Sufficient Conditions and Duality Theorems for Nondifferentiable Minimax Fractional Programming

Abstract

We consider nondifferentiable minimax fractional programming problems involving 𝐵-(𝑝, 𝑟)-invex functions with respect to 𝜂 and 𝑏. Sufficient optimality conditions and duality results for a class of nondifferentiable minimax fractional programming problems are obtained undr 𝐵-(𝑝, 𝑟)-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 𝐵-(𝑝, 𝑟)-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 𝐵-(𝑝,𝑟)-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 𝐵-(𝑝,𝑟)-invexity assumption on objective and constraint functions.

In this paper, we are inspired to extend the result of Lai et al. [10] to 𝐵-(𝑝,𝑟)-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 𝐵-(𝑝,𝑟)-invex with respect to the same function 𝜂 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 𝐵-(𝑝,𝑟)-invex with respect to the same function 𝜂 and with respect to, not necessarily, the same function 𝑏.

2. Some Notations and Preliminary Results

Let ℝ^𝑛 be the 𝑛-dimensional Euclidean space and ℝ^𝑛+ its nonnegative orthant. Throughout the paper, let 𝑋 be a nonempty open set of ℝ^𝕟.

The following definition can be found in [15].

Definition 2.1 (see [15]). Let 𝑝 and 𝑟 be any real numbers. The differentiable function 𝑓∶𝑋→ℝ is said to be (strictly) 𝐵-(𝑝,𝑟)-invex with respect to 𝜂 and 𝑏 at 𝑢∈𝑋 on a nonempty set 𝑋⊂ℝ^𝑛 if, there exist a function 𝜂∶𝑋×𝑋→ℝ^𝑛 and a function 𝑏∶𝑋×𝑋→ℝ₊⧵{0} such that, for all 𝑥∈𝑋, the inequalities 1𝑟𝑏(𝑥,𝑢)𝑒^{𝑟(𝑓(𝑥)−𝑓(𝑢))}−1≥1𝑝∇𝑓(𝑢)𝑒^{𝑝𝜂(𝑥,𝑢)}𝑏(𝑥,𝑢)(𝑓(𝑥)−𝑓(𝑢))≥∇1𝑓(𝑢)𝜂(𝑥,𝑢)−𝟏((>>iiff𝑥𝑥≠≠𝑢𝑢))ffoorr𝑝𝑝≠=00,,𝑟𝑟≠≠00,,𝑝∇𝑓(𝑢)𝑒^{𝑝𝜂(𝑥,𝑢)}∇𝑓(𝑢)𝜂(𝑥,𝑢)−𝟏((>>iiff𝑥𝑥≠≠𝑢𝑢))ffoorr𝑝𝑝≠=00,,𝑟𝑟==00,(2.1) hold.

𝑓 is said to be 𝐵-(𝑝,𝑟)-invex (strictly 𝐵-(𝑝,𝑟)-invex) with respect to 𝜂 and 𝑏 on 𝑋 if it is 𝐵-(𝑝,𝑟)-invex with respect to the same 𝜂 and 𝑏 at each 𝑢∈𝑋.

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)∈ℝ^𝑛.

We consider the following nondifferentiable minimax fractional programming problem: minimize𝐹(𝑥)=sup_𝑦∈𝑌𝑓(𝑥,𝑦)+𝑥^⊤𝐵𝑥^1/2ℎ(𝑥,𝑦)−𝑥^⊤𝐷𝑥^1/2subjectto𝑔(𝑥)≤0,(P) where 𝑌 is a compact subset of ℝ^𝑚, 𝑓(⋅,⋅)∶ℝ^𝑛×ℝ^𝑚↦ℝ, ℎ(⋅,⋅)∶ℝ^𝑛×ℝ^𝑚↦ℝ, and 𝑔(⋅)∶ℝ^𝑛↦ℝ^𝑝 are 𝐶¹-functions, 𝐵 and 𝐷 are 𝑛×𝑛 positive semidefinite matrices, 𝑓(𝑥,𝑦)+(𝑥^⊤𝐵𝑥)^1/2≥0, and ℎ(𝑥,𝑦)−(𝑥^⊤𝐷𝑥)^1/2>0 for each (𝑥,𝑦) in 𝑋^∘×𝑌, where 𝑋^∘ is the set of feasible solutions of problem (P); that is, 𝑋^∘={𝑥∈𝑋∶𝑔(𝑥)≤0}. This is a nondifferentiable programming problem if either 𝐵 or 𝐷 is nonzero. If 𝐵 and 𝐷 are null matrices, then problem (P) is a minimax fractional programming problem.

For each (𝑥,𝑦)∈ℝ^𝑛×ℝ^𝑚 define 𝜙(𝑥,𝑦)=𝑓(𝑥,𝑦)+𝑥^⊤𝐵𝑥^1/2ℎ(𝑥,𝑦)−𝑥^⊤𝐷𝑥^1/2.(2.2) We let𝐽(𝑥𝐽)=={1𝑗,∈2,𝐽…∣,𝑔𝑝_𝑗},𝑌(𝑥)=𝑦∈𝑌∣𝜙((𝑥𝑥),=𝑦)0=s,up_𝑧∈𝑌𝐾(𝑥)=(𝑠,𝑡,𝑦)∈ℕ×ℝ_𝑠+𝜙(𝑥,𝑧),×ℝ^𝑚𝑠∣1≤𝑠≤𝑛+1,𝑡=𝑡₁,𝑡₂,…,𝑡_𝑠∈ℝ^𝑠+with^𝑠_𝑖=1𝑦=𝑦₁𝑡_𝑖=1,,𝑦₂,…,𝑦_𝑠with𝑦_𝑖∈𝑌(𝑥),𝑖=1,2,…,𝑠.(2.3)

Because 𝑓 and ℎ are contionuous differentiable and 𝑌 is compact subset of ℝ^𝑚, we see that for each 𝑥₀∈𝑋^∘, 𝑌(𝑥₀)≠∅, and for any 𝑦_𝑖∈𝑌(𝑥₀), we have a postive constant 𝑘₀=𝜙𝑥₀,𝑦_𝑖=𝑓𝑥⁰,𝑦_𝑖+𝑥^⊤0𝐵𝑥₀^1/2ℎ𝑥₀,𝑦_𝑖−𝑥^⊤0𝐷𝑥₀^1/2.(2.4)

We will use the generalized Schwarz inequality 𝑥^⊤𝐵𝑣≤𝑥^⊤𝐵𝑥^1/2𝑣^⊤𝐵𝑣^1/2,for𝑥,𝑣∈ℝ^𝑛;(2.5) the equality holds when 𝐵𝑥=𝜆𝐵𝑣, for some 𝜆≥0.

Hence if 𝑣^⊤𝐵𝑣≤1, we have 𝑥^⊤𝐵𝑣≤𝑥^⊤𝐵𝑥^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 𝑥₀ be a (𝑃)-optimal solution and satisfying 𝑥^⊤0𝐵𝑥₀>0, 𝑥^⊤0𝐷𝑥₀>0, and ∇𝑔_𝑗(𝑥₀),𝑗∈𝐽(𝑥₀) is linearly independent. Then there exist (𝑠,𝑡^∗,𝑦)∈𝐾(𝑥₀), 𝑘₀∈ℝ₊, 𝑤,𝑣∈ℝ^𝑛, and 𝜇^∗∈ℝ^𝑝+ such that ^𝑠_𝑖=1𝑡^∗𝑖∇𝑓𝑥₀,𝑦_𝑖+𝐵𝑤−𝑘₀∇ℎ𝑥₀,𝑦_𝑖−𝐷𝑣+∇^𝑝_𝑗=1𝑓𝑥𝜇^∗𝑗𝑔_𝑗𝑥₀=0,(2.7)₀,𝑦_𝑖+𝑥^⊤0𝐵𝑥₀^1/2−𝑘₀ℎ𝑥⁰,𝑦_𝑖−𝑥^⊤0𝐷𝑥₀^1/2=0,𝑖=1,2,…,𝑠,(2.8)_𝑝_𝑗=1𝑡𝜇^∗𝑗𝑔_𝑗𝑥₀=0,(2.9)_∗𝑖≥0,^𝑠_𝑖=1𝑤𝑡^∗𝑖=1,(2.10)_⊤𝑥𝐵𝑤≤1,𝑣^⊤𝐷𝑣≤1,_⊤0𝑥_⊤0𝐵𝑤=𝑥^⊤0𝐵𝑥₀^1/2,𝐷𝑣=𝑥^⊤0𝐵𝑥₀^1/2.(2.11)

It should be noted that both the matrices 𝐵 and 𝐷 are positive definite at the solution 𝑥₀ in the above theorem. If one of 𝑥^⊤0𝐵𝑥₀ and 𝑥^⊤0𝐷𝑥₀ is zero, or both 𝐵 and 𝐷 are singular at 𝑥₀, then, for (𝑠,𝑡^∗,𝑦)∈𝐾(𝑥₀), we define a set 𝑍_𝑦(𝑥₀) by 𝑍=_𝑦𝑥𝑧₀∈ℝ_𝑛∣𝑧^⊤∇𝑔_𝑗𝑥₀≤0,𝑗∈𝐽𝑥₀,withanyoneofthenextconditions(i)-(iii)holds.(2.12)

Here conditions (i)–(iii) are given as follows:(i)if 𝑥^⊤0𝐵𝑥₀>0 and 𝑥^⊤0𝐷𝑥₀=0, then 𝑧^⊤^𝑠_𝑖=1𝑡^∗𝑖∇𝑓𝑥⁰,𝑦_𝑖+𝐵𝑥⁰𝑥^⊤0𝐵𝑥₀^1/2−𝑘₀∇ℎ𝑥₀,𝑦_𝑖+𝑧^⊤𝑘²⁰𝐷𝑧^1/2<0,(2.13)(ii)if 𝑥^⊤0𝐵𝑥₀=0 and 𝑥^⊤0𝐷𝑥₀>0, then 𝑧^⊤^𝑠_𝑖=1𝑡^∗𝑖∇𝑓𝑥⁰,𝑦_𝑖−𝑘₀∇ℎ𝑥⁰,𝑦_𝑖−𝐷𝑥⁰𝑥^⊤0𝐷𝑥₀^1/2+𝑧^⊤𝐵𝑧^1/2<0,(2.14)(iii)if 𝑥^⊤0𝐵𝑥₀=0 and 𝑥^⊤0𝐷𝑥₀=0, then 𝑧^⊤^𝑠_𝑖=1𝑡^∗𝑖∇𝑓𝑥₀,𝑦_𝑖−𝑘₀∇ℎ𝑥₀,𝑦_𝑖+𝑧^⊤𝑘₀𝐷𝑧^1/2+𝑧^⊤𝐵𝑧^1/2<0.(2.15)

If we take condition 𝑍_𝑦(𝑥₀)=∅ 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 𝐵-(𝑝,𝑟)-invexity. All theorems in this work will be proved only in the case when 𝑝≠0, 𝑟≠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 𝐵-(𝑝,𝑟)-invex functions with respect to 𝜂 and 𝑏 for given 𝑝 and 𝑟). The proofs of the other cases are easier than this one.

We would establish the sufficient conditions under the 𝐵-(𝑝,𝑟)-invex function.

Theorem 3.1 (sufficient optimality conditions). Let 𝑥₀∈𝑋^∘ be a feasible solution of (P). There exist a positive interger 𝑠, 1≤𝑠≤𝑛+1, 𝑡^∗∈ℝ^𝑠, 𝑦_𝑖∈𝑌(𝑥₀)(𝑖=1,2,…,𝑠), 𝑘₀∈ℝ₊, (𝑤,𝑣)∈ℝ^𝑛×ℝ^𝑛, and 𝜇^∗∈ℝ^𝑝+ to satisfy relations (2.7)~(2.11). Furthermore suppose that any one of conditions (a) and (b) holds: (a)Υ(⋅)=∑^𝑠𝑖=1𝑡^∗𝑖((𝑓(⋅,𝑦_𝑖)+(⋅)^⊤𝐵𝑤)−𝑘₀(ℎ(⋅,𝑦_𝑖)−(⋅)^⊤𝐷𝑣)) is 𝐵-(𝑝,𝑟)-invex with respect to 𝜂 and 𝑏 at 𝑥₀, and ∑^𝑝𝑗=1𝜇^∗𝑗𝑔_𝑗(⋅) is 𝐵-(𝑝,𝑟)-invex with respect to the same function 𝜂 and another function 𝑏₁ at 𝑥₀ on 𝑋^∘, not necessarily, equal to 𝑏, (b)Υ₁(⋅)=∑^𝑠𝑖=1𝑡^∗𝑖((𝑓(⋅,𝑦_𝑖)+(⋅)^⊤𝐵𝑤)−𝑘₀(ℎ(⋅,𝑦_𝑖)−(⋅)^⊤𝐷𝑣))+∑^𝑝𝑗=1𝜇^∗𝑗𝑔_𝑗(⋅) is 𝐵-(𝑝,𝑟)-invex with respect to 𝜂 and 𝑏 at 𝑥₀ on 𝑋^∘,then 𝑥₀ is an optimal solution of (P).

Proof. Suppose that 𝑥₀ is not an optimal solution of (P). Then there exists a (𝑃)-feasible solution 𝑥₁ such that sup_𝑦∈𝑌𝑓𝑥₁,𝑦+𝑥^⊤1𝐵𝑥₁^1/2ℎ𝑥₁,𝑦−𝑥^⊤1𝐷𝑥₁^1/2<sup_𝑦∈𝑌𝑓𝑥₀,𝑦+𝑥^⊤0𝐵𝑥₀^1/2ℎ𝑥₀,𝑦−𝑥^⊤0𝐷𝑥₀^1/2.(3.1) We note that sup_𝑦∈𝑌𝑓𝑥₀,𝑦+𝑥^⊤0𝐵𝑥₀^1/2ℎ𝑥₀,𝑦−𝑥^⊤0𝐷𝑥₀^1/2=𝑓𝑥⁰,𝑦_𝑖+𝑥^⊤0𝐵𝑥₀^1/2ℎ𝑥₀,𝑦_𝑖−𝑥^⊤0𝐷𝑥₀^1/2=𝑘₀,(3.2) for 𝑦_𝑖∈𝑌(𝑥₀), 𝑖=1,2,…,𝑠, and 𝑓𝑥₁,𝑦_𝑖+𝑥^⊤1𝐵𝑥₁^1/2ℎ𝑥₁,𝑦_𝑖−𝑥^⊤1𝐷𝑥₁^1/2≤sup_𝑦∈𝑌𝑓𝑥₁,𝑦+𝑥^⊤1𝐵𝑥₁^1/2ℎ𝑥₁,𝑦−𝑥^⊤1𝐷𝑥₁^1/2.(3.3) Then, we obtain 𝑓𝑥₁,𝑦_𝑖+𝑥^⊤1𝐵𝑥₁^1/2ℎ𝑥₁,𝑦_𝑖−𝑥^⊤1𝐷𝑥₁^1/2<𝑘₀,for𝑖=1,2,…,𝑠.(3.4) It follows that 𝑓𝑥₁,𝑦_𝑖+𝑥^⊤1𝐵𝑥₁^1/2−𝑘₀ℎ𝑥¹,𝑦_𝑖−𝑥^⊤1𝐷𝑥₁^1/2<0,for𝑖=1,2,…,𝑠.(3.5) From relations (2.6), (2.11), (2.8), (2.10), and (3.5), we have Υ𝑥₁=^𝑠_𝑖=1≤𝑡^∗𝑖𝑓𝑥₁,𝑦_𝑖+𝑥^⊤1𝐵𝑤−𝑘₀ℎ𝑥₁,𝑦_𝑖−𝑥^⊤1𝐷𝑣_𝑠_𝑖=1=𝑡^∗𝑖𝑓𝑥¹,𝑦_𝑖+𝑥^⊤1𝐵𝑥₁^1/2−𝑘₀ℎ𝑥¹,𝑦_𝑖−𝑥^⊤1𝐷𝑥₁^1/2<0_𝑠_𝑖=1=𝑡^∗𝑖𝑓𝑥⁰,𝑦_𝑖+𝑥^⊤0𝐵𝑥₀^1/2−𝑘₀ℎ𝑥⁰,𝑦_𝑖−𝑥^⊤0𝐷𝑥₀^1/2_𝑠_𝑖=1𝑡^∗𝑖𝑓𝑥₀,𝑦_𝑖+𝑥^⊤0𝐵𝑤−𝑘₀ℎ𝑥₀,𝑦_𝑖−𝑥^⊤0𝐷𝑣=Υ𝑥₀.(3.6) That is, Υ𝑥₁<Υ𝑥₀.(3.7) From relations (P) and (2.9), we obtain ^𝑝_𝑗=1𝜇^∗𝑗𝑔_𝑗𝑥₁≤0=^𝑝_𝑗=1𝜇^∗𝑗𝑔_𝑗𝑥₀.(3.8) If hypothesis (a) holds, from the 𝐵-(𝑝,𝑟)-invexity with respect to 𝜂 and 𝑏₁ at 𝑥₀ of ∑^𝑝𝑗=1𝜇^∗𝑗𝑔_𝑗(⋅), we have 1𝑟𝑏¹𝑥₁,𝑥₀𝑒^{∑𝑟(𝑝𝑗=1𝜇∗𝑗𝑔𝑗(𝑥1∑)−𝑝𝑗=1𝜇∗𝑗𝑔𝑗(𝑥0))}−1≥1𝑝∇^𝑝_𝑗=1𝜇^∗𝑗𝑔_𝑗𝑥₀𝑒^{𝑝𝜂(𝑥1,𝑥0)}−𝟏.(3.9) From the inequalities (3.8) and (3.9), we get 1𝑝∇^𝑝_𝑗=1𝜇^∗𝑗𝑔_𝑗𝑥₀𝑒^{𝑝𝜂(𝑥1,𝑥0)}−𝟏≤0.(3.10) Now, multiplying equality (2.7) by (1/𝑝)(𝑒^{𝑝𝜂(𝑥1,𝑥0)}−𝟏), we know 1𝑝𝑒^{𝑝𝜂(𝑥1,𝑥0)}−𝟏^𝑠_𝑖=1𝑡^∗𝑖∇𝑓𝑥₀,𝑦_𝑖+𝐵𝑤−𝑘₀∇ℎ𝑥₀,𝑦_𝑖−𝐷𝑣+∇^𝑝_𝑗=1𝜇^∗𝑗𝑔_j𝑥₀=0.(3.11) From relations (3.10) and (3.11), we have 1𝑝𝑒^{𝑝𝜂(𝑥1,𝑥0)}−𝟏^𝑠_𝑖=1𝑡^∗𝑖∇𝑓𝑥₀,𝑦_𝑖+𝐵𝑤−𝑘₀∇ℎ𝑥₀,𝑦_𝑖−𝐷𝑣≥0.(3.12) From the 𝐵-(𝑝,𝑟)-invexity with respect to the same function 𝜂 and the function 𝑏 at 𝑥₀ of Υ(⋅), 1𝑟𝑏(𝑥,𝑢)𝑒^{𝑟(Υ(𝑥1)−Υ(𝑥0))}−1≥1𝑝∇Υ𝑥⁰𝑒^{𝑝𝜂(𝑥1,𝑥0)}−𝟏.(3.13) From inequality (3.12) and the above inequality, we obtain Υ𝑥₁≥Υ𝑥₀,(3.14) which contradicts (3.7), and proves that 𝑥₀ is an optimal solution to (P).
If hypothesis (b) holds, from the 𝐵-(𝑝,𝑟)-invexity with respect to 𝜂 and 𝑏 at 𝑥₀ of Υ₁(⋅), then 1𝑟𝑏(𝑥,𝑢)𝑒^{𝑟(Υ1(𝑥1)−Υ1(𝑥0))}−1≥1𝑝∇Υ¹𝑥₀𝑒^{𝑝𝜂(𝑥1,𝑥0)}−𝟏.(3.15) The above inequality along with (2.7) yields Υ₁𝑥₁≥Υ₁𝑥₀,(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,𝑡,𝑦)∈𝐾(𝑧)}sup_{(𝑧,𝜇,𝑘,𝑤,𝑣)∈𝐻1(𝑠,𝑡,𝑦)}𝑘,(D) where 𝐻₁(𝑠,𝑡,𝑦) denotes the set of (𝑧,𝜇,𝑘,𝑤,𝑣)∈ℝ^𝑛×ℝ^𝑝+×ℝ₊×ℝ^𝑛×ℝ^𝑛 satisfying ^𝑠_𝑖=1𝑡_𝑖∇𝑓𝑧,𝑦_𝑖+𝐵𝑤−𝑘∇ℎ𝑧,𝑦_𝑖−𝐷𝑣+∇^𝑝_𝑗=1𝜇_𝑗𝑔_𝑗(𝑧)=0,(4.1)_𝑠_𝑖=1𝑡_𝑖𝑓𝑧,y_𝑖+𝑧^⊤𝐵𝑤−𝑘ℎ𝑧,𝑦_𝑖−𝑧^⊤𝐷𝑣≥0,(4.2)_𝑝_𝑗=1𝑤𝜇_𝑗𝑔_𝑗(𝑧)≥0,(4.3)_⊤𝐵𝑤(𝑠,≤𝑡,1𝑦,)∈𝐾𝑣^⊤(𝑧𝐷)𝑣.≤1,(4.4) If for a triplet (𝑠,𝑡,𝑦)∈𝐾(𝑧) the set 𝐻₁(𝑠,𝑡,𝑦) is empty, then we define the supremum over it to be −∞.

Let Γ denote the set of all feasible points of (D). Moreover, we denote 𝑝𝑟_𝑋Γ={𝑧∈𝑋∣(𝑧,𝜇,𝑠,𝑡,𝑦,𝑤,𝑣,𝑘)∈Γ}.

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

Theorem 4.1 (weak duality). Let 𝑥 and (𝑧,𝜇,𝑠,𝑡,𝑦,𝑤,𝑣,𝑘) be (𝑃)-feasible and (𝐷)-feasible, respectively. Suppose that any one of the following conditions (𝑎) and (𝑏) holds: (a)Υ₂(⋅)=∑^𝑠𝑖=1𝑡_𝑖((𝑓(⋅,𝑦_𝑖)+(⋅)^⊤𝐵𝑤)−𝑘(ℎ(⋅,𝑦_𝑖)−(⋅)^⊤𝐷𝑣)) is 𝐵-(𝑝,𝑟)-invex with respect to 𝜂 and 𝑏 at 𝑧 and ∑^𝑝𝑗=1𝜇_𝑗𝑔_𝑗(⋅) is 𝐵-(𝑝,𝑟)-invex with respect to the same function 𝜂 and another function 𝑏₂ at 𝑧 on 𝑋^∘∪𝑝𝑟_𝑋Γ, not necessarily, equal to 𝑏, (b)Υ₃(⋅)=∑^𝑠𝑖=1𝑡_𝑖((𝑓(⋅,𝑦_𝑖)+(⋅)^⊤𝐵𝑤)−𝑘(ℎ(⋅,𝑦_𝑖)−(⋅)^⊤𝐷𝑣))+∑^𝑝𝑗=1𝜇_𝑗𝑔_𝑗(⋅) is (𝑝,𝑟)-invex with respect to 𝜂 and 𝑏 at 𝑧 on 𝑋^∘∪𝑝𝑟_𝑋Γ. Then sup_𝑦∈𝑌𝑓(𝑥,𝑦)+𝑥^⊤𝐵𝑥^1/2ℎ(𝑥,𝑦)−𝑥^⊤𝐷𝑥^1/2≥𝑘.(4.5)

Proof. Suppose on the contrary that sup_𝑦∈𝑌𝑓(𝑥,𝑦)+𝑥^⊤𝐵𝑥^1/2ℎ(𝑥,𝑦)−𝑥^⊤𝐷𝑥^1/2<𝑘.(4.6) Then, we have an inequality 𝑓(𝑥,𝑦)+𝑥^⊤𝐵𝑥^1/2−𝑘ℎ(𝑥,𝑦)−𝑥^⊤𝐷𝑥^1/2<0∀𝑦∈𝑌.(4.7) It follows that for 𝑡_𝑖≥0, 𝑖=1,2,…,𝑠 with ∑^𝑠𝑖=1𝑡_𝑖=1, we have 𝑡_𝑖𝑓𝑥,𝑦^𝑖+𝑥^⊤𝐵𝑥^1/2−𝑘ℎ𝑥,𝑦_𝑖−𝑥^⊤𝐷𝑥^1/2≤0,(4.8) with at least one strict inequality because 𝑡=(𝑡₁,𝑡₂,…,𝑡_𝑠)≠0. From relations (2.6), (4.4), (4.8), and (4.2), we obtain Υ₂(𝑥)=^𝑠_𝑖=1≤𝑡_𝑖𝑓𝑥,𝑦_𝑖+𝑥^⊤𝐵𝑤−𝑘ℎ𝑥,𝑦_𝑖−𝑥^⊤𝐷𝑣_𝑠_𝑖=1≤𝑡_𝑖𝑓𝑥,𝑦^𝑖+𝑥^⊤𝐵𝑥^1/2−𝑘ℎ𝑥,𝑦_𝑖−𝑥^⊤𝐷𝑥^1/2<0_𝑠_𝑖=1𝑡_𝑖𝑓𝑧,𝑦_𝑖+𝑧^⊤𝐵𝑤−𝑘ℎ𝑧,𝑦_𝑖−𝑧^⊤𝐷𝑣=Υ₂(𝑧).(4.9) That is, Υ₂(𝑥)<Υ₂(𝑧).(4.10) From relations (P) and (4.3), we have ^𝑝_𝑗=1𝜇_𝑗𝑔_𝑗(𝑥)≤0≤^𝑝_𝑗=1𝜇_𝑗𝑔_𝑗(𝑧),(4.11) If hypothesis (a) holds, from 𝐵-(𝑝,𝑟)-invexity with respect to 𝜂 and 𝑏 at 𝑧 of Υ₂(⋅), we get 1𝑟𝑏(𝑥,𝑧)e^{𝑟(Υ2(𝑥)−Υ2(𝑧))}−1≥1𝑝∇Υ²(𝑧)𝑒^{𝑝𝜂(𝑥,𝑧)}−𝟏.(4.12) From the above inequality together with relation (4.10), we have 1𝑝∇Υ²(𝑧)𝑒^{𝑝𝜂(𝑥,𝑧)}−𝟏<0.(4.13) Multiplying (4.1) by (1/𝑝)(𝑒^{𝑝𝜂(𝑥,𝑧)}−𝟏), we obtain 1𝑝𝑒^{𝑝𝜂(𝑥,𝑧)}−𝟏^𝑠_𝑖=1𝑡_𝑖∇𝑓𝑧,𝑦_𝑖+𝐵𝑤−𝑘∇ℎ𝑧,𝑦_𝑖−𝐷𝑣+∇^𝑝_𝑗=1𝜇_𝑗𝑔_𝑗(𝑧)=0.(4.14) From the above equality and inequality (4.13), we get 1𝑝𝑒^{𝑝𝜂(𝑥,𝑧)}−𝟏^𝑝_𝑗=1𝜇_𝑗∇𝑔_𝑗(𝑧)>0.(4.15) Using the 𝐵-(𝑝,𝑟)-invexity of ∑^𝑝𝑗=1𝜇_𝑗𝑔_𝑗(⋅) with respect to the same function 𝜂 and the function 𝑏₂ at 𝑧 and inequality (4.15), we get ^𝑝_𝑗=1𝜇_𝑗𝑔_𝑗(𝑥)>^𝑝_𝑗=1𝜇_𝑗𝑔_𝑗(𝑧).(4.16) which contradicts (4.11) and proves that sup_𝑦∈𝑌((𝑓(𝑥,𝑦)+(𝑥^⊤𝐵𝑥)^1/2)/(ℎ(𝑥,𝑦)−(𝑥^⊤𝐷x)^1/2))≥𝑘.
If hypothesis (b) holds, from the 𝐵-(𝑝,𝑟)-invexity with respect to 𝜂 and 𝑏 at 𝑧 of Υ₃(⋅), then 1𝑟𝑏(𝑥,𝑧)𝑒^{𝑟(Υ3(𝑥)−Υ3(𝑧))}−1≥1𝑝∇Υ³(𝑧)𝑒^{𝑝𝜂(𝑥,𝑧)}−𝟏.(4.17) By the above inequality and equality (4.1), we have Υ₃(𝑥)≥Υ₃(𝑧).(4.18) From relations (4.10) and (4.11), we obtain Υ₃(𝑥)<Υ₃(𝑧),(4.19) which contradicts inequality (4.18). Thus, the proof is complete.

Theorem 4.2 (strong duality). Let 𝑥^∗ be an optimal solution of (P), and let 𝑥^∗ satisfy a constraint qualification for (P). Then there exist (𝑠^∗,𝑡^∗,𝑦^∗)∈𝐾(𝑥^∗) and (𝑥^∗,𝜇^∗,𝑘^∗,𝑤^∗,𝑣^∗)∈𝐻₁(𝑠^∗,𝑡^∗,𝑦^∗) such that (𝑥^∗,𝜇^∗,𝑠^∗,𝑡^∗,𝑦^∗,𝑘^∗,𝑤^∗,𝑣^∗) is a feasible solution of (D). If in addition the hypothesis of Theorem 4.1 holds, then (𝑥^∗,𝜇^∗,𝑠^∗,𝑡^∗,𝑦^∗,𝑘^∗,𝑤^∗,𝑣^∗) 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 (𝑠^∗,𝑡^∗,𝑦^∗)∈𝐾(𝑥^∗) and (𝑥^∗,𝜇^∗,𝑘^∗,𝑤^∗,𝑣^∗)∈𝐻₁(𝑠^∗,𝑡^∗,𝑦^∗) such that (𝑥^∗,𝜇^∗,𝑠^∗,𝑡^∗,𝑦^∗,𝑘^∗,𝑤^∗,𝑣^∗) is feasible for (D), and 𝑘^∗=𝑓𝑥^∗,𝑦^∗𝑖+𝑥^∗⊤𝐵𝑥^∗^1/2ℎ𝑥^∗,𝑦^∗𝑖−𝑥^∗⊤𝐷𝑥^∗^1/2.(4.20) The optimality of this feasible solution for (D) follows from Theorem 4.1.

Theorem 4.3 (strict converse duality). Let 𝑥 and (𝑧^∗,𝜇^∗,𝑠^∗,𝑡^∗,𝑦^∗,𝑘^∗,𝑤^∗𝑣^∗) 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)Υ₄(⋅)=∑^𝑠∗_𝑖=1𝑡^∗𝑖{𝑓(⋅,𝑦^∗𝑖)+(⋅)^⊤𝐵𝑤^∗−𝑘^∗(ℎ(⋅,𝑦^∗𝑖)−(⋅)^⊤𝐷𝑣^∗)} is strictly 𝐵-(𝑝,𝑟)-invex with respect to 𝜂 and 𝑏 at 𝑧^∗ and ∑^𝑝𝑗=1𝜇^∗𝑗𝑔_𝑗(⋅) is (𝑝,𝑟)-invex with respect to the same function 𝜂 and another function 𝑏₄ at 𝑧^∗ on 𝑋^∘∪𝑝𝑟_𝑋Γ, not necessarily, equal to 𝑏, (b)Υ₅(⋅)=∑^𝑠∗_𝑖=1𝑡^∗𝑖{𝑓(⋅,𝑦^∗𝑖)+(⋅)^⊤𝐵𝑤^∗−𝑘^∗(ℎ(⋅,𝑦^∗𝑖)−(⋅)^⊤𝐷𝑣^∗)}+∑^𝑝𝑗=1𝜇^∗𝑗𝑔_𝑗(⋅) is strictly (𝑝,𝑟)-invex with respect to 𝜂 and 𝑏 at 𝑧^∗ on 𝑋^∘∪𝑝𝑟_𝑋Γ.Then 𝑥=𝑧^∗, that is, 𝑧^∗ solves (P) and sup_𝑦∈𝑌((𝑓(𝑧^∗,𝑦)+((𝑧^∗)^⊤𝐵𝑧^∗)^1/2)/(ℎ(𝑧^∗,𝑦)−((𝑧^∗)^⊤𝐷𝑧^∗)^1/2))=𝑘^∗.

Proof. We shall assume that 𝑥≠𝑧^∗ and reach a contradiction. From Theorem 4.2, we know that there exist (𝑠,𝑡,𝑦)∈𝐾(𝑥) and (𝑥,𝜇,𝑘,𝑤,𝑣)∈𝐻₁(𝑠,𝑡,𝑦) such that (𝑥,𝜇,𝑠,𝑡,𝑦,𝑘,𝑤,𝑣) is an optimal solution for (D) with the optimal value sup_𝑦∈𝑌𝑓𝑥,𝑦+𝑥^⊤𝐵𝑥^1/2ℎ𝑥,𝑦−𝑥^⊤𝐷𝑥^1/2=𝑘.(4.21) Now like the proof of Theorem 4.1 by 𝑥 replacing by 𝑥 and (𝑧,𝜇,𝑠,𝑡,𝑦,𝑤,𝑣,𝑘) by (𝑧^∗,𝜇^∗,𝑠^∗,𝑡^∗,𝑦^∗,𝑘^∗,𝑤^∗,𝑣^∗), we obtain sup_𝑦∈𝑌𝑓𝑥,𝑦+𝑥^⊤𝐵𝑥^1/2ℎ𝑥,𝑦−𝑥^⊤𝐷𝑥^1/2>𝑘^∗.(4.22) The above inequality contradicts sup_𝑦∈𝑌𝑓𝑥,𝑦+𝑥^⊤𝐵𝑥^1/2ℎ𝑥,𝑦−𝑥^⊤𝐷𝑥^1/2=𝑘^∗=𝑘.(4.23) Therefore, we conclude that 𝑥=𝑧^∗. Here, the proof of the theorem is complete.

Remark 4.4. In Theorem 4.3, if ∑^𝑝𝑗=1𝜇^∗𝑗𝑔_𝑗(⋅) is a strictly 𝐵-(𝑝,𝑟)-invex function with respect to 𝜂 and 𝑏₄ and Υ₄(⋅) is a 𝐵-(𝑝,𝑟)-invex function with respect to the same function 𝜂 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,𝑡,𝑦)∈𝐾(𝑧)}sup_{(𝑧,𝜇,𝑤,𝑣)∈𝐻2(𝑠,𝑡,𝑦)}ℱ(𝑧),(MWD) where 𝐻₂(𝑠,𝑡,𝑦) denotes the set of (𝑧,𝜇,𝑤,𝑣)∈ℝ^𝑛×ℝ^𝑝+×ℝ^𝑛×ℝ^𝑛 satisfying^𝑠_𝑖=1𝑡_𝑖−ℎ𝑓𝑧,𝑧𝑦,_𝑖𝑦_𝑖−𝑧^⊤𝐷𝑧^1/2∇𝑓𝑧,𝑦^𝑖+𝐵𝑤+𝑧^⊤𝐵𝑧^1/2∇ℎ𝑧,𝑦^𝑖−𝐷𝑣+^𝑝_𝑗=1𝜇_𝑗∇𝑔_𝑗(𝑧)=0,(5.1)_𝑝_𝑗=1𝑤𝜇_𝑗𝑔_𝑗(𝑧)≥0,(5.2)_⊤𝐵𝑧𝑤𝐵𝑤≤=1,𝑧_⊤𝑣^⊤𝐷𝑣≤1,𝐵𝑧^1/2𝑧𝐷𝑣=𝑧^⊤𝐵𝑧^1/2,(5.3) where ℱ(𝑧)=sup_𝑦∈𝑌𝑓(𝑧,𝑦)+𝑧^⊤𝐵𝑧^1/2ℎ(𝑧,𝑦)−𝑧^⊤𝐷𝑧^1/2=sup_𝑦∈𝑌𝜙(𝑧,𝑦).(5.4)

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

Let Γ denote the set of all feasible points of (MWD). Moreover, we denote 𝑝𝑟_𝑋Γ={𝑧∈𝑋∣(𝑧,𝜇,𝑠,𝑡,𝑦,𝑤,𝑣)∈Γ}.

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 𝑥 and (𝑧,𝜇,s,𝑡,𝑦,𝑤,𝑣) be (𝑃)-feasible and (𝑀𝑊𝐷)-feasible, respectively. Suppose that any one of the following conditions (a) and (b) holds: (a)Υ₆(⋅)=∑^𝑠𝑖=1𝑡_𝑖{(ℎ(𝑧,𝑦_𝑖)−𝑧^⊤𝐷𝑣)(𝑓(⋅,𝑦_𝑖)+(⋅)^⊤𝐵𝑤)−(𝑓(𝑧,𝑦_𝑖)+𝑧^⊤𝐵𝑤)(ℎ(⋅,𝑦_𝑖)−(⋅)^⊤𝐷𝑣)} is 𝐵-(𝑝,𝑟)-invex, respect to 𝜂 and 𝑏 at 𝑧 and ∑^𝑝𝑗=1𝜇_𝑗𝑔_𝑗(⋅) is 𝐵-(𝑝,𝑟)-invex with respect to the same function 𝜂 and another function 𝑏₆ at 𝑧 on 𝑋^∘∪Γ, not necessarily, equal to 𝑏, (b)Υ₇=∑^𝑠𝑖=1𝑡_𝑖{(ℎ(𝑧,𝑦_𝑖)−𝑧^⊤𝐷𝑣)(𝑓(⋅,𝑦_𝑖)+(⋅)^⊤𝐵𝑤)−(𝑓(𝑧,𝑦_𝑖)+𝑧^⊤𝐵𝑤)(ℎ(⋅,𝑦_𝑖)−(⋅)^⊤𝐷𝑣)}+∑^𝑝𝑗=1𝜇_𝑗𝑔_𝑗(⋅) is 𝐵-(𝑝,𝑟)-invex with respect to 𝜂 and 𝑏 at 𝑧 on 𝑋^∘∪Γ. Then sup_𝑦∈𝑌𝑓(𝑥,𝑦)+𝑥^⊤𝐵𝑥^1/2ℎ(𝑥,𝑦)−𝑥^⊤𝐷𝑥^1/2≥ℱ(𝑧).(5.5)

Proof. On the contrary, if possible, suppose that for each 𝑥∈𝑋^∘, sup_𝑦∈𝑌𝑓(𝑥,𝑦)+𝑥^⊤𝐵𝑥^1/2ℎ(𝑥,𝑦)−𝑥^⊤𝐷𝑥^1/2<𝐹(𝑧).(5.6) From the above inequality and 𝑦_𝑖∈𝑌(𝑧), 𝑖=1,2,…,𝑠, we obtain 𝑓𝑥,𝑦_𝑖+𝑥^⊤𝐵𝑥^1/2ℎ𝑥,𝑦_𝑖−𝑥^⊤𝐷𝑥^1/2<𝐹(𝑧)=𝑓𝑧,𝑦^𝑖+𝑧^⊤𝐵𝑧^1/2ℎ𝑧,𝑦_𝑖−𝑧^⊤𝐷𝑧^1/2,𝑖=1,2,…,𝑠.(5.7) By the above inequality, we know that 𝑓−𝑥,𝑦ℎ^𝑖+𝑥,𝑦𝑥^⊤_𝑖𝐵𝑥^1/2ℎ𝑧,𝑦^𝑖−𝑧^⊤𝐷𝑧^1/2−𝑥^⊤𝐷𝑥^1/2𝑓𝑧,𝑦^𝑖+𝑧^⊤𝐵𝑧^1/2<0,(5.8) for all 𝑖=1,2,…,𝑠 and 𝑦_𝑖∈𝑌(𝑧).
Multiplying the above inequality by 𝑡_𝑖∈ℝ^𝑠+ with ∑^𝑠𝑖=1𝑡_𝑖=1, we have ^𝑠_𝑖=1𝑡_𝑖−𝑓ℎ𝑥,𝑥𝑦,^𝑖𝑦_𝑖+𝑥^⊤𝐵𝑥^1/2ℎ𝑧,𝑦^𝑖−𝑧^⊤𝐷𝑧^1/2−𝑥^⊤𝐷𝑥^1/2𝑓𝑧,𝑦^𝑖+𝑧^⊤𝐵𝑧^1/2<0.(5.9) From relations (2.6), (5.3), and (5.9), we get Υ₆(𝑥)=^𝑠_𝑖=1≤𝑡_𝑖𝑓𝑥,𝑦_𝑖+𝑥^⊤𝐵𝑤ℎ𝑧,𝑦_𝑖−𝑧^⊤𝐷𝑣−ℎ𝑥,𝑦_𝑖−𝑥^⊤𝐷𝑣𝑓𝑧,𝑦_𝑖+𝑧^⊤𝐵𝑤_𝑠_𝑖=1𝑡_𝑖−𝑓ℎ𝑥𝑥,,𝑦𝑦^𝑖_𝑖+𝑥^⊤𝐵𝑥^1/2ℎ𝑧,𝑦^𝑖−𝑧^⊤𝐷𝑧^1/2<0=Υ₆−𝑥^⊤𝐷𝑥^1/2𝑓𝑧,𝑦^𝑖+𝑧^⊤𝐵𝑧^1/2(𝑧).(5.10) By relations (P) and (5.2), we have ^𝑝_𝑗=1𝜇_𝑗𝑔_𝑗(𝑥)≤0≤^𝑝_𝑗=1𝜇_𝑗𝑔_𝑗(𝑧).(5.11) Now, if condition (a) holds, from 𝐵-(𝑝,𝑟)-invexity with respect to 𝜂 and 𝑏₆ at 𝑧 of ∑^𝑝𝑗=1𝜇_𝑗𝑔_𝑗(⋅), we get 1𝑟𝑏¹𝑥₁,𝑥₀𝑒^{∑𝑟(𝑝𝑗=1𝜇∗𝑗𝑔𝑗∑(𝑥)−𝑝𝑗=1𝜇∗𝑗𝑔𝑗(𝑧))}−1≥1𝑝∇^𝑝_𝑗=1𝜇^∗𝑗𝑔_𝑗(𝑧)𝑒^{𝑝𝜂(𝑥,𝑧)}−𝟏.(5.12) From the above inequality and inequality (5.11), we obtain 1𝑝∇^𝑝_𝑗=1𝜇_𝑗𝑔_𝑗(𝑧)𝑒^{𝑝𝜂(𝑥,𝑧)}−𝟏≤0.(5.13) Multiplying (5.1) by (1/𝑝)(𝑒^{𝑝𝜂(𝑥,𝑧)}−𝟏), we have 1𝑝𝑒^{𝑝𝜂(𝑥,𝑧)}−𝟏^𝑠_𝑖=1−𝑡_𝑖𝑓ℎ𝑧,𝑧𝑦,_𝑖𝑦^𝑖−𝑧^⊤𝐷𝑧^1/2∇𝑓𝑧,𝑦^𝑖+𝐵𝑤+𝑧^⊤𝐵𝑧^1/2∇ℎ𝑧,𝑦^𝑖−𝐷𝑣+^𝑝_𝑗=1𝜇_𝑗∇𝑔_𝑗(𝑧)=0.(5.14) By the above equality and inequality (5.13), we obtain 1𝑝𝑒^{𝑝𝜂(𝑥,𝑧)}−𝟏^𝑠_𝑖=1−𝑡_𝑖𝑓ℎ𝑧,𝑧𝑦,_𝑖𝑦^𝑖−𝑧^⊤𝐷𝑧^1/2∇𝑓𝑧,𝑦^𝑖+𝐵𝑤+𝑧^⊤𝐵𝑧^1/2∇ℎ𝑧,𝑦^𝑖−𝐷𝑣≥0.(5.15) Using the 𝐵-(𝑝,𝑟)-invexity with respect to the same function 𝜂 and the function 𝑏 at 𝑧 of Υ₆(⋅) and the above inequality, we have Υ₆(𝑥)≥Υ₆(𝑧),(5.16) which contradicts (5.10) and proves that sup_𝑦∈𝑌((𝑓(𝑥,𝑦)+(𝑥^⊤𝐵𝑥)^1/2)/(ℎ(𝑥,𝑦)−(𝑥^⊤𝐷𝑥)^1/2))≥ℱ(𝑧).
If hypothesis (b) holds, from the 𝐵-(𝑝,𝑟)-invexity with respect to 𝜂 and 𝑏 at 𝑧 of Υ₇(⋅) and the equality (5.1), then Υ₇(𝑥)≥Υ₇(𝑧).(5.17) From relations (5.10) and (5.11), we have ^𝑠_𝑖=1𝑡_𝑖+ℎ𝑧,𝑦_𝑖−𝑧^⊤𝐷𝑣𝑓𝑥,𝑦_𝑖+𝑥^⊤𝐵𝑤−𝑓𝑧,𝑦_𝑖+𝑧^⊤𝐵𝑤ℎ𝑥,𝑦_𝑖−𝑥^⊤𝐷𝑣_𝑝_𝑗=1<𝜇_𝑗𝑔_𝑗(𝑥)_𝑠_𝑖=1+𝑡_𝑖ℎ𝑧,𝑦_𝑖−𝑧^⊤𝐷𝑣𝑓𝑧,𝑦_𝑖+𝑧^⊤𝐵𝑤−𝑓𝑧,𝑦_𝑖+𝑧^⊤𝐵𝑤ℎ𝑧,𝑦_𝑖−𝑧^⊤𝐷𝑣_𝑝_𝑗=1𝜇_𝑗𝑔_𝑗(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 𝑥^∗ be an optimal solution of (P) and let 𝑥^∗ satisfy a constraint qualification for (P). Then there exist (𝑠^∗,𝑡^∗,𝑦^∗)∈𝐾(𝑥^∗) and (𝑥^∗,𝜇^∗,𝑤^∗,𝑣^∗)∈𝐻₁(𝑠^∗,𝑡^∗,𝑦^∗) such that (𝑥^∗,𝜇^∗,𝑠^∗,𝑡^∗,𝑦^∗,𝑤^∗,𝑣^∗) is a feasible solution of (MWD). If in addition the hypothesis of Theorem 5.1 holds, then (𝑥^∗,𝜇^∗,𝑠^∗,𝑡^∗,𝑦^∗,𝑤^∗,𝑣^∗) 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 𝑥 and (𝑧^∗,𝜇^∗,𝑠^∗,𝑡^∗,𝑦^∗,𝑤^∗,𝑣^∗) 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)Υ₈(⋅)=∑^𝑠∗_𝑖=1𝑡^∗𝑖{(ℎ(𝑧^∗,𝑦^∗𝑖)−(𝑧^∗)^⊤𝐷𝑣)(𝑓(⋅,𝑦^∗𝑖)+(⋅)^⊤𝐵𝑤)−(𝑓(𝑧^∗,𝑦^∗𝑖)+(𝑧^∗)^⊤𝐵𝑤)(ℎ(⋅,𝑦^∗𝑖)−(⋅)^⊤𝐷𝑣)} is strictly 𝐵-(𝑝,𝑟)-invex with respect to 𝜂 and 𝑏 at 𝑧^∗ and ∑^𝑝𝑗=1𝜇^∗𝑗𝑔_𝑗(⋅) is 𝐵-(𝑝,𝑟)-invex with respect to the same function 𝜂 and another function 𝑏₈ at 𝑧^∗ on 𝑋^∘∪Γ, not necessarily, equal to 𝑏, (b)Υ₉(⋅)=∑^𝑠∗_𝑖=1𝑡^∗𝑖{(ℎ(𝑧^∗,𝑦^∗𝑖)−(𝑧^∗)^⊤𝐷𝑣)(𝑓(⋅,𝑦^∗𝑖)+(⋅)^⊤𝐵𝑤)−(𝑓(𝑧^∗,𝑦^∗𝑖)+(𝑧^∗)^⊤𝐵𝑤)(ℎ(⋅,𝑦^∗𝑖)−(⋅)^⊤𝐷𝑣)}+∑^𝑝𝑗=1𝜇^∗𝑗𝑔_𝑗(⋅) is strictly 𝐵-(𝑝,𝑟)-invex with respect to 𝜂 and 𝑏 at 𝑧^∗ on 𝑋^∘∪Γ.Then 𝑥=𝑧^∗, that is, 𝑧^∗ solves (P) and sup_𝑦∈𝑌((𝑓(𝑧^∗,𝑦)+((𝑧^∗)^⊤𝐵𝑧^∗)^1/2)/(ℎ(𝑧^∗,𝑦)−((𝑧^∗)^⊤𝐷𝑧^∗)^1/2))=ℱ(𝑧^∗).

Proof. We suppose on the contrary that if 𝑥≠𝑧^∗, then sup_𝑦∈𝑌𝑓𝑥,𝑦+𝑥^⊤𝐵𝑥^1/2ℎ𝑥,𝑦−𝑥^⊤𝐷𝑥^1/2≤ℱ𝑧^∗.(5.19) Since 𝑦^∗𝑖∈𝑌(𝑧^∗), for 𝑖=1,2,…,𝑠^∗, 𝑓𝑧^∗,𝑦^∗𝑖+𝑧^∗^⊤𝐵𝑧^∗^1/2ℎ𝑧^∗,𝑦^∗𝑖−(𝑧^∗)^⊤𝐷𝑧^∗^1/2=ℱ𝑧^∗,for𝑖=1,2,…,𝑠^∗.(5.20) It follows that 𝑓𝑥,𝑦+𝑥^⊤𝐵𝑥^1/2ℎ𝑧^∗,𝑦^∗𝑖−𝑧^∗^⊤𝐷𝑧^∗^1/2−ℎ𝑥,𝑦−𝑥^⊤𝐷𝑥^1/2𝑓𝑧^∗,𝑦^∗𝑖+𝑧^∗^⊤𝐵𝑧^∗^1/2≤0,for𝑖=1,2,…,𝑠^∗,𝑦∈𝑌.(5.21) From 𝑦^∗𝑖∈𝑌(𝑧^∗)⊂𝑌 and 𝑡∈ℝ^𝑠∗₊ with ∑^𝑠∗_𝑖=1𝑡^∗𝑖=1, we have ^𝑠∗_𝑖=1𝑡^∗𝑖ℎ𝑧^∗,𝑦^∗𝑖−𝑧^∗^⊤𝐷𝑧^∗^1/2𝑓𝑥,𝑦^∗𝑖+𝑥^⊤𝐵𝑥^1/2−ℎ𝑥,𝑦^∗𝑖−𝑥^⊤𝐷𝑥^1/2𝑓𝑧^∗,𝑦^∗𝑖+𝑧^∗^⊤𝐵𝑧^∗^1/2≤0.(5.22) From relations (2.6), (5.3), and inequality (5.22), we obtain that Υ₈𝑥=^𝑠∗_𝑖=1𝑡^∗𝑖−ℎ𝑓𝑧𝑧^∗,𝑦^∗𝑖−𝑧^∗^⊤𝐷𝑣𝑓𝑥,𝑦^∗𝑖+𝑥^⊤𝐵𝑤_∗≤,𝑦^∗𝑖+𝑧^∗^⊤𝐵𝑤ℎ𝑥,𝑦^∗𝑖−𝑥^⊤𝐷𝑣_𝑠∗_𝑖=1𝑡^∗𝑖ℎ𝑧^∗,𝑦^∗𝑖−𝑥^⊤𝐷𝑥^1/2𝑓𝑧^∗,𝑦^∗𝑖+𝑧^∗^⊤𝐵𝑧^∗^1/2−ℎ𝑥,𝑦^∗𝑖−𝑥^⊤𝐷𝑥^1/2𝑓𝑧^∗,𝑦^∗𝑖+𝑧^∗^⊤𝐵𝑧^∗^1/2≤0=Υ₈𝑧^∗.(5.23) By relations (P) and (5.2), we get ^𝑝_𝑗=1𝜇^∗𝑗𝑔_𝑗𝑥≤0≤^𝑝_𝑗=1𝜇^∗𝑗𝑔_𝑗𝑧^∗.(5.24) Now, if condition (a) holds, from 𝐵-(𝑝,𝑟)-invexity with respect to 𝜂 and 𝑏₈ at 𝑧 of ∑^𝑝𝑗=1𝜇^∗𝑗𝑔_𝑗(⋅), then 1𝑟𝑏¹𝑥,𝑧^∗𝑒^{∑𝑟(𝑝𝑗=1𝜇∗𝑗𝑔𝑗(∑𝑥)−𝑝𝑗=1𝜇∗𝑗𝑔𝑗(𝑧∗))}−1≥1𝑝∇^𝑝_𝑗=1𝜇^∗𝑗𝑔_𝑗𝑧^∗𝑒^{𝑝𝜂(𝑥,𝑧∗)}−𝟏.(5.25) From the above inequality and inequality (5.24), we know that 1𝑝∇^𝑝_𝑗=1𝜇^∗𝑗𝑔_𝑗𝑧^∗𝑒^{𝑝𝜂(𝑥,𝑧∗)}−𝟏≤0.(5.26) Multiplying (MWD) by (1/𝑝)(𝑒^{𝑝𝜂(𝑥,𝑧∗)}−1), we have 1𝑝𝑒^{𝑝𝜂(𝑥,𝑧∗)}−𝟏⎧⎪⎨⎪⎩^𝑠∗_𝑖=1𝑡^∗𝑖ℎ𝑧^∗,𝑦^∗𝑖−𝑧^∗^⊤𝐷𝑧^∗^1/2∇𝑓𝑧^∗−𝑓𝑧_∗,𝑦^∗𝑖+𝐵𝑤^∗,𝑦^∗𝑖+𝑧^∗^⊤𝐵𝑧^∗^1/2∇ℎ𝑧^∗+,𝑦^∗𝑖−𝐷𝑣_𝑝_𝑗=1𝜇^∗𝑗∇𝑔_𝑗𝑧^∗=0.(5.27) Basing on the above inequality and (5.26), we get the inequality 1𝑝𝑒^{𝑝𝜂(𝑥,𝑧∗)}−𝟏⎧⎪⎨⎪⎩^𝑠∗_𝑖=1𝑡^∗𝑖ℎ𝑧^∗,𝑦^∗𝑖−𝑧^∗^⊤𝐷𝑧^∗^1/2∇𝑓𝑧^∗−𝑓𝑧_∗,𝑦^∗𝑖+𝐵𝑤^∗,𝑦^∗𝑖+𝑧^∗^⊤𝐵𝑧^∗^1/2∇ℎ𝑧^∗,𝑦^∗𝑖−𝐷𝑣⎫⎪⎬⎪⎭≥0.(5.28) Since Υ₈(⋅) is strictly 𝐵-(𝑝,𝑟)-invex with respect to the same function 𝜂 and the function 𝑏 at 𝑧^∗ and the above inequality, we obtain Υ₈𝑧^∗<Υ₈𝑥,(5.29) which contradicts relations (5.23). Hence 𝑥=𝑧^∗ and 𝑧^∗ is an optimal solution of (P).
If hypothesis (b) holds, from the strict 𝐵-(𝑝,𝑟)-invexity with respect to 𝜂 and 𝑏 at 𝑧^∗ of Υ₉(⋅) and equality (5.1), then Υ₉𝑥>Υ₉𝑧^∗.(5.30) Relations (5.23) together with (5.24) yield ^𝑠∗_𝑖=1𝑡^∗𝑖−ℎ𝑓𝑧^∗𝑧,𝑦^∗𝑖−𝑧^∗^⊤𝐷𝑣𝑓𝑥,𝑦^∗𝑖+𝑥^⊤𝐵𝑤_∗,𝑦^∗𝑖+𝑧^⊤𝐵𝑤ℎ𝑥,𝑦^∗𝑖−𝑥^⊤𝐷𝑣+^𝑝_𝑗=1≤_𝑠∗𝜇^∗𝑗𝑔_𝑗𝑥_𝑖=1𝑡^∗𝑖−ℎ𝑓𝑧𝑧^∗,𝑦^∗𝑖−𝑧^∗^⊤𝐷𝑣𝑓𝑧^∗,𝑦^∗𝑖+𝑧^∗^⊤𝐵𝑤_∗,𝑦^∗𝑖+𝑧^∗^⊤𝐵𝑤ℎ𝑧^∗,𝑦^∗𝑖−𝑧^∗^⊤𝐷𝑣+^𝑝_𝑗=1𝜇^∗𝑗𝑔_𝑗𝑧^∗,(5.31) which contradicts (5.30). Hence 𝑧^∗=𝑥 is also an optimal solution of (P). The proof is complete.

Remark 5.4. In Theorem 5.3, if ∑^𝑝𝑗=1𝜇^∗𝑗𝑔_𝑗(⋅) is a strictly 𝐵-(𝑝,𝑟)-invex function with respect to 𝜂 and 𝑏₈ and Υ₈(⋅) is a 𝐵-(𝑝,𝑟)-invex function with respect to the same function 𝜂 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 𝑥₀ be a (𝑃)-optimal solution, and let ∇𝑔_𝑗(𝑥₀), 𝑗∈𝐽(𝑥₀) be linearly independent. Then there exist (𝑠^∗,𝑡^∗,𝑦^∗)∈𝐾(𝑥₀),(𝑤,𝑣)∈ℝ^𝑛×ℝ^𝑛, and 𝜇^∗∈ℝ^𝑝+ such that ∇⎛⎜⎜⎝∑^{𝑠∗𝑖=1}𝑡^∗𝑖𝑓𝑥₀,𝑦^∗𝑖+𝑥^⊤0𝐵𝑤+∑^𝑝𝑗=1𝜇^∗𝑗𝑔_𝑗𝑥₀∑^𝑠∗_𝑖=1𝑡^∗𝑖ℎ𝑥₀,𝑦^∗𝑖−𝑥^⊤0𝐷𝑣⎞⎟⎟⎠=0,_𝑝_𝑗=1𝑡𝜇^∗𝑗𝑔_𝑗𝑥₀=0,_∗𝑖≥0,^𝑠∗_𝑖=1𝑤𝑡^∗𝑖=1,_⊤𝑥𝐵𝑤≤1,𝑣^⊤𝐷𝑣≤1,_⊤0𝑥_⊤0𝐵𝑤=𝑥^⊤0𝐵𝑥₀^1/2,𝐷𝑣=𝑥^⊤0𝐵𝑥₀^1/2.(6.1)

In this section, we present the Wolfe dual (WD) to the minimax program (P): max_{(𝑠,𝑡,𝑦)∈𝐾(𝑧)}sup_{(𝑧,𝜇,𝑤,𝑣)∈𝐻3(𝑠,𝑡,𝑦)}𝒜(𝑧),(WD) where 𝐻₃(𝑠,𝑡,𝑦) denotes the set of (𝑧,𝜇,𝑤,𝑣)∈ℝ^𝑛×ℝ^𝑝+×ℝ^𝑛×ℝ^𝑛 satisfying ∇∑^𝑠𝑖=1𝑡_𝑖𝑓𝑧,𝑦_𝑖+𝑧^⊤𝐵𝑤+∑^𝑝𝑗=1𝜇_𝑗𝑔_𝑗(𝑧)∑^𝑠𝑖=1𝑡_𝑖ℎ𝑤𝑧,𝑦_𝑖−𝑧^⊤𝐷𝑣=0,(𝑠,𝑡,𝑦)∈𝐾,(6.2)_⊤𝐵𝑤𝑧≤1,𝑣^⊤𝐷𝑣≤1,_⊤𝑧_⊤𝐵𝑤=𝑧^⊤𝐵𝑧^1/2,𝒜(𝑧)=∑_𝑠𝑖=1𝐷𝑣=𝑧^⊤𝐵𝑧^1/2,(6.3)𝑡_𝑖𝑓𝑧,𝑦_𝑖+𝑧^⊤𝐵𝑤+∑^𝑝𝑗=1𝜇_𝑗𝑔_𝑗(𝑧)∑^𝑠𝑖=1𝑡_𝑖ℎ𝑧,𝑦_𝑖−𝑧^⊤𝐷𝑣.(6.4) If for a triplet (𝑠,𝑡,𝑦)∈𝐾(𝑧) the set 𝐻₃(𝑠,𝑡,𝑦) is empty, then we define the supremum over it to be −∞.

Let Γ denote the set of all feasible points of (WD). Moreover, we denote 𝑝𝑟_𝑋Γ={𝑧∈𝑋∣(𝑧,𝜇,𝑠,𝑡,𝑦,𝑤,𝑣)∈Γ}.

We assume throughout this section that ∑^𝑠𝑖=1𝑡_𝑖(𝑓(𝑧,𝑦_𝑖)+𝑧^⊤𝐵𝑤)+∑^𝑝𝑗=1𝜇_𝑗𝑔_𝑗(𝑧)≥0 and ∑^𝑠𝑖=1𝑡_𝑖(ℎ(𝑧,𝑦_𝑖)−𝑧^⊤𝐷𝑣)>0.

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

Theorem 6.2 (weak duality). Let 𝑥 and (𝑧,𝜇,𝑠,𝑡,𝑦,𝑤,𝑣) be (𝑃)-feasible and (𝑊𝐷)-feasible, respectively, and assume that Υ₁₀(⋅)=^𝑠_𝑖=1𝑡_𝑖ℎ𝑧,𝑦_𝑖−𝑧^⊤𝐷𝑣^𝑠_𝑖=1𝑡_𝑖𝑓⋅,𝑦_𝑖+(⋅)^⊤𝐵𝑤+^𝑝_𝑗=1−𝜇_𝑗𝑔_𝑗(⋅)_𝑠_𝑖=1𝑡_𝑖𝑓𝑧,𝑦_𝑖+𝑧^⊤𝐵𝑤+^𝑝_𝑗=1𝜇_𝑗𝑔_𝑗(𝑧)^𝑠_𝑖=1𝑡_𝑖ℎ⋅,𝑦_𝑖−(⋅)^⊤𝐷𝑣(6.5) is 𝐵-(𝑝,𝑟)-invex with respect to 𝜂 and 𝑏 at 𝑧 on 𝑋^∘⋃𝑝𝑟_𝑋Γ; then sup_y∈𝑌𝑓(𝑥,𝑦)+𝑥^⊤𝐵𝑥^1/2ℎ(𝑥,𝑦)−𝑥^⊤𝐷𝑥^1/2≥𝒜(𝑧).(6.6)

Proof. Suppose on the contrary that sup_𝑦∈𝑌𝑓(𝑥,𝑦)+𝑥^⊤𝐵𝑥^1/2ℎ(𝑥,𝑦)−𝑥^⊤𝐷𝑥^1/2<𝐴(𝑧).(6.7) Hence, we have an inequality 𝑓(x,𝑦)+𝑥^⊤𝐵𝑥^1/2ℎ(𝑥,𝑦)−𝑥^⊤𝐷𝑥^1/2<𝐴(𝑧)∀𝑦∈𝑌.(6.8) Then, we get 𝑓(−𝑥,𝑦ℎ)+(𝑥,𝑥𝑦)^⊤𝐵−𝑥𝑥^1/2_⊤^𝑠_𝑖=1𝑡_𝑖ℎ𝑧,𝑦_𝑖−𝑧^⊤𝐷𝑣𝐷𝑥^1/2^𝑠_𝑖=1𝑡_𝑖𝑓𝑧,𝑦_𝑖+𝑧^⊤𝐵𝑤+^𝑝_𝑖=1𝜇_𝑗𝑔_𝑗(𝑧)<0,(6.9) for all 𝑦∈𝑌.
If 𝑦 is replaced by 𝑦_𝑖 in the above inequality and is multiplied by 𝑡_𝑖, then summing up, we get ^𝑠_𝑖=1𝑡_𝑖𝑓𝑥,𝑦^𝑖+𝑥^⊤𝐵𝑥^1/2^𝑠_𝑖=1−𝑡_𝑖ℎ𝑧,𝑦_𝑖−𝑧^⊤𝐷𝑣_𝑠_𝑖=1𝑡_𝑖ℎ𝑥,𝑦^𝑖−𝑥^⊤𝐷𝑥^1/2^𝑠_𝑖=1𝑡_𝑖𝑓𝑧,𝑦_𝑖+𝑧^⊤𝐵𝑤+^𝑝_𝑖=1𝜇_𝑗𝑔_𝑗(𝑧)<0.(6.10) By (2.6) and (6.3), we have Υ₁₀(𝑥)≤^𝑠_𝑖=1𝑡_𝑖𝑓𝑥,𝑦^𝑖+𝑥^⊤𝐵𝑥^1/2+^𝑝_𝑖=1𝜇_𝑗𝑔_𝑗(𝑥)^𝑠_𝑖=1−𝑡_𝑖ℎ𝑧,𝑦_𝑖−𝑧^⊤𝐷𝑣_𝑠_𝑖=1𝑡_𝑖ℎ𝑥,𝑦^𝑖−𝑥^⊤𝐷𝑥^1/2^𝑠_𝑖=1𝑡_𝑖𝑓𝑧,𝑦_𝑖+𝑧^⊤𝐵𝑤+^𝑝_𝑖=1<𝜇_𝑗𝑔_𝑗(𝑧)_𝑝_𝑖=1𝜇_𝑗𝑔_𝑗(𝑥)^𝑠_𝑖=1𝑡_𝑖ℎ𝑧,𝑦_𝑖−𝑧^⊤𝐷𝑣.(6.11) Since ∑^𝑠𝑖=1𝑡_𝑖(ℎ(𝑧,𝑦_𝑖)−𝑧^⊤𝐷𝑣)>0 and ∑^𝑝𝑖=1𝜇_𝑗𝑔_𝑗(𝑥)≤0, it follows that Υ₁₀(𝑥)<0=Υ₁₀(𝑧).(6.12) Using the 𝐵-(𝑝,𝑟)-invexity with respect to 𝜂 and 𝑏 at 𝑧 of Υ₁₀(⋅), we have the inequality 1𝑝∇Υ¹⁰(𝑧)𝑒^{𝑝𝜂(𝑥,𝑧)}−𝟏<0.(6.13) This contradicts the equality of (6.2) ×(1/𝑝)(𝑒^{𝑝𝜂(𝑥,𝑧)}−𝟏). Here the proof is complete.

As a consequence of Theorems 6.1 and 6.2, we obtain Theorem 6.3. By a similar way, we can prove the strong duality and strictly converse duality theorems with respect to (P) and (WD) which we state as follows.

Theorem 6.3 (strong duality). Let 𝑥^∗ be an optimal solution of (P) satisfying the hypothesis of Theorem 6.2. Then there exist (𝑠^∗,𝑡^∗,𝑦^∗)∈𝐾(𝑥^∗) and (𝑥^∗,𝜇^∗,𝑤^∗,𝑣^∗)∈𝐻₃(𝑠^∗,𝑡^∗,𝑦^∗) such that (𝑥^∗,𝜇^∗,𝑠^∗,𝑡^∗,𝑦^∗,𝑤^∗,𝑣^∗) is feasible for (WD). If any of the conditions of Theorem 6.2 hold, then (𝑥^∗,𝜇^∗,𝑠^∗,𝑡^∗,𝑦^∗,𝑤^∗,𝑣^∗) is an optimal solution of (WD) and the two problems (P) and (WD) have the same extremal values.

Theorem 6.4 (strict converse duality). Let 𝑥 and (𝑧^∗,𝜇^∗,𝑠^∗,𝑡^∗,𝑦^∗,𝑤^∗,𝑣^∗) be optimal solutions of (P) and (WD), respectively. Suppose that the assumptions of Theorem 6.3 are fulfilled and Υ₁₁(⋅)=⎡⎢⎢⎣^𝑠∗_𝑖=1𝑡^∗𝑖ℎ𝑧^∗,𝑦^∗𝑖−𝑧^∗^⊤𝐷𝑣^∗⎤⎥⎥⎦⎡⎢⎢⎣^𝑠∗_𝑖=1𝑡^∗𝑖𝑓⋅,𝑦^∗𝑖+(⋅)^⊤𝐵𝑤^∗+^𝑝_𝑗=1−⎡⎢⎢⎣𝜇^∗𝑗𝑔_𝑗(⋅)_𝑠∗⎤⎥⎥⎦_𝑖=1𝑡^∗𝑖𝑓𝑧^∗,𝑦^∗𝑖+𝑧^∗^⊤𝐵𝑤^∗+^𝑝_𝑗=1𝜇^∗𝑗𝑔_𝑗𝑧^∗⎤⎥⎥⎦^𝑠_𝑖=1𝑡^∗𝑖ℎ⋅,𝑦^∗𝑖−(⋅)^⊤𝐷𝑣^∗(6.14) is strictly 𝐵-(𝑝,𝑟)-invex with respect to 𝜂 and 𝑏 at 𝑧^∗ on 𝑋^∘∪𝑝𝑟_𝑋Γ.

Then 𝑥=𝑧^∗, that is, 𝑧^∗ is an optimal solution of (P) and sup_𝑦∈𝑌((𝑓(𝑧^∗,𝑦)+((𝑧^∗)^⊤𝐵𝑧^∗)^1/2)/(ℎ(𝑧^∗,𝑦)−((𝑧^∗)^⊤𝐷𝑧^∗)^1/2))=𝒜(𝑧^∗).

Acknowledgment

The author is partly supported by NSC, Taiwan