·π‘§,π‘¦π‘–ξ€Έβˆ’π‘§βŠ€π‘“ξ€·π·π‘£ξ€Έξ€·π‘₯,𝑦𝑖+π‘₯βŠ€ξ€Έβˆ’ξ€·π‘“ξ€·π΅π‘€π‘§,𝑦𝑖+π‘§βŠ€β„Žξ€·π΅π‘€ξ€Έξ€·π‘₯,π‘¦π‘–ξ€Έβˆ’π‘₯⊀+𝐷𝑣𝑝𝑗=1πœ‡π‘—π‘”π‘—<(π‘₯)𝑠𝑖=1π‘‘π‘–β„Žξ€·ξ€½ξ€·π‘§,π‘¦π‘–ξ€Έβˆ’π‘§βŠ€π‘“ξ€·π·π‘£ξ€Έξ€·π‘§,𝑦𝑖+π‘§βŠ€ξ€Έβˆ’ξ€·π‘“ξ€·π΅π‘€π‘§,𝑦𝑖+π‘§βŠ€β„Žξ€·π΅π‘€ξ€Έξ€·π‘§,π‘¦π‘–ξ€Έβˆ’π‘§βŠ€+𝐷𝑣𝑝𝑗=1πœ‡π‘—π‘”π‘—(𝑧),(5.18) which contradicts inequality (5.17). Hence, the proof is complete.

Similar to the proof of Theorem 4.2, we can establish Theorem 5.2.

Theorem 5.2 (strong duality). Let π‘₯βˆ— be an optimal solution of (P) and let π‘₯βˆ— satisfy a constraint qualification for (P). Then there exist (π‘ βˆ—,π‘‘βˆ—,π‘¦βˆ—)∈𝐾(π‘₯βˆ—) and (π‘₯βˆ—,πœ‡βˆ—,π‘€βˆ—,π‘£βˆ—)∈𝐻1(π‘ βˆ—,π‘‘βˆ—,π‘¦βˆ—) such that (π‘₯βˆ—,πœ‡βˆ—,π‘ βˆ—,π‘‘βˆ—,π‘¦βˆ—,π‘€βˆ—,π‘£βˆ—) is a feasible solution of (MWD). If in addition the hypothesis of Theorem 5.1 holds, then (π‘₯βˆ—,πœ‡βˆ—,π‘ βˆ—,π‘‘βˆ—,π‘¦βˆ—,π‘€βˆ—,π‘£βˆ—) is an optimal solution of (MWD) and the two problems (P) and (MWD) have the same optimal value.

Theorem 5.3 (strict converse duality). Let π‘₯ and (π‘§βˆ—,πœ‡βˆ—,π‘ βˆ—,π‘‘βˆ—,π‘¦βˆ—,π‘€βˆ—,π‘£βˆ—) be optimal solutions of (P) and (MWD), respectively, and assume that the hypothesis of Theorem 5.2 is fulfilled. Suppose that any one of the following conditions (a) and (b) holds: (a)Ξ₯8βˆ‘(β‹…)=π‘ βˆ—π‘–=1π‘‘βˆ—π‘–{(β„Ž(π‘§βˆ—,π‘¦βˆ—π‘–)βˆ’(π‘§βˆ—)βŠ€π·π‘£)(𝑓(β‹…,π‘¦βˆ—π‘–)+(β‹…)βŠ€π΅π‘€)βˆ’(𝑓(π‘§βˆ—,π‘¦βˆ—π‘–)+(π‘§βˆ—)βŠ€π΅π‘€)(β„Ž(β‹…,π‘¦βˆ—π‘–)βˆ’(β‹…)βŠ€π·π‘£)} is strictly 𝐡-(𝑝,π‘Ÿ)-invex with respect to πœ‚ and 𝑏 at π‘§βˆ— and βˆ‘π‘π‘—=1πœ‡βˆ—π‘—π‘”π‘—(β‹…) is 𝐡-(𝑝,π‘Ÿ)-invex with respect to the same function πœ‚ and another function 𝑏8 at π‘§βˆ— on π‘‹βˆ˜βˆͺΓ, not necessarily, equal to 𝑏, (b)Ξ₯9βˆ‘(β‹…)=π‘ βˆ—π‘–=1π‘‘βˆ—π‘–{(β„Ž(π‘§βˆ—,π‘¦βˆ—π‘–)βˆ’(π‘§βˆ—)βŠ€π·π‘£)(𝑓(β‹…,π‘¦βˆ—π‘–)+(β‹…)βŠ€π΅π‘€)βˆ’(𝑓(π‘§βˆ—,π‘¦βˆ—π‘–)+(π‘§βˆ—)βŠ€π΅π‘€)(β„Ž(β‹…,π‘¦βˆ—π‘–)βˆ’(β‹…)βŠ€βˆ‘π·π‘£)}+𝑝𝑗=1πœ‡βˆ—π‘—π‘”π‘—(β‹…) is strictly 𝐡-(𝑝,π‘Ÿ)-invex with respect to πœ‚ and 𝑏 at π‘§βˆ— on π‘‹βˆ˜βˆͺΓ.Then π‘₯=π‘§βˆ—, that is, π‘§βˆ— solves (P) and supπ‘¦βˆˆπ‘Œ((𝑓(π‘§βˆ—,𝑦)+((π‘§βˆ—)βŠ€π΅π‘§βˆ—)1/2)/(β„Ž(π‘§βˆ—,𝑦)βˆ’((π‘§βˆ—)βŠ€π·π‘§βˆ—)1/2))=β„±(π‘§βˆ—).

Proof. We suppose on the contrary that if π‘₯β‰ π‘§βˆ—, then supπ‘¦βˆˆπ‘Œπ‘“ξ€·ξ€Έ+ξ‚€π‘₯,𝑦π‘₯⊀𝐡π‘₯1/2β„Žξ€·ξ€Έβˆ’ξ‚€π‘₯,𝑦π‘₯⊀𝐷π‘₯1/2ξ€·π‘§β‰€β„±βˆ—ξ€Έ.(5.19) Since π‘¦βˆ—π‘–βˆˆπ‘Œ(π‘§βˆ—), for 𝑖=1,2,…,π‘ βˆ—, π‘“ξ€·π‘§βˆ—,π‘¦βˆ—π‘–ξ€Έ+ξ‚€ξ€·π‘§βˆ—ξ€ΈβŠ€π΅π‘§βˆ—ξ‚1/2β„Žξ€·π‘§βˆ—,π‘¦βˆ—π‘–ξ€Έβˆ’ξ€·(π‘§βˆ—)βŠ€π·π‘§βˆ—ξ€Έ1/2𝑧=β„±βˆ—ξ€Έ,for𝑖=1,2,…,π‘ βˆ—.(5.20) It follows that 𝑓+ξ‚€π‘₯,𝑦π‘₯⊀𝐡π‘₯1/2β„Žξ€·π‘§ξ‚Άξ‚΅βˆ—,π‘¦βˆ—π‘–ξ€Έβˆ’ξ‚€ξ€·π‘§βˆ—ξ€ΈβŠ€π·π‘§βˆ—ξ‚1/2ξ‚Άβˆ’ξ‚΅β„Žξ€·ξ€Έβˆ’ξ‚€π‘₯,𝑦π‘₯⊀𝐷π‘₯1/2ξ‚Άξ‚€π‘“ξ€·π‘§βˆ—,π‘¦βˆ—π‘–ξ€Έ+ξ€·π‘§βˆ—ξ€ΈβŠ€π΅π‘§βˆ—ξ‚1/2≀0,for𝑖=1,2,…,π‘ βˆ—,π‘¦βˆˆπ‘Œ.(5.21) From π‘¦βˆ—π‘–βˆˆπ‘Œ(π‘§βˆ—)βŠ‚π‘Œ and π‘‘βˆˆβ„π‘ βˆ—+ with βˆ‘π‘ βˆ—π‘–=1π‘‘βˆ—π‘–=1, we have π‘ βˆ—ξ“π‘–=1π‘‘βˆ—π‘–β„Žξ€·π‘§ξ‚»ξ‚΅βˆ—,π‘¦βˆ—π‘–ξ€Έβˆ’ξ‚€ξ€·π‘§βˆ—ξ€ΈβŠ€π·π‘§βˆ—ξ‚1/2𝑓π‘₯,π‘¦βˆ—π‘–ξ€Έ+ξ‚€π‘₯⊀𝐡π‘₯1/2ξ‚Άβˆ’ξ‚΅β„Žξ€·π‘₯,π‘¦βˆ—π‘–ξ€Έβˆ’ξ‚€π‘₯⊀𝐷π‘₯1/2π‘“ξ€·π‘§ξ‚Άξ‚΅βˆ—,π‘¦βˆ—π‘–ξ€Έ+ξ‚€ξ€·π‘§βˆ—ξ€ΈβŠ€π΅π‘§βˆ—ξ‚1/2≀0.(5.22) From relations (2.6), (5.3), and inequality (5.22), we obtain that Ξ₯8ξ€·π‘₯ξ€Έ=π‘ βˆ—ξ“π‘–=1π‘‘βˆ—π‘–β„Žξ€·π‘§ξ‚†ξ‚€βˆ—,π‘¦βˆ—π‘–ξ€Έβˆ’ξ€·π‘§βˆ—ξ€ΈβŠ€π‘“ξ€·π·π‘£ξ‚ξ‚€π‘₯,π‘¦βˆ—π‘–ξ€Έ+π‘₯βŠ€ξ‚βˆ’ξ‚€π‘“ξ€·π‘§π΅π‘€βˆ—,π‘¦βˆ—π‘–ξ€Έ+ξ€·π‘§βˆ—ξ€ΈβŠ€β„Žξ€·π΅π‘€ξ‚ξ‚€π‘₯,π‘¦βˆ—π‘–ξ€Έβˆ’π‘₯βŠ€β‰€π·π‘£ξ‚ξ‚‡π‘ βˆ—ξ“π‘–=1π‘‘βˆ—π‘–β„Žξ€·π‘§ξ‚»ξ‚΅βˆ—,π‘¦βˆ—π‘–ξ€Έβˆ’ξ‚€π‘₯⊀𝐷π‘₯1/2π‘“ξ€·π‘§ξ‚Άξ‚΅βˆ—,π‘¦βˆ—π‘–ξ€Έ+ξ‚€ξ€·π‘§βˆ—ξ€ΈβŠ€π΅π‘§βˆ—ξ‚1/2ξ‚Άβˆ’ξ‚΅β„Žξ€·π‘₯,π‘¦βˆ—π‘–ξ€Έβˆ’ξ‚€π‘₯⊀𝐷π‘₯1/2π‘“ξ€·π‘§ξ‚Άξ‚΅βˆ—,π‘¦βˆ—π‘–ξ€Έ+ξ‚€ξ€·π‘§βˆ—ξ€ΈβŠ€π΅π‘§βˆ—ξ‚1/2≀0=Ξ₯8ξ€·π‘§βˆ—ξ€Έ.(5.23) By relations (P) and (5.2), we get 𝑝𝑗=1πœ‡βˆ—π‘—π‘”π‘—ξ€·π‘₯≀0≀𝑝𝑗=1πœ‡βˆ—π‘—π‘”π‘—ξ€·π‘§βˆ—ξ€Έ.(5.24) Now, if condition (a) holds, from 𝐡-(𝑝,π‘Ÿ)-invexity with respect to πœ‚ and 𝑏8 at 𝑧 of βˆ‘π‘π‘—=1πœ‡βˆ—π‘—π‘”π‘—(β‹…), then 1π‘Ÿπ‘1ξ€·π‘₯,π‘§βˆ—ξ€Έξ‚€π‘’βˆ‘π‘Ÿ(𝑝𝑗=1πœ‡βˆ—π‘—π‘”π‘—(βˆ‘π‘₯)βˆ’π‘π‘—=1πœ‡βˆ—π‘—π‘”π‘—(π‘§βˆ—))β‰₯1βˆ’1π‘βˆ‡π‘ξ“π‘—=1πœ‡βˆ—π‘—π‘”π‘—ξ€·π‘§βˆ—ξ€Έξ‚€π‘’π‘πœ‚(π‘₯,π‘§βˆ—).βˆ’πŸ(5.25) From the above inequality and inequality (5.24), we know that 1π‘βˆ‡π‘ξ“π‘—=1πœ‡βˆ—π‘—π‘”π‘—ξ€·π‘§βˆ—ξ€Έξ‚€π‘’π‘πœ‚(π‘₯,π‘§βˆ—)ξ‚βˆ’πŸβ‰€0.(5.26) Multiplying (MWD) by (1/𝑝)(π‘’π‘πœ‚(π‘₯,π‘§βˆ—)βˆ’1), we have 1π‘ξ‚€π‘’π‘πœ‚(π‘₯,π‘§βˆ—)ξ‚βŽ§βŽͺ⎨βŽͺβŽ©βˆ’πŸπ‘ βˆ—ξ“π‘–=1π‘‘βˆ—π‘–β„Žξ€·π‘§ξ‚»ξ‚΅βˆ—,π‘¦βˆ—π‘–ξ€Έβˆ’ξ‚€ξ€·π‘§βˆ—ξ€ΈβŠ€π·π‘§βˆ—ξ‚1/2ξ‚Άξ€·ξ€·π‘§βˆ‡π‘“βˆ—,π‘¦βˆ—π‘–ξ€Έ+π΅π‘€βˆ—ξ€Έβˆ’ξ‚΅π‘“ξ€·π‘§βˆ—,π‘¦βˆ—π‘–ξ€Έ+ξ‚€ξ€·π‘§βˆ—ξ€ΈβŠ€π΅π‘§βˆ—ξ‚1/2ξ‚Άξ€·ξ€·π‘§βˆ‡β„Žβˆ—,π‘¦βˆ—π‘–ξ€Έξ€Έξ‚Ό+βˆ’π·π‘£π‘ξ“π‘—=1πœ‡βˆ—π‘—βˆ‡π‘”π‘—ξ€·π‘§βˆ—ξ€Έξƒ°=0.(5.27) Basing on the above inequality and (5.26), we get the inequality 1π‘ξ‚€π‘’π‘πœ‚(π‘₯,π‘§βˆ—)ξ‚βŽ§βŽͺ⎨βŽͺβŽ©βˆ’πŸπ‘ βˆ—ξ“π‘–=1π‘‘βˆ—π‘–β„Žξ€·π‘§ξ‚»ξ‚΅βˆ—,π‘¦βˆ—π‘–ξ€Έβˆ’ξ‚€ξ€·π‘§βˆ—ξ€ΈβŠ€π·π‘§βˆ—ξ‚1/2ξ‚Άξ€·ξ€·π‘§βˆ‡π‘“βˆ—,π‘¦βˆ—π‘–ξ€Έ+π΅π‘€βˆ—ξ€Έβˆ’ξ‚΅π‘“ξ€·π‘§βˆ—,π‘¦βˆ—π‘–ξ€Έ+ξ‚€ξ€·π‘§βˆ—ξ€ΈβŠ€π΅π‘§βˆ—ξ‚1/2ξ‚Άξ€·ξ€·π‘§βˆ‡β„Žβˆ—,π‘¦βˆ—π‘–ξ€Έξ€Έξ‚ΌβŽ«βŽͺ⎬βŽͺβŽ­βˆ’π·π‘£β‰₯0.(5.28) Since Ξ₯8(β‹…) is strictly 𝐡-(𝑝,π‘Ÿ)-invex with respect to the same function πœ‚ and the function 𝑏 at π‘§βˆ— and the above inequality, we obtain Ξ₯8ξ€·π‘§βˆ—ξ€Έ<Ξ₯8ξ€·π‘₯ξ€Έ,(5.29) which contradicts relations (5.23). Hence π‘₯=π‘§βˆ— and π‘§βˆ— is an optimal solution of (P).
If hypothesis (b) holds, from the strict 𝐡-(𝑝,π‘Ÿ)-invexity with respect to πœ‚ and 𝑏 at π‘§βˆ— of Ξ₯9(β‹…) and equality (5.1), then Ξ₯9ξ€·π‘₯ξ€Έ>Ξ₯9ξ€·π‘§βˆ—ξ€Έ.(5.30) Relations (5.23) together with (5.24) yield π‘ βˆ—ξ“π‘–=1π‘‘βˆ—π‘–β„Žξ€·π‘§ξ‚†ξ‚€βˆ—,π‘¦βˆ—π‘–ξ€Έβˆ’ξ€·π‘§βˆ—ξ€ΈβŠ€π‘“ξ€·π·π‘£ξ‚ξ‚€π‘₯,π‘¦βˆ—π‘–ξ€Έ+π‘₯βŠ€ξ‚βˆ’ξ€·π‘“ξ€·π‘§π΅π‘€βˆ—,π‘¦βˆ—π‘–ξ€Έ+π‘§βŠ€ξ€Έξ‚€β„Žξ€·π΅π‘€π‘₯,π‘¦βˆ—π‘–ξ€Έβˆ’π‘₯⊀+𝐷𝑣𝑝𝑗=1πœ‡βˆ—π‘—π‘”π‘—ξ€·π‘₯ξ€Έβ‰€π‘ βˆ—ξ“π‘–=1π‘‘βˆ—π‘–β„Žξ€·π‘§ξ‚†ξ‚€βˆ—,π‘¦βˆ—π‘–ξ€Έβˆ’ξ€·π‘§βˆ—ξ€ΈβŠ€π‘“ξ€·π‘§π·π‘£ξ‚ξ‚€βˆ—,π‘¦βˆ—π‘–ξ€Έ+ξ€·π‘§βˆ—ξ€ΈβŠ€ξ‚βˆ’ξ‚€π‘“ξ€·π‘§π΅π‘€βˆ—,π‘¦βˆ—π‘–ξ€Έ+ξ€·π‘§βˆ—ξ€ΈβŠ€β„Žξ€·π‘§π΅π‘€ξ‚ξ‚€βˆ—,π‘¦βˆ—π‘–ξ€Έβˆ’ξ€·π‘§βˆ—ξ€ΈβŠ€+𝐷𝑣𝑝𝑗=1πœ‡βˆ—π‘—π‘”π‘—ξ€·π‘§βˆ—ξ€Έ,(5.31) which contradicts (5.30). Hence π‘§βˆ—=π‘₯ is also an optimal solution of (P). The proof is complete.

Remark 5.4. In Theorem 5.3, if βˆ‘π‘π‘—=1πœ‡βˆ—π‘—π‘”π‘—(β‹…) is a strictly 𝐡-(𝑝,π‘Ÿ)-invex function with respect to πœ‚ and 𝑏8 and Ξ₯8(β‹…) is a 𝐡-(𝑝,π‘Ÿ)-invex function with respect to the same function πœ‚ and the function 𝑏, not necessarily, equal to 𝑏8, then Theorem 5.3 also holds.

6. Wolfe Dual-Type Model

Based on the result of Theorem 2.2, we can rewrite Theorem 2.2 as follows.

Theorem 6.1 ((necessary conditions) Lai and Lee [11, Theorem 4]). Let π‘₯0 be a (𝑃)-optimal solution, and let βˆ‡π‘”π‘—(π‘₯0), π‘—βˆˆπ½(π‘₯0) be linearly independent. Then there exist (π‘ βˆ—,π‘‘βˆ—,π‘¦βˆ—)∈𝐾(π‘₯0),(𝑀,𝑣)βˆˆβ„π‘›Γ—β„π‘›, and πœ‡βˆ—βˆˆβ„π‘+ such that βˆ‡βŽ›βŽœβŽœβŽβˆ‘π‘ βˆ—π‘–=1π‘‘βˆ—π‘–π‘“ξ€·π‘₯0,π‘¦βˆ—π‘–ξ€Έ+π‘₯⊀0βˆ‘π΅π‘€+𝑝𝑗=1πœ‡βˆ—π‘—π‘”π‘—ξ€·π‘₯0ξ€Έβˆ‘π‘ βˆ—π‘–=1π‘‘βˆ—π‘–ξ€·β„Žξ€·π‘₯0,π‘¦βˆ—π‘–ξ€Έβˆ’π‘₯⊀0ξ€ΈβŽžβŽŸβŽŸβŽ π·π‘£=0,𝑝𝑗=1πœ‡βˆ—π‘—π‘”π‘—ξ€·π‘₯0𝑑=0,βˆ—π‘–β‰₯0,π‘ βˆ—ξ“π‘–=1π‘‘βˆ—π‘–π‘€=1,βŠ€π΅π‘€β‰€1,π‘£βŠ€π‘₯𝐷𝑣≀1,⊀0ξ€·π‘₯𝐡𝑀=⊀0𝐡π‘₯0ξ€Έ1/2,π‘₯⊀0ξ€·π‘₯𝐷𝑣=⊀0𝐡π‘₯0ξ€Έ1/2.(6.1)

In this section, we present the Wolfe dual (WD) to the minimax program (P): max(𝑠,𝑑,𝑦)∈𝐾(𝑧)sup(𝑧,πœ‡,𝑀,𝑣)∈𝐻3(𝑠,𝑑,𝑦)π’œ(𝑧),(WD) where 𝐻3(𝑠,𝑑,𝑦) denotes the set of (𝑧,πœ‡,𝑀,𝑣)βˆˆβ„π‘›Γ—β„π‘+×ℝ𝑛×ℝ𝑛 satisfying βˆ‡ξƒ©βˆ‘π‘ π‘–=1𝑑𝑖𝑓𝑧,𝑦𝑖+π‘§βŠ€ξ€Έ+βˆ‘π΅π‘€π‘π‘—=1πœ‡π‘—π‘”π‘—(𝑧)βˆ‘π‘ π‘–=1π‘‘π‘–ξ€·β„Žξ€·π‘§,π‘¦π‘–ξ€Έβˆ’π‘§βŠ€ξ€Έξƒͺ𝑀𝐷𝑣=0,(𝑠,𝑑,𝑦)∈𝐾,(6.2)βŠ€π΅π‘€β‰€1,π‘£βŠ€π‘§π·π‘£β‰€1,βŠ€ξ€·π‘§π΅π‘€=βŠ€ξ€Έπ΅π‘§1/2,π‘§βŠ€ξ€·π‘§π·π‘£=βŠ€ξ€Έπ΅π‘§1/2,βˆ‘(6.3)π’œ(𝑧)=𝑠𝑖=1𝑑𝑖𝑓𝑧,𝑦𝑖+π‘§βŠ€ξ€Έ+βˆ‘π΅π‘€π‘π‘—=1πœ‡π‘—π‘”π‘—(𝑧)βˆ‘π‘ π‘–=1π‘‘π‘–ξ€·β„Žξ€·π‘§,π‘¦π‘–ξ€Έβˆ’π‘§βŠ€ξ€Έ.𝐷𝑣(6.4) If for a triplet (𝑠,𝑑,𝑦)∈𝐾(𝑧) the set 𝐻3(𝑠,𝑑,𝑦) is empty, then we define the supremum over it to be βˆ’βˆž.

Let Ξ“ denote the set of all feasible points of (WD). Moreover, we denote π‘π‘Ÿπ‘‹Ξ“={π‘§βˆˆπ‘‹βˆ£(𝑧,πœ‡,𝑠,𝑑,𝑦,𝑀,𝑣)βˆˆΞ“}.

We assume throughout this section that βˆ‘π‘ π‘–=1𝑑𝑖(𝑓(𝑧,𝑦𝑖)+π‘§βŠ€βˆ‘π΅π‘€)+𝑝𝑗=1πœ‡π‘—π‘”π‘—(𝑧)β‰₯0 and βˆ‘π‘ π‘–=1𝑑𝑖(β„Ž(𝑧,𝑦𝑖)βˆ’π‘§βŠ€π·π‘£)>0.

We establish the weak, strong, and strict converse duality theorems for (π‘Šπ·) with respect to the primal problem (P).

Theorem 6.2 (weak duality). Let π‘₯ and (𝑧,πœ‡,𝑠,𝑑,𝑦,𝑀,𝑣) be (𝑃)-feasible and (π‘Šπ·)-feasible, respectively, and assume that Ξ₯10(β‹…)=𝑠𝑖=1π‘‘π‘–ξ€·β„Žξ€·π‘§,π‘¦π‘–ξ€Έβˆ’π‘§βŠ€ξ€Έπ·π‘£ξƒ­ξƒ¬π‘ ξ“π‘–=1𝑑𝑖𝑓⋅,𝑦𝑖+(β‹…)βŠ€ξ€Έ+𝐡𝑀𝑝𝑗=1πœ‡π‘—π‘”π‘—ξƒ­βˆ’ξƒ¬(β‹…)𝑠𝑖=1𝑑𝑖𝑓𝑧,𝑦𝑖+π‘§βŠ€ξ€Έ+𝐡𝑀𝑝𝑗=1πœ‡π‘—π‘”π‘—(𝑧)ξƒ