Generalized Minimax Programming with Nondifferentiable (𝐺, 𝛽)-Invexity

Abstract

We consider the generalized minimax programming problem (P) in which functions are locally Lipschitz (𝐺, 𝛽)-invex. Not only 𝐺-sufficient but also 𝐺-necessary optimality conditions are established for problem (P). With 𝐺-necessary optimality conditions and (𝐺, 𝛽)-invexity on hand, we construct dual problem (DI) for the primal one (P) and prove duality results between problems (P) and (DI). These results extend several known results to a wider class of programs.

1. Introduction

Convexity plays a central role in many aspects of mathematical programming including analysis of stability, sufficient optimality conditions, and duality. Based on convexity assumptions, nonlinear programming problems can be solved efficiently. There have been many attempts to weaken the convexity assumptions in order to treat many practical problems. Therefore, many concepts of generalized convex functions have been introduced and applied to mathematical programming problems in the literature [1]. One of these concepts, invexity, was introduced by Hanson in [2]. Hanson has shown that invexity has a common property in mathematical programming with convexity that Karush-Kuhn-Tucker conditions are sufficient for global optimality of nonlinear programming under the invexity assumptions. Ben-Israel and Mond [3] introduced the concept of preinvex functions which is a special case of invexity.

Recently, Antczak extended further invexity to 𝐺-invexity [4] for scalar differentiable functions and introduced new necessary optimality conditions for differentiable mathematical programming problem. Antczak also applied the introduced 𝐺-invexity notion to develop sufficient optimality conditions and new duality results for differentiable mathematical programming problems. Furthermore, in the natural way, Antczak’s definition of 𝐺-invexity was also extended to the case of differentiable vector-valued functions. In [5], Antczak defined vector 𝐺-invex (𝐺-incave) functions with respect to 𝜂 and applied this vector 𝐺-invexity to develop optimality conditions for differentiable multiobjective programming problems with both inequality and equality constraints. He also established the so-called 𝐺-Karush-Kuhn-Tucker necessary optimality conditions for differentiable vector optimization problems under the Kuhn-Tucker constraint qualification [5]. With this vector 𝐺-invexity concept, Antczak proved new duality results for nonlinear differentiable multiobjective programming problems [6]. A number of new vector duality problems such as 𝐺-Mond-Weir, 𝐺-Wolfe, and 𝐺-mixed dual vector problems to the primal one were also defined in [6].

In the last few years, many concepts of generalized convexity, which include (𝑝,𝑟)-invexity [7], (𝐹,𝜌)-convexity [8], (𝐹,𝛼,𝜌,𝑑)-convexity [9], (𝐶,𝛼,𝜌,𝑑)-convexity [10], (𝜙,𝜌)-invexity [11], 𝑉-𝑟-invexity [12], and their extensions, have been introduced and applied to different mathematical programming problems. In particular, they have also been applied to deal with minimax programming; see [13–17] for details. However, we have not found a paper which deals with generalized minimax programming problem (P) under 𝐺-invexity or its generalizations assumptions.

Note that the function 𝐺∘𝑓 may not be differentiable even if the function 𝐺 is differentiable. Yuan et al. [18] introduced the (𝐺_𝑓,𝛽_𝑓)-invexity concept for the locally Lipschtiz function 𝑓. This (𝐺_𝑓,𝛽_𝑓)-invexity extended Antczak’s 𝐺-invexity concept to the nonsmooth case. In this paper, we deal with nondifferentiable generalized minimax programming problem (P) with the vector (𝐺,𝛽)-invexity proposed in [18]. Here, the generalized minimax programming problem (P) is presented as follows: minsup_𝑦∈𝑌𝜙𝑥,𝑦subjectto𝑔_𝑗(𝑥)(≤0,𝑗=1,…,𝑚,P) where 𝑌 is a compact subset of ℝ^𝑝, 𝜙(⋅,⋅)∶ℝ^𝑛×ℝ^𝑝→ℝ, and 𝑔_𝑗(⋅)∶ℝ^𝑛→ℝ(𝑗∈𝑀). Let 𝐸_𝑃 be the set of feasible solutions of problem (P); in other words, 𝐸_𝑃={𝑥∈ℝ^𝑛∣𝑔_𝑗(𝑥)≤0,𝑗∈𝑀}. For convenience, let us define the following sets for every 𝑥∈𝐸: 𝐽(𝑥)=𝑗∈𝑀∣𝑔_𝑗(𝑥),𝑌(𝑥)==0𝑦∈𝑌∣𝜙𝑥,𝑦=sup_𝑧∈𝑌𝜙(𝑥,𝑧).(1)

The rest of the paper is organized as follows. In Section 2, we present concepts in regards to nondifferentiable vector (𝐺,𝛽)-invexity. In Section 3, we present not only 𝐺-sufficient but also 𝐺-necessary optimality conditions for problem (P). When the 𝐺-necessary optimality conditions and the (𝐺,𝛽)-invexity concept are utilized, dual problem (DI) is formulated for the primal one (P) and duality results between them are presented in Section 4.

2. Notations and Preliminaries

In this section, we provide some definitions and results that we will use in the sequel. The following convention for equalities and inequalities will be used throughout the paper. For any 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛)^𝑇, 𝑦=(𝑦₁,𝑦₂,…,𝑦_𝑛)^𝑇, we define the following: 𝑥>𝑦ifandonlyif𝑥_𝑖>𝑦_𝑖,for𝑖=1,2,…,𝑛,𝑥≧𝑦ifandonlyif𝑥_𝑖≥𝑦_𝑖,for𝑖=1,2,…,𝑛,𝑥⩾𝑦ifandonlyif𝑥_𝑖≥𝑦_𝑖,for𝑖=1,2,…,𝑛,but𝑥̸𝑥≠𝑦,>𝑦isthenegationof𝑥𝑥>𝑦,_╱⩾𝑦isthenegationof𝑥⩾𝑦.(2) Let ℝ^𝑛₊={𝑥∈ℝ^𝑛|𝑥≧0}, ℝ̇^𝑛₊={𝑥∈ℝ^𝑛|𝑥>0} and 𝑋 be a subset of ℝ^𝑛. For our convenience, denote 𝑄∶={1,2,…,𝑞}, 𝑄^∗∶={1,2,…,𝑞^∗}, 𝐾∶={1,2,…,𝑘}, and 𝑀∶={1,2,…,𝑚}. Further, we recall some definitions and a lemma.

Definition 1 (see [19]). Let 𝑑∈ℝ^𝑛, 𝑋 be a nonempty set of ℝ^𝑛 and 𝑓∶𝑋→ℝ. If 𝑓⁰(𝑥;𝑑)∶=lim_{𝑦→𝑥𝜇↓0}1sup𝜇𝑓𝑦+𝜇𝑑−𝑓𝑦(3) exists, then 𝑓⁰(𝑥;𝑑) is called the Clarke derivative of 𝑓 at 𝑥 in the direction 𝑑. If this limit superior exists for all 𝑑∈ℝ^𝑛, then 𝑓 is called the Clarke differentiable at 𝑥. The set 𝜕𝑓(𝑥)=𝜁∣𝑓⁰(𝑥;𝑑)≥⟨𝜁,𝑑⟩,∀𝑑∈ℝ^𝑛(4) is called the Clarke subdifferential of 𝑓 at 𝑥.

Note that if a given function 𝑓 is locally Lipschitz, then the Clarke subdifferential 𝜕𝑓(𝑥) exists.

Lemma 2 (see [18]). Let 𝜓 be a real-valued Lipschitz continuous function defined on 𝑋 and denote the image of 𝑋 under 𝜓 by 𝐼_𝜓(𝑋); let 𝜑∶𝐼_𝜓(𝑋)→ℝ be a differentiable function such that 𝜑^′(𝛾) is continuous on 𝐼_𝜓(𝑋) and 𝜑^′(𝛾)≥0 for each 𝛾∈𝐼_𝜓(𝑋). Then the chain rule 𝜑∘𝜓⁰(𝑥,𝑑)=𝜑^′𝜓(𝑥)𝜓⁰(𝑥,𝑑),(5) holds for each 𝑑∈ℝ^𝑛. Therefore, 𝜕𝜑∘𝜓(𝑥)=𝜑^′𝜓(𝑥)𝜕𝜓(𝑥).(6)

Definition 3. Let 𝑓=(𝑓₁,…,𝑓_𝑘) be a vector-valued locally Lipschitz function defined on a nonempty set 𝑋⊂ℝ^𝑛. Consider the functions 𝜂∶𝑋×𝑋→ℝ^𝑛, 𝐺_𝑓𝑖∶𝐼_𝑓𝑖(𝑋)→ℝ, and 𝛽^𝑓_𝑖∶𝑋×𝑋→ℝ₊ for 𝑖∈𝐾. Moreover, 𝐺_𝑓𝑖 is strictly increasing on its domain 𝐼_𝑓𝑖(𝑋) for each 𝑖∈𝐾. If 𝐺_𝑓𝑖∘𝑓_𝑖(𝑥)−𝐺_𝑓𝑖∘𝑓_𝑖(𝑢)≥(>)𝛽^𝑓_𝑖(𝑥,𝑢)𝐺^′_𝑓𝑖𝑓_𝑖(𝑢)𝜁_𝑖,𝜂(𝑥,𝑢),∀𝜁_𝑖∈𝜕𝑓_𝑖(𝑢),(7) holds for all 𝑥∈𝑋(𝑥≠𝑢) and 𝑖∈𝐾, then 𝑓 is said to be a (strictly) nondifferentiable vector (𝐺_𝑓,𝛽_𝑓)-invex at 𝑢 on 𝑋 (with respect to 𝜂) (or shortly, (𝐺_𝑓,𝛽_𝑓)-invex at 𝑢 on 𝑋), where 𝐺_𝑓=(𝐺_𝑓1,…,𝐺_𝑓𝑘) and 𝛽∶=(𝛽^𝑓₁,𝛽^𝑓₂,…,𝛽^𝑓_𝑘). If 𝑓 is a (strictly) nondifferentiable vector (𝐺_𝑓,𝛽_𝑓)-invex at 𝑢 on 𝑋(with respect to 𝜂) for all 𝑢∈𝑋, then 𝑓 is a (strictly) nondifferential vector (𝐺_𝑓,𝛽_𝑓)-invex on 𝑋(with respect to 𝜂).

3. Optimality Conditions

In this section, we firstly establish the 𝐺-necessary optimality conditions for problem (P) involving functions which are locally Lipschitz with respect to the variable 𝑥. For this purpose, we will need some additional assumptions with respect to problem (P).

Condition 4. Assume the following: (a) the set 𝑌 is compact;
(b) 𝜙(𝑥,𝑦) and 𝜕𝜙_𝑥(𝑥,𝑦) are upper semicontinuous at (𝑥,𝑦);
(c) 𝜙(𝑥,𝑦) is locally Lipschitz in 𝑥 and this Lipschitz continuity is uniform for 𝑦 in 𝑌;
(d) 𝜙(𝑥,𝑦) is regular in 𝑥; that is, 𝜙^∘_𝑥(𝑥,𝑦;⋅)=𝜙^′_𝑥(𝑥,𝑦;⋅);
(e) 𝑔_𝑗,𝑗∈𝑀 are regular and locally Lipschitz at 𝑥₀.

Condition 5. For each 𝜂=(𝜂₁,…,𝜂_𝑚)∈ℝ^𝑚 satisfying the conditions 𝜂_𝑗=0,∀𝑗∈𝑀⧵𝐽𝑥^∗,𝜂_𝑗≥0,∀𝑗∈𝐽𝑥^∗,(8) the following implication holds: 𝑧^∗_𝑗∈𝜕𝑔_𝑗𝑥^∗∀𝑗∈𝑀_𝑚𝑗=1,𝜂_𝑗𝑧^∗_𝑗=0⇒𝜂_𝑗=0,𝑗∈𝑀.(9)

We will also use the following auxiliary programming problem (G-P): minsup_𝑦∈𝑌𝐺_𝜙∘𝜙𝑥,𝑦s.t.𝐺_𝑔(𝑥)𝐺∘𝑔∶=_𝑔1∘𝑔₁(𝑥),𝐺_𝑔2∘𝑔₂(𝑥),…,𝐺_𝑔𝑚∘𝑔_𝑚(𝑥)≦𝐺_𝑔(0),(G-P) where 𝐺_𝑔(0)∶=(𝐺_𝑔1(0),𝐺_𝑔2(0),…,𝐺_𝑔𝑚(0)). We denote by 𝐸_𝐺-𝑃={𝑥∈ℝ^𝑛∣𝐺_𝑔∘𝑔(𝑥)≦𝐺_𝑔(0)}, 𝐽^′(𝑥)∶={𝑗∈𝑀∶𝐺_𝑔𝑗∘𝑔_𝑗(𝑥)=𝐺_𝑔𝑗(0)}. If function 𝐺_𝑔𝑗 is strictly increasing on 𝐼_𝑔𝑗(𝑋) for each 𝑗∈M, then 𝐸_𝑃=𝐸_𝐺-𝑃 and 𝐽(𝑥)=𝐽^′(𝑥). So, we represent the set of all feasible solutions and the set of constraint active indices for either (P) or (G-P) by the notations 𝐸 and 𝐽(𝑥), respectively.

The following necessary optimality conditions are presented in [20].

Theorem 6 (necessary optimality conditions). Let 𝑥^∗ be an optimal solution of (P). One also assumes that Conditions 4 and 5 hold. Then there exist positive integer 𝑞^∗ and vectors 𝑦_𝑖∈𝑌(𝑥^∗) together with scalars 𝜆^∗_𝑖(𝑖∈𝑄^∗) and 𝜇^∗_𝑗(𝑗∈𝑀) such that 0∈^𝑞∗_𝑖=1𝜆^∗_𝑖𝜕_𝑥𝜙𝑥^∗,𝑦_𝑖+^𝑚_𝑗=1𝜇^∗_𝑗𝜕𝑔_𝑗𝑥^∗,𝜇^∗_𝑗𝑔_𝑗𝑥^∗=0,𝜇^∗_𝑗≥0,𝑗∈𝑀,_{𝑞∗𝑖=1}𝜆^∗_𝑖=1,𝜆^∗_𝑖>0,𝑖∈𝑄^∗.(10) Furthermore, if 𝛼 is the number of nonzero 𝜆^∗_𝑖 and 𝛽 is the number of nonzero 𝜇^∗_𝑗, then 1≤𝛼+𝛽≤𝑛+1.(11)

Making use of Theorem 6, we can derive the following 𝐺-necessary conditions theorem for problem (P), see Theorem 7, here we require the scalars 𝜆^∗_𝑖(𝑖=1,…,𝑞^∗) to be positive.

Theorem 7 (𝐺-necessary optimality conditions). Let problem (P) satisfy Conditions 4 and 5; let 𝑥^∗ be an optimal solution of problem (P). Assume that 𝐺_𝜙 is both continuously differentiable and strictly increasing on 𝐼_𝜙(𝑋,𝑌). If 𝐺_𝑔𝑗 is both continuously differentiable and strictly increasing on 𝐼_𝑔𝑗(𝑋) with 𝐺^′_𝑔𝑗(𝑔_𝑗(𝑥^∗))>0 for each 𝑗∈𝑀, then there exist positive integer 𝑞^∗(1≤𝑞^∗≤𝑛+1) and vectors 𝑦_𝑖∈𝑌(𝑥^∗) together with scalars 𝜆^∗_𝑖(𝑖∈𝑄^∗) and 𝜇^∗_𝑗(𝑗∈𝑀) such that 0∈^𝑞∗_𝑖=1𝜆^∗_𝑖𝐺^′_𝜙𝜙𝑥^∗,𝑦_𝑖𝜕_𝑥𝜙𝑥^∗,𝑦_𝑖+^𝑚_𝑗=1𝜇^∗_𝑗𝐺^′_𝑔𝑗𝑔_𝑗𝑥^∗𝜕𝑔_𝑗𝑥^∗,(12)𝜇^∗_𝑗𝐺_𝑔𝑗∘𝑔_𝑗𝑥^∗−𝐺_𝑔𝑗(0)=0,𝜇^∗_𝑗≥0,𝑗∈𝑀,(13)_{𝑞∗𝑖=1}𝜆^∗_𝑖=1,𝜆^∗_𝑖≥0,𝑖∈𝑄^∗.(14)

Proof. Since 𝑥^∗ is an optimal solution to problem (P), it is easy to see that 𝑥^∗ is an optimal solution to problem (G-P). Consider problem (G-P), it is easy to check that problem (G-P) satisfies the assumptions of Theorem 6. Therefore, we choose 𝑦_𝑖∈𝑌(𝑥^∗) and 𝑖∈𝑄^∗ with 𝑞^∗≤𝑛+1, such that they satisfy Theorem 6.
Now, for each 𝑦_𝑖, we consider the scalar programming 𝐺-𝑃_𝑦𝑖 as follows: min𝐺_𝜙∘𝜙𝑥,𝑦_𝑖subjectto𝐺_𝑔𝑗∘𝑔_𝑗(𝑥)𝐺≤0,𝑗∈𝑀.-𝑃_𝑦𝑖 It is easy to see that 𝑥^∗ is an optimal solution to problem 𝐺-𝑃_𝑦𝑖. Thus, there exist 𝜆_𝑖>0 and 𝜇_𝑗𝑖≥0 for 𝑗∈𝑀 such that 0∈𝜆_𝑖𝜕_𝑥𝐺_𝜙∘𝜙𝑥^∗,𝑦_𝑖+^𝑚_𝑗=1𝜇^∗_𝑗𝑖𝜕𝐺_𝑔𝑗∘𝑔_𝑗𝑥^∗,(15)𝜇_𝑗𝑖𝐺_𝑔𝑗∘𝑔_𝑗𝑥^∗−𝐺_𝑔𝑗(0)=0,𝜇_𝑗𝑖≥0,𝑗∈𝑀.(16) So, we obtain from (15) that 0∈^𝑞∗_𝑖=1𝜆_𝑖𝜕_𝑥𝐺_𝜙∘𝜙𝑥^∗,𝑦_𝑖+^𝑚_𝑗=1^𝑞∗_𝑖=1𝜇_𝑗𝑖𝜕𝐺_𝑔𝑗∘𝑔_𝑗𝑥^∗,(17) or 0∈^𝑞∗_𝑖=1𝜆^∗_𝑖𝐺_𝜙∘𝜙𝑥^∗,𝑦_𝑖+^𝑚_𝑗=1𝜇^∗_𝑗𝜕𝐺_𝑔𝑗∘𝑔_𝑗𝑥^∗,(18) where 𝜆^∗_𝑖=𝜆_𝑖/∑^𝑞∗_𝑗=1𝜆_𝑗, 𝜇^∗_𝑗=∑^𝑞∗_𝑖=1𝜇_𝑗𝑖/∑^𝑞∗_𝑖=1𝜆_𝑖. By Lemma 2, we have 𝜕_𝑥𝐺_𝜙∘𝜙𝑥^∗,𝑦_𝑖=𝐺^′_𝜙𝜙𝑥^∗,𝑦_𝑖𝜕_𝑥𝜙𝑥^∗,𝑦_𝑖,𝑖∈𝑄^∗,𝜕𝐺_𝑔𝑗∘𝑔_𝑗𝑥^∗=𝐺^′_𝑔𝑗𝑔_𝑗𝑥^∗𝜕𝑔_𝑗𝑥^∗,𝑗∈𝑀.(19) Now, from (18), we can deduce the required results.

Next, we derive 𝐺-sufficient optimality conditions for problem (P) under the assumption of (𝐺,𝛽)-invexity proposed in [18].

Theorem 8 (𝐺-sufficient optimality conditions). Let (𝑥^∗,𝜇^∗,𝜈^∗,𝑞^∗,𝜆^∗,𝑦) satisfy conditions (12)–(14), where 𝜈^∗=𝜙(𝑥^∗,𝑦₁)=⋯=𝜙(𝑥^∗,𝑦_𝑞∗); let 𝐺_𝜙 be both continuously differentiable and strictly increasing on 𝐼_𝜙(𝑋,𝑌); let 𝐺_𝑔𝑗 be both continuously differentiable and strictly increasing on 𝐼_𝑔𝑗(𝑋) for each 𝑗∈𝑀. Assume that 𝜙(⋅,𝑦_𝑖) is (𝐺_𝜙,𝛽^𝜙_𝑖)-invex at 𝑥^∗ on 𝐸 for each 𝑖∈𝑄^∗ and 𝑔_𝑗 is (𝐺^𝑔_𝑗,𝛽^𝑔_𝑗)-invex at 𝑥^∗ on 𝐸 for each 𝑗∈𝑀. Then, 𝑥^∗ is an optimal solution to (P).

Proof. Suppose, contrary to the result, that 𝑥^∗ is not an optimal solution for problem (P). Hence, there exists 𝑥₀∈𝐸 such that sup_𝑦∈𝑌𝜙𝑥₀,𝑦<𝜙𝑥^∗,𝑦_𝑖,𝑖∈𝑄^∗.(20) By the monotonicity of 𝐺_𝜙, we have 𝐺_𝜙∘𝜙𝑥₀,𝑦_𝑖<𝐺_𝜙∘𝜙𝑥^∗,𝑦_𝑖,𝑖∈𝑄^∗.(21) Employing (13), (14), and the fact that 𝐺_𝑔𝑗∘𝑔_𝑗𝑥₀≤𝐺_𝑔𝑗(0)=𝐺_𝑔𝑗∘𝑔_𝑗𝑥^∗,𝑗∈𝐽𝑥^∗,(22) we can write the following statement ∑^𝑞∗_𝑖=1𝜆^∗_𝑖𝐺_𝜙∘𝜙𝑥₀,𝑦_𝑖−𝐺_𝜙∘𝜙𝑥^∗,𝑦_𝑖𝛽^𝜙_𝑖𝑥₀,𝑥^∗+∑^𝑚_𝑗=1𝜇^∗_𝑗G_𝑔𝑗∘𝑔_𝑗𝑥₀−𝐺_𝑔𝑗∘𝑔_𝑗𝑥^∗𝛽^𝑔_𝑗𝑥₀,𝑥^∗<0.(23) By the generalized invexity assumptions of 𝜙(⋅,𝑦_𝑖) and 𝑔_𝑗, we have 𝐺_𝜙∘𝜙𝑥₀,𝑦_𝑖−𝐺_𝜙∘𝜙𝑥^∗,𝑦_𝑖≥(>)𝛽^𝜙_𝑖𝑥₀,𝑥^∗𝐺^′_𝜙𝜙𝑥^∗,𝑦_𝑖𝜉^𝜙_𝑖,𝜂𝑥₀,𝑥^∗,∀𝜉^𝜙_𝑖∈𝜕_𝑥𝜙𝑥^∗,𝑦_𝑖,𝑖∈𝑄^∗,𝐺_𝑔𝑗∘𝑔_𝑗𝑥₀−𝐺_𝑔𝑗∘𝑔_𝑗𝑥^∗≥(>)𝛽^𝑔_𝑗𝑥₀,𝑥^∗𝐺^′_𝑔𝑗𝑔_𝑗𝑥^∗𝜉^𝑔_𝑗,𝜂𝑥₀,𝑥^∗,∀𝜉^𝑔_𝑗∈𝜕𝑔_𝑗𝑥^∗,𝑗∈𝑀.(24) Employing (24) to (23), we have ^𝑞∗_𝑖=1𝜆^∗_𝑖𝐺^′_𝜙𝜙𝑥^∗,𝑦_𝑖𝜉^𝜙_𝑖,𝜂𝑥₀,𝑥^∗+^𝑚_𝑗=1𝜇^∗_𝑗𝐺^′_𝑔𝑗𝑔_𝑗𝑥^∗⟨𝜉^𝑔_𝑗,𝜂𝑥₀,𝑥^∗⟩<0(25) or ^𝑞∗_𝑖=1𝜆^∗_𝑖𝐺^′_𝜙𝜙𝑥^∗,𝑦_𝑖𝜉^𝜙_𝑖+^𝑚_𝑗=1𝜇^∗_𝑗𝐺^′_𝑔𝑗𝑔_𝑗𝑥^∗𝜉^𝑔_𝑗,𝜂𝑥₀,𝑥^∗<0,(26) which implies that 0∈^╱𝑞∗_𝑖=1𝜆^∗_𝑖𝐺^′_𝜙𝜙𝑥^∗,𝑦_𝑖𝜕_𝑥𝜙𝑥^∗,𝑦_𝑖+^𝑚_𝑗=1𝜇^∗_𝑗𝐺^′_𝑔𝑗𝑔_𝑗𝑥^∗𝜕𝑔_𝑗𝑥^∗.(27) This is a contradiction to condition (12).

Example 9. Let 𝑌=[0,1]. Define 𝜙=𝑒𝑥,𝑦^𝑥+2𝑦𝑒,𝑦≥𝑥_2𝑥+𝑦𝑔(𝑥)=3,𝑥<𝑦,𝑥−2−12.(28) Then, 𝜙(⋅,𝑦) is (log,1)-invex at 𝑥=1 for each 𝑦∈𝑌, 𝑔 is 1-invex at 𝑥=1, and 𝑓(𝑥)∶=sup_𝑦∈𝑌𝜙=𝑒𝑥,𝑦^𝑥+2𝑒,𝑦≥𝑥_2𝑥+1,𝑥<𝑦.(29) Since log𝜙𝑥,𝑦=𝑥+2𝑦,𝑦≥𝑥2𝑥+𝑦,𝑥<𝑦,(30) then 𝜕_𝑥log𝜙𝑥,𝑦=[]1,𝑦≥𝑥1,2,𝑥=𝑦2,𝑥<𝑦.(31) Consider 𝑥₀=1. Since 𝑌(𝑥₀)={1}, then we can assume that 𝑞=1. Therefore, 0∈𝜆𝐺^′_𝜙𝜙(1,1)𝜕_𝑥𝜙(1,1)−𝜇𝑔^′_𝑗(1)=𝜆[1,2]−𝜇,(32) where  𝜆=𝜇=1. Now, from Theorem 8, we can say that 𝑥₀=1 is an optimal solution to (P).

4. Duality

Making use of the optimality conditions of the preceding section, we present dual problem (DI) to the primal one (P) and establish 𝐺-weak, 𝐺-strong, and 𝐺-strict converse duality theorems. For convenience, we use the following notations: 𝐾(𝑥)=𝑞,𝜆,𝑦∈N×ℝ̇^𝑞₊×ℝ^𝑝𝑞𝜆∣1≤𝑞≤𝑛+1,𝜆=₁,…,𝜆_𝑞∈ℝ̇^𝑞₊with^𝑞_𝑖=1𝜆_𝑖=1,𝑦=𝑦₁,…𝑦_𝑞with𝑦_𝑖∈𝑌(𝑥),𝑖=1,…𝑞.(33)𝐻₁(𝑞,𝜆,𝑦) denotes the set of all triplets (𝑧,𝜇,𝜈)∈ℝ^𝑛×ℝ^𝑚₊×ℝ₊ satisfying 0∈^𝑞_𝑖=1𝜆_𝑖𝐺^′_𝜙𝜙𝑧,𝑦_𝑖𝜕_𝑧𝜙𝑧,𝑦_𝑖+^𝑚_𝑗=1𝜇_𝑗𝐺^′_𝑔𝑗𝑔_𝑗(𝑧)𝜕𝑔_𝑗(𝑧),(34)𝜙𝑧,𝑦_𝑖≥𝜈,𝑖=1,2,…,𝑞,(35)𝜇_𝑗𝑔_𝑗(𝑧)≥0,𝑗∈𝑀,(36)𝑦_𝑖∈𝑌(𝑧),𝑞,𝜆,𝑦∈𝐾(𝑧).(37)

Our dual problem (DI) can be stated as follows: max_{(𝑞,𝜆,𝑦)∈𝐾(𝑧)}sup_{(𝑧,𝜇,𝜈)∈𝐻1(𝑞,𝜆,𝑦)}(𝜈.DI) Note that if 𝐻₁(𝑞,𝜆,𝑦) is empty for some triplet (𝑞,𝜆,𝑦)∈𝐾(𝑧), then define sup_{(𝑧,𝜇,𝜈)∈𝐻1(𝑞,𝜆,𝑦)}𝜈=−∞.

Theorem 10 10 (𝐺-weak duality). Let 𝑥 and (𝑧,𝜇,𝜈,𝑞,𝜆,𝑦) be (𝑃)-feasible and (𝐷𝐼)-feasible, respectively; let 𝐺_𝜙 be both continuously differentiable and strictly increasing on 𝐼_𝜙(𝑋,𝑌); let 𝐺_𝑔𝑗 be both continuously differentiable and strictly increasing on 𝐼_𝑔𝑗(𝑋) for each 𝑗∈𝑀. If 𝜙(⋅,𝑦_𝑖) is (𝐺_𝜙,𝛽^𝜙_𝑖)-invex at 𝑧 for each 𝑖∈𝑄 and 𝑔_𝑗 is (𝐺_𝑔𝑗,𝛽^𝑔_𝑗)-invex at 𝑧 for each 𝑗∈𝑀, then sup_𝑦∈𝑌𝜙𝑥,𝑦⩾𝜈.(38)

Proof. Suppose to the contrary that sup_𝑦∈𝑌𝜙(𝑥,𝑦)<𝜈. Therefore, we obtain 𝜙𝑥,𝑦<𝜈≤𝜙𝑧,𝑦_𝑖,𝑦_𝑖∈𝑌(𝑧),∀𝑦∈𝑌,𝑖∈𝑄.(39) Thus, we obtain from the monotonicity assumption of 𝐺_𝜙 that 𝐺_𝜙∘𝜙𝑥,𝑦_𝑖<𝐺_𝜙∘𝜙𝑧,𝑦_𝑖,𝑦_𝑖∈𝑌(𝑧),𝑖∈𝑄.(40) Again, we obtain from the monotonicity assumption of 𝐺_𝑔𝑗 and the fact 𝑔_𝑗(𝑥)≤0,𝜇_𝑗𝑔_𝑗(𝑧)≥0,𝜇_𝑗≥0,𝑗∈𝑀(41) that 𝐺_𝑔𝑗∘𝑔_𝑗(𝑥)≤𝐺_𝑔𝑗∘𝑔_𝑗(𝑧),𝑗∈𝑀.(42) Hence, ^𝑞_𝑖=1𝜆_𝑖𝐺_𝜙∘𝜙𝑥,𝑦_𝑖−𝐺_𝜙∘𝜙𝑧,𝑦_𝑖𝛽^𝜙_𝑖()+𝑥,𝑧_𝑚𝑗=1𝜇_𝑗𝐺_𝑔𝑗∘𝑔_𝑗(𝑥)−𝐺_𝑔𝑗∘𝑔_𝑗(𝑧)𝛽^𝑔_𝑗(𝑥,𝑧)<0.(43) Similar to the proof of Theorem 8, by (43) and the generalized invexity assumptions of 𝜙(⋅,𝑦_𝑖) and 𝑔_𝑗, we have ^𝑞_𝑖=1𝜆_𝑖𝐺^′_𝜙𝜙𝑧,𝑦_𝑖𝜉^𝜙_𝑖+^𝑚_𝑗=1𝜇_𝑗𝐺^′_𝑔𝑗𝑔_𝑗(𝑧)𝜉^𝑔_𝑗,𝜂(𝑥,𝑧)<0,(44) which follows that 0∈^╱𝑞_𝑖=1𝜆_𝑖𝐺^′_𝜙𝜙𝑧,𝑦_𝑖𝜕_𝑥𝜙𝑧,𝑦_𝑖+^𝑚_𝑗=1𝜇_𝑗𝐺^′_𝑔𝑗𝑔_𝑗(𝑧)𝜕𝑔_𝑗(𝑧).(45) Thus, we have a contradiction to (34). So sup_𝑦∈𝑌𝜙(𝑥,𝑦)⩾𝜈.

Theorem 11 (𝐺-strong duality). Let problem (P) satisfy Conditions 4 and 5; let 𝑥^∗ be an optimal solution of problem (P). Suppose that 𝐺_𝜙 is both continuously differentiable and strictly increasing on 𝐼_𝜙(𝑋,𝑌) and 𝐺_𝑔𝑗 is both continuously differentiable and strictly increasing on 𝐼_𝑔𝑗(𝑋) with 𝐺^′_𝑔𝑗(𝑔_𝑗(𝑥^∗))>0 for each 𝑗∈𝑀. If the hypothesis of Theorem 10 holds for all (𝐷𝐼)-feasible points (𝑧,𝜇,𝜈,𝑞,𝜆,𝑦), then there exist (𝑞^∗,𝜆^∗,𝑦^∗)∈𝐾(𝑧) and (𝑥^∗,𝜇^∗,𝜈^∗)∈𝐻₁(𝑞^∗,𝜆^∗,𝑦^∗) such that (𝑞^∗,𝜆^∗,𝑦^∗,𝑥^∗,𝜇^∗,𝜈^∗) is a (DI) optimal solution, and the two problems (P) and (DI) have the same optimal values.

Proof. By Theorem 7, there exists 𝜈^∗=𝜙(𝑥^∗,𝑦^∗_𝑖),𝑖=1,…,𝑞^∗, satisfying the requirements specified in the theorem, such that (𝑞^∗,𝜆^∗,𝑦^∗,𝑥^∗,𝜇^∗,𝜈^∗) is a (DI) feasible solution, then the optimality of this feasible solution for (DI) follows from Theorem 10.

Theorem 12 (𝐺-strict converse duality). Let 𝑥 and (𝑧,𝜇,𝜈,𝑞,𝜆,𝑦) be optimal solutions for (P) and (DI), respectively. Suppose that 𝐺_𝜙 is both continuously differentiable and strictly increasing on 𝐼_𝜙(𝑋,𝑌) and 𝐺_𝑔𝑗 is both continuously differentiable and strictly increasing on 𝐼_𝑔𝑗(𝑋) for each 𝑗∈𝑀. If 𝜙(⋅,𝑦_𝑖) is (𝐺_𝜙,𝛽^𝜙_𝑖)-invex at 𝑧 for each 𝑖∈𝑄 and 𝑔_𝑗 is (𝐺_𝑔𝑗,𝛽^𝑔_𝑗)-invex at 𝑧 for each 𝑗∈𝑀, then 𝑥=𝑧; that is, 𝑧 is a (𝑃)-optimal solution and sup_𝑦∈𝑌𝜙(𝑥,𝑦)=𝜈.

Proof. Suppose to the contrary that 𝑥≠𝑧. Similar to the arguments as in the proof of Theorem 8, there exist 𝜉^𝜙_𝑖∈𝜙(𝑧,𝑦_𝑖) and 𝜉^𝑔_𝑗∈𝑔_𝑗(𝑧) such that 0=^𝑞_𝑖=1𝜆_𝑖𝐺^′_𝜙𝜙𝑧,𝑦_𝑖𝜉^𝜙_𝑖+^𝑚_𝑗=1𝜇_𝑗𝐺^′_𝑔𝑗𝑔_𝑗(𝑧)𝜉^𝑔_𝑗,𝜂(<𝑥,𝑧)_𝑞𝑖=1𝜆_𝑖𝐺_𝜙∘𝜙𝑥,𝑦_𝑖−𝐺_𝜙∘𝜙𝑧,𝑦_𝑖𝛽^𝜙_𝑖()+𝑥,𝑧_𝑚𝑗=1𝜇_𝑗𝐺_𝑔𝑗∘𝑔_𝑗(𝑥)−𝐺_𝑔𝑗∘𝑔_𝑗(𝑧)𝛽^𝑔_𝑗(),𝑥,𝑧_𝑚𝑗=1𝜇_𝑗𝐺_𝑔𝑗∘𝑔_𝑗(𝑥)−𝐺_𝑔𝑗∘𝑔_𝑗(𝑧)𝛽^𝑔_𝑗(𝑥,𝑧)≤0.(46) Therefore, ^𝑞_𝑖=1𝜆_𝑖𝐺_𝜙∘𝜙𝑥,𝑦_𝑖−𝐺_𝜙∘𝜙𝑧,𝑦_𝑖𝛽^𝜙_𝑖(𝑥,𝑧)>0.(47) From the above inequality, we can conclude that there exists 𝑖₀∈𝑄, such that 𝐺_𝜙∘𝜙𝑥,𝑦_𝑖0−𝐺_𝜙∘𝜙𝑧,𝑦_𝑖0>0(48) or 𝜙𝑥,𝑦_𝑖0>𝜙𝑧,𝑦_𝑖0.(49) It follows that sup_𝑦∈𝑌𝜙𝑥,𝑦≥𝜙𝑥,𝑦_𝑖0>𝜙𝑧,𝑦_𝑖0>𝜈.(50) On the other hand, we know from Theorem 10 that sup_𝑦∈𝑌𝜙𝑥,𝑦=𝜈.(51) This contradicts to (50).

5. Conclusion

In this paper, we have discussed the applications of (𝐺,𝛽)-invexity for a class of nonsmooth minimax programming problem (P). Firstly, we established 𝐺-necessary optimality conditions for problem (P). Under the nondifferential (𝐺,𝛽)-invexity assumptions, we have also derived the sufficiency of the 𝐺-necessary optimality conditions for the same problem. Further, we have constructed a dual model (DI) and derived 𝐺-duality results between problems (P) and (DI). Note that many researchers are interested in dealing with the minimax programming under generalized invexity assumptions; see [1, 10, 11, 14–17]. However, we have not found results for minimax programming problems under the 𝐺-invexity or its extension assumptions. Hence, this work extends the applications of 𝐺-invexity to the generalized minimax programming as well as to the nonsmooth case.

Acknowledgments

The authors are grateful to the referee for his valuable suggestions that helped to improve this paper in its present form. This research is supported by Science Foundation of Hanshan Normal University (LT200801).