Nondifferentiable Minimax Programming Problems in Complex Spaces Involving Generalized Convex Functions

Abstract

We start our discussion with a class of nondifferentiable minimax programming problems in complex space and establish sufficient optimality conditions under generalized convexity assumptions. Furthermore, we derive weak, strong, and strict converse duality theorems for the two types of dual models in order to prove that the primal and dual problems will have no duality gap under the framework of generalized convexity for complex functions.

1. Introduction

The literature of the mathematical programming is crowded with necessary and sufficient conditions for a point to be an optimal solution to the optimization problem. Levinson [1] was the first to study mathematical programming in complex space who extended the basic theorems of linear programming over complex space. In particular, using a variant of the Farkas lemma from real space to complex space, he generalized duality theorems from real linear programming. Since then, linear fractional, nonlinear, and nonlinear fractional complex programming problems were studied by many researchers (see [2–5]).

Minimax problems are encountered in several important contexts. One of the major context is zero sum games, where the objective of the first player is to minimize the amount given to the other player and the objective of the second player is to maximize this amount. Ahmad and Husain [6] established sufficient optimality conditions for a class of nondifferentiable minimax fractional programming problems involving (𝐹,𝛼,𝜌,𝑑)-convexity. Later on, Jayswal et al. [7] extended the work of Ahmad and Husain [6] to establish sufficient optimality conditions and duality theorems for the nondifferentiable minimax fractional problem under the assumptions of generalized (𝐹,𝛼,𝜌,𝑑)-convexity. Recently, Jayswal and Kumar [8] established sufficient optimality conditions and duality theorems for a class of nondifferentiable minimax fractional programming problems under the assumptions of (𝐶,𝛼,𝜌,𝑑)-convexity. Lai et al. [9] established several sufficient optimality conditions for minimax programming in complex spaces under the assumptions of generalized convexity of complex functions. Subsequently, they applied the optimality conditions to formulate parametric dual and derived weak, strong, and strict converse duality theorems.

The first work on fractional programming in complex space appeared in 1970, when Swarup and Sharma [10] generalized the results of Charnes and Cooper [11] to the complex space. Lai and Huang [12] showed that a minimax fractional programming problem is equivalent to a minimax nonfractional parametric problem for a given parameter in complex space and established the necessary and sufficient optimality conditions for nondifferentiable minimax fractional programming problem with complex variables under generalized convexity assumptions.

Recently, Lai and Liu [13] considered a nondifferentiable minimax programming problem in complex space and established the appropriate duality theorems for parametric dual and parameter free dual models. They showed that there is no duality gap between the two dual problems with respect to the primal problem under some generalized convexities of complex functions in the complex programming problem.

In this paper, we focus our study on nondifferentiable minimax programming over complex spaces. The paper is organized as follows. In Section 2, we recall some notations and definitions in complex spaces. In Section 3, we establish sufficient optimality conditions under generalized convexity assumptions. Weak, strong, and strict converse duality theorems related to nondifferentiable minimax programming problems in complex spaces for two types of dual models are established in Sections 4 and 5 followed by the conclusion in Section 6.

2. Notations and Preliminaries

We use the following notations that appear in most works on mathematical programming in complex space: 𝐶^𝑛(𝑅^𝑛) = 𝑛-dimensional vector space of complex (real) numbers, 𝐶^𝑚×𝑛(𝑅^𝑚×𝑛) = the set of 𝑚×𝑛 complex (real) matrices, 𝑅^𝑛₊={𝑥∈𝑅^𝑛,𝑥_𝑗≥0,𝑗=1,2,…,𝑛} = the nonnegative orthant of 𝑅^𝑛, 𝐴^𝐻=𝐴^𝑇 = the conjugate transpose of 𝐴=[𝑎_𝑖𝑗], ⟨𝑧,𝑢⟩=𝑢^𝐻𝑧= the inner product of 𝑢,𝑧 in 𝐶^𝑛.

Now, we recall some definitions related to mathematical programming in complex space that are used in the sequel of the paper.

Definition 1 (see [5]). A subset 𝑆⊆𝐶^𝑛 is polyhedral cone if there is 𝑘∈𝑁 and 𝐴∈𝐶^𝑛×𝑘 such that 𝑆=𝐴𝑅^𝑘₊={𝐴𝑥∣𝑥∈𝑅^𝑘₊}; that is, 𝑆 is generated by a finite number of vectors (the columns of 𝐴).
Equivalently, 𝑆⊆𝐶^𝑛 is said to be a polyhedral cone if it is the intersection of a finite number of closed half-spaces having the origin on the boundary; that is, there is a natural number 𝑝 and 𝑝-points 𝑢₁,𝑢₂,…,𝑢_𝑝 such that 𝑆=^𝑝_𝑘=1𝐷𝑢_𝑘=𝑧∈𝐶^𝑛∣Re𝑧,𝑢_𝑘≥0,𝑘=1,2,…,𝑝,(1) where 𝐷(𝑢_𝑘), 𝑘=1,2,…,𝑝 are closed half-spaces involving the point 𝑢_𝑘.

Definition 2 (see [5]). If ∅≠𝑆⊂𝐶^𝑛, then 𝑆^∗={𝑦∈𝐶^𝑛∣forall𝑧∈𝑆⇒Re(𝑦^𝐻𝑧)≥0} constitute the dual (polar) of 𝑆.
If Θ∶𝐶^𝑛→𝐶 is analytic in a neighbourhood of 𝑧₀∈𝐶^𝑛, then ∇_𝑧Θ(𝑧₀)=[𝜕Θ(𝑧₀)/𝜕𝑧_𝑖], 𝑖=1,2,…,𝑛, is the gradient of function Θ at 𝑧₀. Similarly, if the complex function Θ(𝑤¹,𝑤²) is analytic in 2𝑛 variables (𝑤¹,𝑤²) and (𝑧₀,𝑧₀)∈𝐶^2𝑛, we define the gradients by ∇_𝑧Θ𝑧₀,𝑧₀=𝜕Θ𝑧₀,𝑧₀𝜕𝑤¹_𝑗∇,𝑗=1,2,…,𝑛,_𝑧Θ𝑧₀,𝑧₀=𝜕Θ𝑧₀,𝑧₀𝜕𝑤²_𝑗,𝑗=1,2,…,𝑛.(2) In this paper, we consider the following complex programming problem: min_𝜁∈𝑋sup_𝜂∈𝑌Re𝑓𝜁,𝜂+𝑧^𝐻𝐴𝑧^1/2subjectto𝜁∈𝑋=𝜁=(𝑧,𝑧)∈𝐶^2𝑛(𝜁),(P)∣−ℎ∈𝑆
where 𝑌={𝜂=(𝑤,𝑤)∣𝑤∈𝐶^𝑚} is a compact subset in 𝐶^2𝑚, 𝐴∈𝐶^𝑛×𝑛 is a positive semidefinite Hermitian matrix, 𝑆 is a polyhedral cone in 𝐶^𝑝,𝑓(⋅,⋅) is continuous, and, for each 𝜂∈𝑌, 𝑓(⋅,𝜂)∶𝐶^2𝑛→𝐶 and ℎ(⋅)∶𝐶^2𝑛→𝐶^𝑝 are analytic in 𝑄={(𝑧,𝑧)∣𝑧∈𝐶^𝑛}⊂𝐶^2𝑛, where 𝑄 is a linear manifold over a real field. In order to have a convex real part for a nonlinear analytic function, the complex functions need to be defined on the linear manifold over 𝑅; that is, 𝑄={𝜁=(𝑧,𝑧)∈𝐶^2𝑛∣𝑧∈𝐶^𝑛}.

Special Cases. (i) If problem (P) is a real programming problem with two variables nondifferentiable minimax problem, it may be expressed as minsup_𝑦∈𝑌𝑓𝑥,𝑦+𝑥^𝑇𝐵𝑥^1/2s.t.𝑔(𝑥)≤0,𝑥∈𝑅^𝑛,(3)

where 𝑌 is compact subset of 𝑅^𝑙,𝑓(⋅,⋅)∶𝑅^𝑛×𝑅^𝑙→𝑅 and 𝑔(⋅)∶𝑅^𝑛→𝑅^𝑚 are continuously differentiable functions at 𝑥∈𝑅^𝑛, and 𝐵 is a positive semidefinite symmetric matrix. This problem was studied by Ahmad et al. [14, 15].

(ii) If 𝑌 vanishes in (P), then problem (P) reduces to the problem considered by Mond and Craven [16]; that is, minRe𝑓(𝜁)+𝑧^𝐻𝐴𝑧^1/2s.t.𝜁∈𝑋=𝜁∈𝐶^2𝑛∣−ℎ(𝜁)∈𝑆,𝜁=(𝑧,𝑧),𝑧∈𝐶^𝑛.P₁

(iii) If 𝐴=0, then (P) becomes a differentiable complex minimax programming problem studied by Datta and Bhatia [3]; that is, min_𝜁∈𝑋sup_𝜂∈𝑌Re𝑓𝜁,𝜂s.t.𝜁∈𝑋=𝜁∈𝐶^2𝑛(𝜁).P∣−ℎ∈𝑆⁰

Definition 3. A functional 𝐹∶𝐶^𝑛×𝐶^𝑛×𝐶^𝑛→𝑅 is said to be sublinear in its third variable, if, for any 𝑧₁,𝑧₂∈𝐶^𝑛, the following conditions are satisfied: (i)𝐹(𝑧₁,𝑧₂;𝑢₁+𝑢₂)≤𝐹(𝑧₁,𝑧₂;𝑢₁)+𝐹(𝑧₁,𝑧₂;𝑢₂), (ii)𝐹(𝑧₁,𝑧₂;𝛼𝑢)=𝛼𝐹(𝑧₁,𝑧₂;𝑢), for any 𝛼≥0 in 𝑅₊ and 𝑢₁,𝑢₂,𝑢∈𝐶^𝑛. From (ii), it is clear that 𝐹(𝑧₁,𝑧₂;0)=0.
Let 𝐹∶𝐶^𝑛×𝐶^𝑛×𝐶^𝑛→𝑅 be sublinear on the third variable, 𝜃∶𝐶^𝑛×𝐶^𝑛→𝑅₊ with 𝜃(𝑧₁,𝑧₂)=0, if 𝑧₁=𝑧₂ and 𝛼∶𝐶^𝑛×𝐶^𝑛×→𝑅₊⧵{0}. Let 𝑓 and ℎ be analytic functions and 𝜌 let be a real number. Now, we introduce the following definitions, which are extensions of the definitions given by Lai et al. [9] and Mishra and Rueda [17].

Definition 4. The real part Re[𝑓] of analytic function 𝑓∶𝑄⊂𝐶^2𝑛→𝐶 is said to be (𝐹,𝛼,𝜌,𝜃)-convex (strict (𝐹,𝛼,𝜌,𝜃)-convex) with respect to 𝑅₊ on the manifold 𝑄={𝜁=(𝑧,𝑧)∣𝑧∈𝐶^𝑛}⊂𝐶^2𝑛, if, for any 𝜁=(𝑧,𝑧), 𝜁₀=(𝑧₀,𝑧₀)∈𝑄, one has Re𝑓(𝑧,𝑧)−𝑓𝑧₀,𝑧₀≥(>)𝐹𝑧,𝑧₀;𝛼𝑧,𝑧₀×∇_𝑧𝑓𝑧₀,𝑧₀+∇_𝑧𝑓𝑧₀,𝑧₀+𝜌𝜃²𝑧,𝑧₀.(4)

Definition 5. The real part Re[𝑓] of analytic function 𝑓∶𝑄⊂𝐶^2𝑛→𝐶 is said to be (𝐹,𝛼,𝜌,𝜃)-quasiconvex (strict (𝐹,𝛼,𝜌,𝜃)-quasiconvex) with respect to 𝑅₊ on the manifold 𝑄={𝜁=(𝑧,𝑧)∣𝑧∈𝐶^𝑛}⊂𝐶^2𝑛, if, for any 𝜁=(𝑧,𝑧), 𝜁₀=(𝑧₀,𝑧₀)∈𝑄, one has Re𝑓(𝑧,𝑧)−𝑓𝑧₀,𝑧₀≤(<)0⟹𝐹𝑧,𝑧₀;𝛼𝑧,𝑧₀×∇_𝑧𝑓𝑧₀,𝑧₀+∇_𝑧𝑓𝑧₀,𝑧₀≤−𝜌𝜃²𝑧,𝑧₀.(5)

Definition 6. The real part Re[𝑓] of analytic function 𝑓∶𝑄⊂𝐶^2𝑛→𝐶 is said to be (𝐹,𝛼,𝜌,𝜃)-pseudoconvex (strict (𝐹,𝛼,𝜌,𝜃)-pseudoconvex) with respect to 𝑅₊ on the manifold 𝑄={𝜁=(𝑧,𝑧)∣𝑧∈𝐶^𝑛}⊂𝐶^2𝑛, if, for any 𝜁=(𝑧,𝑧), 𝜁₀=(𝑧₀,𝑧₀)∈𝑄, one has 𝐹𝑧,𝑧₀;𝛼𝑧,𝑧₀∇_𝑧𝑓𝑧₀,𝑧₀+∇_𝑧𝑓𝑧₀,𝑧₀≥−𝜌𝜃²𝑧,𝑧₀⟹Re𝑓(𝑧,𝑧)−𝑓𝑧₀,𝑧₀≥(>)0.(6)

Definition 7. The mapping ℎ∶𝐶^2𝑛→𝐶^𝑝 is said to be (𝐹,𝛼,𝜌,𝜃)-convex (strict (𝐹,𝛼,𝜌,𝜃)-convex) with respect to the polyhedral cone 𝑆⊂𝐶^𝑝 on the manifold 𝑄 if, for any 𝜇∈𝑆 and 𝜁=(𝑧,𝑧), 𝜁₀=(𝑧₀,𝑧₀)∈𝑄, one has Re𝜇,ℎ(𝜁)−ℎ𝜁₀≥(>)𝐹𝑧,𝑧₀;𝛼𝑧,𝑧₀×𝜇^𝑇∇_𝑧ℎ𝜁₀+𝜇^𝐻∇_𝑧ℎ𝜁₀+𝜌𝜃²𝑧,𝑧₀.(7)

Definition 8. The mapping ℎ∶𝐶^2𝑛→𝐶^𝑝 is said to be (𝐹,𝛼,𝜌,𝜃)-quasiconvex (strict (𝐹,𝛼,𝜌,𝜃)-quasiconvex) with respect to the polyhedral cone 𝑆⊂𝐶^𝑝 on the manifold 𝑄 if, for any 𝜇∈𝑆 and 𝜁=(𝑧,𝑧), 𝜁₀=(𝑧₀,𝑧₀)∈𝑄, one has Re𝜇,ℎ(𝜁)−ℎ𝜁₀≤(<)0⟹𝐹𝑧,𝑧₀;𝛼𝑧,𝑧₀×𝜇^𝑇∇_𝑧ℎ𝜁₀+𝜇^𝐻∇_𝑧ℎ𝜁₀≤−𝜌𝜃²𝑧,𝑧₀.(8)

Definition 9. The mapping ℎ∶𝐶^2𝑛→𝐶^𝑝 is said to be (𝐹,𝛼,𝜌,𝜃)-pseudoconvex (strict (𝐹,𝛼,𝜌,𝜃)-pseudoconvex) with respect to the polyhedral cone 𝑆⊂𝐶^𝑝 on the manifold 𝑄 if for any 𝜇∈𝑆 and 𝜁=(𝑧,𝑧), 𝜁₀=(𝑧₀,𝑧₀)∈𝑄, one has 𝐹𝑧,𝑧₀;𝛼𝑧,𝑧₀𝜇^𝑇∇_𝑧ℎ𝜁₀+𝜇^𝐻∇_𝑧ℎ𝜁₀≥𝜌𝜃²𝑧,𝑧₀⟹Re𝜇,ℎ(𝜁)−ℎ𝜁₀≥(>)0.(9)

Remark 10. In the proofs of theorems, sometimes it may be more convenient to use certain alternative but equivalent forms of the above definitions. Consider the following example.
The real part Re[𝑓] of analytic function 𝑓∶𝑄⊂𝐶^2𝑛→𝐶 is said to be (𝐹,𝛼,𝜌,𝜃)-pseudoconvex with respect to 𝑅₊ on the manifold 𝑄={𝜁=(𝑧,𝑧)∣𝑧∈𝐶^𝑛}⊂𝐶^2𝑛, if, for any 𝜁₀=(𝑧₀,𝑧₀)∈𝑄, one has Re𝑓(𝑧,𝑧)−𝑓𝑧₀,𝑧₀<0⟹𝐹𝑧,𝑧₀;𝛼𝑧,𝑧₀×∇_𝑧𝑓𝑧₀,𝑧₀+∇_𝑧𝑓𝑧₀,𝑧₀<−𝜌𝜃²𝑧,𝑧₀.(10)

Remark 11. If we take 𝛼(𝑧,𝑧₀)=1, then the above definitions reduce to that given by Lai et al. [9]. In addition, if we take 𝜌=0, then we obtain the definitions given by Mishra and Rueda [17].
Let 𝐴∈𝐶^𝑛×𝑛 and 𝑧,𝑢∈𝐶^𝑛; then Schwarz inequality can be written as Re𝑧^𝐻𝐴𝑢≤𝑧^𝐻𝐴𝑧^1/2𝑢^𝐻𝐴𝑢^1/2.(11) The equality holds if 𝐴𝑧=𝜆𝐴𝑢 or 𝑧=𝜆𝑢 for 𝜆≥0.

Definition 12 (see [12]). The problem (P) is said to satisfy the constraint qualification at a point 𝜁₀=(𝑧₀,𝑧₀), if, for any nonzero 𝜇∈𝑆^∗⊂𝐶^𝑝, Reℎ^′_𝜁𝜁₀𝜁−𝜁₀,𝜇≠0,for𝜁≠𝜁₀.(12)
In the next section, we recall some notations and discuss necessary and sufficient optimality conditions for problem (P) on the basis of Lai and Liu [18] and Lai and Huang [12].

3. Necessary and Sufficient Conditions

Let 𝑓(𝜁,⋅), 𝜁=(𝑧,𝑧)∈𝐶^2𝑛 be a continuous function defined on 𝑌, where 𝑌⊂𝐶^2𝑚 is a specified compact subset in problem (P). Then the supremum sup_𝜈∈𝑌Re𝑓(𝜁,𝜈) will be attained to its maximum in 𝑌, and the set 𝑌(𝜁)=𝜂∈𝑌∣Re𝑓𝜁,𝜂=sup_𝜈∈𝑌Re𝑓(𝜁,𝜈)(13) is then also a compact set in 𝐶^2𝑚. In particular, if 𝜁=𝜁₀=(𝑧₀,𝑧₀) is an optimal solution of problem (P), there exist a positive integer 𝑘 and finite points 𝜂_𝑖∈𝑌(𝜁₀), 𝜆_𝑖>0, 𝑖=1,2,…,𝑘 with ∑^𝑘_𝑖=1𝜆_𝑖=1 such that the Lagrangian function 𝜙(𝜁)=^𝑘_𝑖=1𝜆_𝑖𝑓𝜁,𝜂_𝑖+ℎ(𝜁),𝜇,𝜇≠0in𝑆^∗,(14) satisfies the Kuhn-Tucker type condition at 𝜁₀. That is, ^𝑘_𝑖=1𝜆_𝑖𝑓^′_𝜁𝜁₀,𝜂_𝑖+ℎ^′_𝜁𝜁₀,𝜇𝜁−𝜁₀ℎ𝜁=0,(15)Re₀,𝜇=0.(16) Equivalent form of expression (15) at 𝜁=𝜁₀∈𝑄 is ^𝑘_𝑖=1𝜆_𝑖∇_𝑧𝑓𝜁₀,𝜂_𝑖+∇_𝑧𝑓𝜁₀,𝜂_𝑖+𝜇^𝑇∇_𝑧ℎ𝜁₀+𝜇^𝐻∇_𝑧ℎ𝜁₀=0.(17) For the integer 𝑘, corresponding a vector 𝜂≡(𝜂₁,𝜂₂,…,𝜂_𝑘)∈𝑌(𝜁₀)^𝑘 and 𝜆_𝑖>0, 𝑖=1,2,…,𝑘 with ∑^𝑘_𝑖=1𝜆_𝑖=1, we define a set as follows: 𝑍𝜂𝜁₀=𝜁∈𝐶^2𝑛∣−ℎ^′_𝜁𝜁₀𝜁∈𝑆−ℎ𝜁₀,𝜁=(𝑧,𝑧)∈𝑄,Re^𝑘_𝑖=1𝜆_𝑖𝑓^′_𝜁𝜁₀,𝜂_𝑖𝜁+⟨𝐴𝑧,𝑧⟩^1/2<0,(18) where the set 𝑆(𝑠₀) is the intersection of closed half-spaces having the point 𝑠₀∈𝑆 on their boundaries.

Theorem 13 (necessary optimality conditions). Let 𝜁₀=(𝑧₀,𝑧₀)∈𝑄 be an optimal solution to (P). Suppose that the constraint qualification is satisfied for (P) at 𝜁₀ and 𝑧^𝐻₀𝐴𝑧₀=⟨𝐴𝑧₀,𝑧₀⟩>0. Then there exist 0≠𝜇∈𝑆^∗⊂𝐶^𝑝, 𝑢∈𝐶^𝑛 and a positive integer 𝑘 with the following properties:(i)𝜂_𝑖∈𝑌(𝜁₀), 𝑖=1,2,…,𝑘,(ii) 𝜆_𝑖>0, 𝑖=1,2,…,𝑘, ∑^𝑘_𝑖=1𝜆_𝑖=1,such that ∑^𝑘_𝑖=1𝜆_𝑖𝑓(𝜁,𝜂_𝑖)+⟨ℎ(𝜁),𝜇⟩+⟨𝐴𝑧,𝑧⟩^1/2 satisfies the following conditions: ^𝑘_𝑖=1𝜆_𝑖∇_𝑧𝑓𝜁₀,𝜂_𝑖+∇_𝑧𝑓𝜁₀,𝜂_𝑖+𝜇+𝐴𝑢_𝑇∇_𝑧ℎ𝜁₀+𝜇^𝐻∇_𝑧ℎ𝜁₀ℎ𝜁=0,(19)Re₀𝑢,𝜇=0,(20)_𝐻𝑧𝐴𝑢≤1,(21)_𝐻0𝐴𝑧₀^1/2=Re𝑧^𝐻₀𝐴𝑢.(22)

Theorem 14 (sufficient optimality conditions). Let 𝜁₀=(𝑧₀,𝑧₀)∈𝑄 be a feasible solution to (P). Suppose that there exists a positive integer 𝑘,𝜆_𝑖>0, 𝜂_𝑖∈𝑌(𝜁₀), 𝑖=1,2,…,𝑘 with ∑^𝑘_𝑖=1𝜆_𝑖=1 and 0≠𝜇∈𝑆^∗⊂𝐶^𝑝 satisfying conditions (19)–(22). Further, if Re[∑^𝑘_𝑖=1𝜆_𝑖[𝑓(𝜁,𝜂_𝑖)+𝑧^𝐻𝐴𝑢]] is (𝐹,𝛼₁,𝜌₁,𝜃)-convex with respect to 𝑅₊ on 𝑄,ℎ(𝜁) is (𝐹,𝛼₂,𝜌₂,𝜃)-convex on 𝑄 with respect to the polyhedral cone 𝑆⊂𝐶^𝑝, and 𝜌₁/𝛼₁(𝑧,𝑧₀)+𝜌₂/𝛼₂(𝑧,𝑧₀)≥0, then 𝜁₀=(𝑧₀,𝑧₀) is an optimal solution to (P).

Proof. We prove this theorem by contradiction. Suppose that there is a feasible solution 𝜁∈𝑄 such that sup_𝜂∈𝑌Re𝑓𝜁,𝜂+𝑧^𝐻𝐴𝑧^1/2<sup_𝜂∈𝑌Re𝑓𝜁₀,𝜂+𝑧^𝐻₀𝐴𝑧₀^1/2.(23) Since 𝜂_𝑖∈𝑌(𝜁₀), 𝑖=1,2,…,𝑘, we have sup_𝜂∈𝑌Re𝑓𝜁₀,𝜂+𝑧^𝐻₀𝐴𝑧₀^1/2=Re𝑓𝜁₀,𝜂_𝑖+𝑧^𝐻₀𝐴𝑧₀^1/2,𝑓for𝑖=1,2,…,𝑘,Re𝜁,𝜂_𝑖+𝑧^𝐻𝐴𝑧^1/2<sup_𝜂∈𝑌Re𝑓𝜁,𝜂+𝑧^𝐻𝐴𝑧^1/2,for𝑖=1,2,…,𝑘.(24) Thus, from the above three inequalities, we obtain Re𝑓𝜁,𝜂_𝑖+𝑧^𝐻𝐴𝑧^1/2<Re𝑓𝜁₀,𝜂_𝑖+𝑧^𝐻₀𝐴𝑧₀^1/2,for𝑖=1,2,…,𝑘.(25) Using (21) and generalized Schwarz inequality, we get Re𝑧^𝐻𝐴𝑢≤𝑧^𝐻𝐴𝑧^1/2𝑢^𝐻𝐴𝑢^1/2≤𝑧^𝐻𝐴𝑧^1/2=Re𝑧^𝐻𝐴𝑧^1/2(26) and inequality (22) yields Re𝑧^𝐻₀𝐴𝑢=Re𝑧^𝐻₀𝐴𝑧₀^1/2.(27) Using (26) and (27) in (25), we have Re𝑓𝜁,𝜂_𝑖+𝑧^𝐻𝐴𝑢<Re𝑓𝜁₀,𝜂_𝑖+𝑧^𝐻₀𝐴𝑢,for𝑖=1,2,…,𝑘.(28) Since 𝜆_𝑖>0 and ∑^𝑘_𝑖=1𝜆_𝑖=1, we have Re^𝑘_𝑖=1𝜆_𝑖𝑓𝜁,𝜂_𝑖+𝑧^𝐻𝐴𝑢−^𝑘_𝑖=1𝜆_𝑖𝑓𝜁₀,𝜂_𝑖+𝑧^𝐻₀𝐴𝑢<0.(29) Since Re[∑^𝑘_𝑖=1𝜆_𝑖[𝑓(𝜁,𝜂_𝑖)+𝑧^𝐻𝐴𝑢]] is (𝐹,𝛼₁,𝜌₁,𝜃)-convex with respect to 𝑅₊ on 𝑄, we have Re^𝑘_𝑖=1𝜆_𝑖𝑓𝜁,𝜂_𝑖+𝑧^𝐻−𝐴𝑢_𝑘𝑖=1𝜆_𝑖𝑓𝜁₀,𝜂_𝑖+𝑧^𝐻₀𝐴𝑢≥𝐹𝑧,𝑧₀;𝛼₁𝑧,𝑧₀×^𝑘_𝑖=1𝜆_𝑖∇_𝑧𝑓𝜁₀,𝜂_𝑖+∇_𝑧𝑓𝜁₀,𝜂_𝑖+𝐴𝑢+𝜌₁𝜃²𝑧,𝑧₀.(30) From (29) and (30), we conclude that 𝐹𝑧,𝑧₀;𝛼₁𝑧,𝑧₀×^𝑘_𝑖=1𝜆_𝑖∇_𝑧𝑓(𝜁₀,𝜂_𝑖)+∇_𝑧𝑓𝜁₀,𝜂_𝑖+𝐴𝑢<−𝜌₁𝜃²𝑧,𝑧₀,(31) which due to sublinearity of 𝐹 can be written as 𝐹𝑧,𝑧₀;^𝑘_𝑖=1𝜆_𝑖∇_𝑧𝑓𝜁₀,𝜂_𝑖+∇_𝑧𝑓𝜁₀,𝜂_𝑖𝜌+𝐴𝑢<−¹𝛼₁𝑧,𝑧₀𝜃²𝑧,𝑧₀.(32) On the other hand, from the feasibility of 𝜁 to (P), we have −ℎ(𝜁)∈𝑆, or Re⟨ℎ(𝜁),𝜇⟩≤0 for 𝜇∈𝑆^∗, which along with (20) yields Reℎ(𝜁),𝜇≤0=Reℎ𝜁₀,𝜇.(33) Since ℎ(𝜁) is (𝐹,𝛼₂,𝜌₂,𝜃)-convex on 𝑄 with respect to the polyhedral cone 𝑆⊂𝐶^𝑝, we have Reℎ(𝜁),𝜇−Reℎ𝜁₀,𝜇≥𝐹𝑧,𝑧₀;𝛼₂𝑧,𝑧₀𝜇^𝑇∇_𝑧ℎ𝜁₀+𝜇^𝐻∇_𝑧ℎ𝜁₀+𝜌₂𝜃²𝑧,𝑧₀.(34) From (33) and (34), it follows that 𝐹𝑧,𝑧₀;𝛼₂𝑧,𝑧₀𝜇^𝑇∇_𝑧ℎ𝜁₀+𝜇^𝐻∇_𝑧ℎ𝜁₀≤−𝜌₂𝜃²𝑧,𝑧₀,(35) which due to sublinearity of 𝐹 can be written as 𝐹𝑧,𝑧₀;𝜇^𝑇∇_𝑧ℎ𝜁₀+𝜇^𝐻∇_𝑧ℎ𝜁₀𝜌≤−²𝛼₂𝑧,𝑧₀𝜃²𝑧,𝑧₀.(36) On adding (32) and (36) and using sublinearity of 𝐹, we get 𝐹𝑧,𝑧₀;^𝑘_𝑖=1𝜆_𝑖∇_𝑧𝑓𝜁₀,𝜂_𝑖+∇_𝑧𝑓𝜁₀,𝜂_𝑖+𝐴𝑢+𝜇^𝑇∇_𝑧ℎ𝜁₀+𝜇^𝐻∇_𝑧ℎ𝜁₀𝜌<−¹𝛼₁𝑧,𝑧₀+𝜌₂𝛼₂𝑧,𝑧₀𝜃²𝑧,𝑧₀.(37) The above inequality, together with the assumption 𝜌₁/𝛼₁(𝑧,𝑧₀)+𝜌₂/𝛼₂(𝑧,𝑧₀)≥0, gives 𝐹𝑧,𝑧₀;^𝑘_𝑖=1𝜆_𝑖∇_𝑧𝑓𝜁₀,𝜂_𝑖+∇_𝑧𝑓𝜁₀,𝜂_𝑖+𝐴𝑢+𝜇^𝑇∇_𝑧ℎ𝜁₀+𝜇^𝐻∇_𝑧ℎ𝜁₀<0,(38) which contradicts (19), hence the theorem.

Theorem 15 (sufficient optimality conditions). Let 𝜁₀=(𝑧₀,𝑧₀)∈𝑄 be a feasible solution to (P). Suppose that there exists a positive integer 𝑘,𝜆_𝑖>0, 𝜂_𝑖∈𝑌(𝜁₀), 𝑖=1,2,…,𝑘 with ∑^𝑘_𝑖=1𝜆_𝑖=1 and 0≠𝜇∈𝑆^∗⊂𝐶^𝑝 satisfying conditions (19)–(22). Further, if Re[∑^𝑘_𝑖=1𝜆_𝑖[𝑓(𝜁,𝜂_𝑖)+𝑧^𝐻𝐴𝑢]] is (𝐹,𝛼₁,𝜌₁,𝜃)-pseudoconvex with respect to 𝑅₊ on 𝑄,ℎ(𝜁) is (𝐹,𝛼₂,𝜌₂,𝜃)-quasiconvex on 𝑄 with respect to the polyhedral cone 𝑆⊂𝐶^𝑝, and 𝜌₁/𝛼₁(𝑧,𝑧₀)+𝜌₂/𝛼₂(𝑧,𝑧₀)≥0, then 𝜁₀=(𝑧₀,𝑧₀) is an optimal solution to (P).

Proof. Proceeding as in Theorem 14, we have Re^𝑘_𝑖=1𝜆_𝑖𝑓𝜁,𝜂_𝑖+𝑧^𝐻𝐴𝑢−^𝑘_𝑖=1𝜆_𝑖𝑓𝜁₀,𝜂_𝑖+𝑧^𝐻₀𝐴𝑢<0,(39) which, by (𝐹,𝛼₁,𝜌₁,𝜃)-pseudoconvexity of Re[∑^𝑘_𝑖=1𝜆_𝑖[𝑓(𝜁,𝜂_𝑖)+𝑧^𝐻𝐴𝑢]] with respect to 𝑅₊ on 𝑄, yields 𝐹𝑧,𝑧₀;𝛼₁𝑧,𝑧₀^𝑘_𝑖=1𝜆_𝑖∇_𝑧𝑓𝜁₀,𝜂_𝑖+∇_𝑧𝑓𝜁₀,𝜂_𝑖+𝐴𝑢<−𝜌₁𝜃²𝑧,𝑧₀.(40) Using the sublinearity of 𝐹, the above inequality can be written as 𝐹𝑧,𝑧₀;^𝑘_𝑖=1𝜆_𝑖∇_𝑧𝑓𝜁₀,𝜂_𝑖+∇_𝑧𝑓𝜁₀,𝜂_𝑖𝜌+𝐴𝑢<−¹𝛼₁𝑧,𝑧₀𝜃²𝑧,𝑧₀.(41) On the other hand, from the feasibility of 𝜁 to (P), we have −ℎ(𝜁)∈𝑆, or Re⟨ℎ(𝜁),𝜇⟩≤0 for 𝜇∈𝑆^∗, which along with (20) yields Reℎ(𝜁),𝜇≤0=Reℎ𝜁₀,𝜇.(42) Since ℎ(𝜁) is (𝐹,𝛼₂,𝜌₂,𝜃)-quasiconvex on 𝑄 with respect to the polyhedral cone 𝑆⊂𝐶^𝑝, the above inequality yields 𝐹𝑧,𝑧₀;𝛼₂𝑧,𝑧₀𝜇^𝑇∇_𝑧ℎ𝜁₀+𝜇^𝐻∇_𝑧ℎ𝜁₀≤−𝜌₂𝜃²𝑧,𝑧₀,(43) which due to sublinearity of 𝐹 can be written as 𝐹𝑧,𝑧₀;𝜇^𝑇∇_𝑧ℎ𝜁₀+𝜇^𝐻∇_𝑧ℎ𝜁₀𝜌≤−²𝛼₂𝑧,𝑧₀𝜃²𝑧,𝑧₀.(44) On adding (41) and (44) and using sublinearity of 𝐹, we get 𝐹𝑧,𝑧₀;^𝑘_𝑖=1𝜆_𝑖∇_𝑧𝑓𝜁₀,𝜂_𝑖+∇_𝑧𝑓𝜁₀,𝜂_𝑖+𝐴𝑢+𝜇^𝑇∇_𝑧ℎ𝜁₀+𝜇^𝐻∇_𝑧ℎ𝜁₀𝜌<−¹𝛼₁𝑧,𝑧₀+𝜌₂𝛼₂𝑧,𝑧₀𝜃²𝑧,𝑧₀.(45) The above inequality, together with the assumption 𝜌₁/𝛼₁(𝑧,𝑧₀)+𝜌₂/𝛼₂(𝑧,𝑧₀)≥0, gives 𝐹𝑧,𝑧₀;^𝑘_𝑖=1𝜆_𝑖∇_𝑧𝑓𝜁₀,𝜂_𝑖+∇_𝑧𝑓𝜁₀,𝜂_𝑖+𝐴𝑢+𝜇^𝑇∇_𝑧ℎ𝜁₀+𝜇^𝐻∇_𝑧ℎ𝜁₀<0,(46) which contradicts (19), hence the theorem.

Theorem 16 (sufficient optimality conditions). Let 𝜁₀=(𝑧₀,𝑧₀)∈𝑄 be a feasible solution to (P). Suppose that there exists a positive integer 𝑘,𝜆_𝑖>0, 𝜂_𝑖∈𝑌(𝜁₀), 𝑖=1,2,…,𝑘 with ∑^𝑘_𝑖=1𝜆_𝑖=1 and 0≠𝜇∈𝑆^∗⊂𝐶^𝑝 satisfying conditions (19)–(22). Further, if Re[∑^𝑘_𝑖=1𝜆_𝑖[𝑓(𝜁,𝜂_𝑖)+𝑧^𝐻𝐴𝑢]] is (𝐹,𝛼₁,𝜌₁,𝜃)-quasiconvex with respect to 𝑅₊ on 𝑄,ℎ(𝜁) is strict (𝐹,𝛼₂,𝜌₂,𝜃)-pseudoconvex on 𝑄 with respect to the polyhedral cone 𝑆⊂𝐶^𝑝, and 𝜌₁/𝛼₁(𝑧,𝑧₀)+𝜌₂/𝛼₂(𝑧,𝑧₀)≥0, then 𝜁₀=(𝑧₀,𝑧₀) is an optimal solution to (P).

Proof. The proof follows on the similar lines of Theorem 15.

4. Parametric Duality

We adopt the following notations in order to simplify the formulation of dual: 𝐾(𝜉)=𝑘,𝜆,𝜂∈𝑁×𝑅^𝑘₊×𝐶^2𝑚𝑘∣𝜆=𝜆₁,𝜆₂,…,𝜆_𝑘with^𝑘_𝑖=1𝜆_𝑖𝜂=1,𝜂=₁,𝜂₂,…,𝜂_𝑘with𝜂_𝑖∈𝑌(𝜉),𝑖=1,2,…,𝑘,(47) for 𝜉=(𝑢,𝑢)∈𝑄⊂𝐶^2𝑛.

Now, we formulate a parametric dual problem (D1) with respect to the complex minimax programming problem (P) as follows: max_𝑘,𝜆_,𝜂₍^𝜉₎^∈𝐾sup_{𝜉,𝜇,𝑤,𝑡∈𝑋𝑘,}𝜆_,𝜂()𝑡,D1 where 𝑋(𝑘,𝜆,𝜂) denotes the set of all (𝜉,𝜇,𝑤,𝑡)∈𝐶^2𝑛×𝐶^𝑝×𝐶^𝑛×𝑅 to satisfy the following conditions: ^𝑘_𝑖=1𝜆_𝑖∇_𝑧𝑓𝜉,𝜂_𝑖+∇_𝑧𝑓𝜉,𝜂_𝑖+𝐴𝑤+𝜇^𝑇∇_𝑧ℎ(𝜉)+𝜇^𝐻∇_𝑧ℎ(𝜉)=0,(48)_𝑘𝑖=1𝜆_𝑖Re𝑓𝜉,𝜂_𝑖+𝑢^𝐻𝐴𝑢^1/2ℎ(𝜉)𝑤−𝑡≥0,(49)Re,𝜇≥0,(50)_𝐻𝑢𝐴𝑤≤1,(51)_𝐻𝐴𝑢^1/2=Re𝑢^𝐻𝐴𝑤,(52)0≠𝜇∈𝑆^∗.(53) If, for a triplet (𝑘,𝜆,𝜂)∈𝐾(𝜉), the set 𝑋(𝑘,𝜆,𝜂)=∅, then we define the supremum over 𝑋(𝑘,𝜆,𝜂) to be −∞ for nonexecption in the formulation of (D1).

Theorem 17 (weak duality). Let 𝜁=(𝑧,𝑧) and (𝑘,𝜆,𝜂,𝜉,𝜇,𝑤,𝑡) be feasible solutions to (P) and (D1), respectively. Further, if Re[∑^𝑘_𝑖=1𝜆_𝑖[𝑓(𝜁,𝜂_𝑖)+𝑧^𝐻𝐴𝑤]] is (𝐹,𝛼₁,𝜌₁,𝜃)-pseudoconvex with respect to 𝑅₊ on 𝑄,ℎ(𝜁) is (𝐹,𝛼₂,𝜌₂,𝜃)-quasiconvex on 𝑄 with respect to the polyhedral cone 𝑆⊂𝐶^𝑝, and 𝜌₁/𝛼₁(𝑧,𝑢)+𝜌₂/𝛼₂(𝑧,𝑢)≥0, Then 𝜂sup_∈𝑌Re𝑓𝜁,𝜂+𝑧^𝐻𝐴𝑧^1/2≥𝑡.(54)

Proof. Suppose, on the contrary, that 𝜂sup_∈𝑌Re𝑓𝜁,𝜂+𝑧^𝐻𝐴𝑧^1/2<𝑡.(55) By compactness of 𝑌(𝜉)⊂𝑌 in 𝐶^𝑝,𝜉∈(𝑢,𝑢)∈𝑄, there exist an integer 𝑘>0 and finite points 𝜂_𝑖∈𝑌(𝜉), 𝜆_𝑖>0, 𝑖=1,2,…,𝑘 with ∑^𝑘_𝑖=1𝜆_𝑖=1 such that (49) holds. From (49) and (55), we have Re^𝑘_𝑖=1𝜆_𝑖𝑓𝜁,𝜂_𝑖+𝑧^𝐻𝐴𝑧^1/2<^𝑘_𝑖=1𝜆_𝑖𝑡≤^𝑘_𝑖=1𝜆_𝑖Re𝑓𝜉,𝜂_𝑖+𝑢^𝐻𝐴𝑢^1/2.(56) From (51) and the generalized Schwarz inequality, we have Re𝑧^𝐻𝐴𝑤≤𝑧^𝐻𝐴𝑧^1/2𝑤^𝐻𝐴𝑤^1/2≤𝑧^𝐻𝐴𝑧^1/2.(57) Using (52) and (57) in (56), we get Re^𝑘_𝑖=1𝜆_𝑖𝑓𝜁,𝜂_𝑖+𝑧^𝐻𝐴𝑤<Re^𝑘_𝑖=1𝜆_𝑖𝑓𝜉,𝜂_𝑖+𝑢^𝐻𝐴𝑤.(58) Since Re[∑^𝑘_𝑖=1𝜆_𝑖[𝑓(𝜁,𝜂_𝑖)+𝑧^𝐻𝐴𝑤]] is (𝐹,𝛼₁,𝜌₁,𝜃)-pseudoconvex with respect to 𝑅₊ on 𝑄, the above inequality implies that 𝐹𝑧,𝑢;𝛼₁(𝑧,𝑢)^𝑘_𝑖=1𝜆_𝑖∇_𝑧𝑓𝜉,𝜂_𝑖+∇_𝑧𝑓𝜉,𝜂_𝑖+𝐴𝑤<−𝜌₁𝜃²(𝑧,𝑢),(59) which due to sublinearity of 𝐹 can be written as 𝐹𝑧,𝑢;^𝑘_𝑖=1𝜆_𝑖∇_𝑧𝑓𝜉,𝜂_𝑖+∇_𝑧𝑓𝜉,𝜂_𝑖𝜌+𝐴𝑤<−¹𝛼₁()𝜃𝑧,𝑢²(𝑧,𝑢).(60) By the feasibility of 𝜁=(𝑧,𝑧) to (P), we have −ℎ(𝜁)∈𝑆, or Re⟨ℎ(𝜁),𝜇⟩≤0, for 𝜇∈𝑆^∗, which along with (50) yields Reℎ(𝜁),𝜇≤0≤Reℎ(𝜉),𝜇.(61) The above inequality, together with the (𝐹,𝛼₂,𝜌₂,𝜃)-quasiconvexity of ℎ(𝜁) on 𝑄 with respect to the polyhedral cone 𝑆⊂𝐶^𝑝, implies 𝐹𝑧,𝑢;𝛼₂(𝑧,𝑢)𝜇^𝑇∇_𝑧ℎ(𝜉)+𝜇^𝐻∇_𝑧ℎ(𝜉)≤−𝜌₂𝜃²(𝑧,𝑢),(62) which due to sublinearity of 𝐹 can be written as 𝐹𝑧,𝑢;𝜇^𝑇∇_𝑧ℎ(𝜉)+𝜇^𝐻∇_𝑧ℎ(𝜉𝜌)≤−²𝛼₂()𝜃𝑧,𝑢²(𝑧,𝑢).(63) On adding (60) and (63) and using sublinearity of 𝐹, we get 𝐹𝑧,𝑢;^𝑘_𝑖=1𝜆_𝑖∇_𝑧𝑓𝜉,𝜂_𝑖+∇_𝑧𝑓𝜉,𝜂_𝑖+𝐴𝑤+𝜇^𝑇∇_𝑧ℎ(𝜉)+𝜇^𝐻∇_𝑧ℎ(𝜉𝜌)<−¹𝛼₁()+𝜌𝑧,𝑢𝛼₂(²)𝜃𝑧,𝑢²(𝑧,𝑢).(64) From the assumption 𝜌₁/𝛼₁(𝑧,𝑢)+𝜌₂/𝛼₂(𝑧,𝑢)≥0, the above inequality yields 𝐹𝑧,𝑢;^𝑘_𝑖=1𝜆_𝑖∇_𝑧𝑓𝜉,𝜂_𝑖+∇_𝑧𝑓𝜉,𝜂_𝑖+𝐴𝑤+𝜇^𝑇∇_𝑧ℎ(𝜉)+𝜇^𝐻∇_𝑧ℎ(𝜉)<0,(65) which contradicts (48), hence the theorem.

Theorem 18 (weak duality). Let 𝜁=(𝑧,𝑧) and (𝑘,𝜆,𝜂,𝜉,𝜇,𝑤,𝑡) be feasible solutions to (P) and (D1), respectively. Further, if Re[∑^𝑘_𝑖=1𝜆_𝑖[𝑓(𝜁,𝜂_𝑖)+𝑧^𝐻𝐴𝑤]] is (𝐹,𝛼₁,𝜌₁,𝜃)-quasiconvex with respect to 𝑅₊ on 𝑄,ℎ(𝜁) is (𝐹,𝛼₂,𝜌₂,𝜃)-pseudoconvex on 𝑄 with respect to the polyhedral cone 𝑆⊂𝐶^𝑝, and 𝜌₁/𝛼₁(𝑧,𝑢)+𝜌₂/𝛼₂(𝑧,𝑢)≥0, then 𝜂sup_∈𝑌Re𝑓𝜁,𝜂+𝑧^𝐻𝐴𝑧^1/2≥𝑡.(66)

Proof. The proof follows the same lines as in Theorem 17.

Theorem 19 (strong duality). Let 𝜁₀=(𝑧₀,𝑧₀) be an optimal solution to the problem (P) at which a constraint qualification is satisfied. Then there exist (𝑘,𝜆,𝜂)∈𝐾(𝜁₀) and (𝜁₀,𝜇,𝑤,𝑡)∈𝑋(𝑘,𝜆,𝜂) such that (𝑘,𝜆,𝜂,𝜁₀,𝜇,𝑤,𝑡) is a feasible solution to the dual problem (D1). If the hypotheses of Theorem 17 or 18 are satisfied, then (𝑘,𝜆,𝜂,𝜁₀,𝜇,𝑤,𝑡) is optimal to (D1), and the two problems (P) and (D1) have the same optimal values.

Proof. The proof follows along the lines of Theorem 6 (Lai and Liu [13]).

Theorem 20 (strict converse duality). Let 𝜁̂ and (𝑘,𝜆,𝜂,𝜉̂,𝜇,𝑤,𝑡)̂ be optimal solutions to (P) and (D1), respectively, and assume that the assumptions of Theorem 19 are satisfied. Further, assume that the following conditions are satisfied: (i) Re∑𝑘_𝑖=1𝜆_𝑖[𝑓(𝜁̂,𝜂_𝑖)+𝑢^𝐻𝐴𝑤] is strict (𝐹,𝛼₁,𝜌₁,𝜃)-pseudoconvex with respect to 𝑅₊ on 𝑄 and ℎ(𝜁̂) is (𝐹,𝛼₂,𝜌₂,𝜃)-quasiconvex on 𝑄 with respect to the polyhedral cone 𝑆⊂𝐶^𝑝;(ii) 𝜌₁/𝛼₁(𝑧,𝑢)+𝜌₂/𝛼₂(𝑧,𝑢)≥0. Then 𝜁̂=𝜉̂; that is, 𝜁̂ is optimal solution to (D1).

Proof. On the contrary, suppose that (𝑧,𝑧)=𝜁̂≠𝜉̂=(𝑢,𝑢).
On applying Theorem 19, we know that ̂𝜂𝑡=sup_∈𝑌Re𝑓𝜁̂,𝜂+𝑧^𝐻𝐴𝑧^1/2.(67) From the feasibility of 𝜁̂∈𝑄 to (P), 𝜇∈𝑆^∗ and (50), we have Reℎ𝜁̂,𝜇≤0≤Reℎ𝜉̂,𝜇.(68) Since ℎ(𝜁̂) is (𝐹,𝛼₂,𝜌₂,𝜃)-quasiconvex on 𝑄 with respect to the polyhedral cone Sin𝐶^𝑝, the above inequality yields 𝐹𝑧,𝑢;𝛼₂(𝑧,𝑢)𝜇^𝑇∇_𝑧ℎ𝜉̂+𝜇^𝐻∇_𝑧ℎ𝜉̂≤−𝜌₂𝜃²(𝑧,𝑢),(69) which by sublinearity of 𝐹 implies 𝐹𝑧,𝑢;𝜇^𝑇∇_𝑧ℎ𝜉̂+𝜇^𝐻∇_𝑧ℎ𝜉̂𝜌≤−²𝛼₂(𝑢)𝜃𝑧,²(𝑧,𝑢).(70) By (48) and the sublinearity of 𝐹, we have 𝐹𝑘𝑧,𝑢;_𝑖=1𝜆_𝑖∇_𝑧𝑓𝜉̂,𝜂_𝑖+∇_𝑧𝑓𝜉̂,𝜂_𝑖𝑤𝜇+𝐴+𝐹𝑧,𝑢;^𝑇∇_𝑧ℎ𝜉̂+𝜇^𝐻∇_𝑧ℎ𝜉̂𝑘≥𝐹𝑧,𝑢;_𝑖=1𝜆_𝑖∇_𝑧𝑓𝜉̂,𝜂_𝑖+∇_𝑧𝑓𝜉̂,𝜂_𝑖𝑤+𝜇+𝐴_𝑇∇_𝑧ℎ𝜉̂+𝜇^𝐻∇_𝑧ℎ𝜉̂=0.(71) The above inequality, together with (70) and 𝜌₁/𝛼₁(𝑧,𝑢)+𝜌₂/𝛼₂(𝑧,𝑢)≥0, gives 𝐹𝑘𝑧,𝑢;_𝑖=1𝜆_𝑖∇_𝑧𝑓𝜉̂,𝜂_𝑖+∇_𝑧𝑓𝜉̂,𝜂_𝑖𝑤𝜇+𝐴≥−𝐹𝑧,𝑢;^𝑇∇_𝑧ℎ𝜉̂+𝜇^𝐻∇_𝑧ℎ𝜉̂≥𝜌₂𝛼₂(𝑢)𝜃𝑧,²(𝑢)𝜌𝑧,≥−¹𝛼₁(𝑢)𝜃𝑧,²(𝑧,𝑢).(72) That is, 𝐹𝑘𝑧,𝑢;_𝑖=1𝜆_𝑖∇_𝑧𝑓𝜉̂,𝜂_𝑖+∇_𝑧𝑓𝜉̂,𝜂_𝑖𝑤𝜌+𝐴≥−¹𝛼₁(𝑢)𝜃𝑧,²(𝑧,𝑢),(73) which by sublinearity of 𝐹 implies 𝐹𝑧,𝑢;𝛼₁(𝑢)𝑘𝑧,_𝑖=1𝜆_𝑖∇_𝑧𝑓𝜉̂,𝜂_𝑖+∇_𝑧𝑓𝜉̂,𝜂_𝑖+𝐴𝑤≥−𝜌₁𝜃²(𝑧,𝑢).(74) Since Re∑𝑘_𝑖=1𝜆_𝑖[𝑓(𝜁̂,𝜂_𝑖)+𝑢^𝐻𝐴𝑤] is strict (𝐹,𝛼₁,𝜌₁,𝜃)-pseudoconvex with respect to 𝑅₊ on 𝑄, the above inequality implies that 𝑘Re_𝑖=1𝜆_𝑖𝑓𝜁̂,𝜂_𝑖+𝑧^𝐻𝐴𝑤𝑘>Re_𝑖=1𝜆_𝑖𝑓𝜉̂,𝜂_𝑖+𝑢^𝐻𝐴𝑤.(75) From (51), (52), and the generalized Schwarz inequality, we have Re𝑧^𝐻𝐴𝑤≤𝑧^𝐻𝐴𝑧^1/2,Re𝑢^𝐻𝐴𝑤=𝑢^𝐻𝐴𝑢^1/2,(76) which on substituting in (75) and by using (49), we obtain 𝑘Re_𝑖=1𝜆_𝑖𝑓𝜁̂,𝜂_𝑖+𝑧^𝐻𝐴𝑧^1/2𝑘>Re_𝑖=1𝜆_𝑖𝑓𝜉̂,𝜂_𝑖+𝑢^𝐻𝐴𝑢^1/2≥_𝑖=𝑘₁𝜆_𝑖𝑡.̂(77) Consequently, there exist certain 𝑖₀ which satisfy Re𝑓𝜁̂,𝜂_𝑖0+𝑧^𝐻𝐴𝑧^1/2>𝑡.̂(78) Hence, 𝜂sup_∈𝑌Re𝑓𝜁̂,𝜂+𝑧^𝐻𝐴𝑧^1/2≥Re𝑓𝜁̂,𝜂_𝑖0+𝑧^𝐻𝐴𝑧^1/2>𝑡,̂(79) which contradicts (67), hence the theorem.

5. Parameter Free Duality

Making use of the optimality conditions, we show that the following formation is a dual (D2) to the complex programming problem (P): max_𝑘,𝜆_,𝜂_{(𝜉)∈𝐾}sup_{𝜉,𝜇,𝑤∈𝑋𝑘,}𝜆_,𝜂Re𝑓𝜉,𝜂+𝑢^𝐻𝐴𝑢^1/2,(D2) where 𝑋(𝑘,𝜆,𝜂) denotes the set of all (𝜉,𝜇,𝑤)∈𝐶^2𝑛×𝐶^𝑝×𝐶^𝑛 to satisfy the following conditions: ^𝑘_𝑖=1𝜆_𝑖∇_𝑧𝑓𝜉,𝜂_𝑖+∇_𝑧𝑓𝜉,𝜂_𝑖+𝐴𝑤+𝜇^𝑇∇_𝑧ℎ(𝜉)+𝜇^𝐻∇_𝑧ℎ(𝜉)ℎ(𝜉)𝑤=0,(80)Re,𝜇≥0,(81)_𝐻𝐴𝑤≤1,(82)(𝑢^𝐻𝐴𝑢)^1/2=Re𝑢^𝐻𝐴𝑤.(83) If, for a triplet (𝑘,𝜆,𝜂)∈𝐾(𝜉), the set 𝑋(𝑘,𝜆,𝜂)=∅, then we define the supremum over 𝑋(𝑘,𝜆,𝜂) to be −∞ for nonexception in the formulation of (D2).

Now, we establish appropriate duality theorems and prove that optimal values of (P) and (D2) are equal under the assumption of generalized convexity in order to show that the problems (P) and (D2) have no duality gap.

Theorem 21 (weak duality). Let 𝜁=(𝑧,𝑧) and (𝑘,𝜆,𝜂,𝜉,𝜇,𝑤) be feasible solutions to (P) and (D2), respectively. Further, if Re[∑^𝑘_𝑖=1𝜆_𝑖[𝑓(𝜁,𝜂_𝑖)+𝑧^𝐻𝐴𝑤]] is (𝐹,𝛼₁,𝜌₁,𝜃)-pseudoconvex with respect to 𝑅₊ on 𝑄,ℎ(𝜁) is (𝐹,𝛼₂,𝜌₂,𝜃)-quasiconvex on 𝑄 with respect to the polyhedral cone 𝑆⊂𝐶^𝑝, and 𝜌₁/𝛼₁(𝑧,𝑢)+𝜌₂/𝛼₂(𝑧,𝑢)≥0, then 𝜂sup_∈𝑌Re𝑓𝜁,𝜂+𝑧^𝐻𝐴𝑧^1/2𝜂≥sup_∈𝑌Re𝑓𝜉,𝜂+𝑢^𝐻𝐴𝑢^1/2.(84)

Proof. On the contrary, we suppose that 𝜂sup_∈𝑌Re𝑓𝜁,𝜂+𝑧^𝐻𝐴𝑧^1/2𝜂<sup_∈𝑌Re𝑓𝜉,𝜂+𝑢^𝐻𝐴𝑢^1/2.(85) Since 𝜂_𝑖∈𝑌(𝜉)⊂𝑌, 𝑖=1,2,…,𝑘, we have 𝜂sup_∈𝑌Re𝑓𝜉,𝜂+𝑢^𝐻𝐴𝑢^1/2=Re𝑓𝜉,𝜂_𝑖+𝑢^𝐻𝐴𝑢^1/2𝑓,𝑖=1,2,…,𝑘,(86)Re𝜁,𝜂_𝑖+𝑧^𝐻𝐴𝑧^1/2𝜂≤sup_∈𝑌Re𝑓𝜁,𝜂+𝑧^𝐻𝐴𝑧^1/2,𝑖=1,2,…,𝑘.(87) Then the above three inequalities give Re𝑓𝜁,𝜂_𝑖+𝑧^𝐻𝐴𝑧^1/2<Re𝑓𝜉,𝜂_𝑖+𝑢^𝐻𝐴𝑢^1/2,𝑖=1,2,…,𝑘.(88) From (82), (83), (88), and the generalized Schwarz inequality, we have Re𝑓𝜁,𝜂_𝑖+𝑧^𝐻𝑓𝐴𝑤<Re𝜉,𝜂_𝑖+𝑢^𝐻𝐴𝑤,𝑖=1,2,…,𝑘.(89) As 𝜆_𝑖>0, 𝑖=1,2,…,𝑘 and ∑^𝑘_𝑖=1𝜆_𝑖=1, we have Re^𝑘_𝑖=1𝜆_𝑖𝑓𝜁,𝜂_𝑖+𝑧^𝐻𝐴𝑤−Re^𝑘_𝑖=1𝜆_𝑖𝑓𝜉,𝜂_𝑖+𝑢^𝐻𝐴𝑤<0.(90) Since Re[∑^𝑘_𝑖=1𝜆_𝑖[𝑓(𝜁,𝜂_𝑖)+𝑧^𝐻𝐴𝑤]] is (𝐹,𝛼₁,𝜌₁,𝜃)-pseudoconvex with respect to 𝑅₊ on 𝑄, the above inequality implies that 𝐹𝑧,𝑢;𝛼₁()×𝑧,𝑢_𝑘𝑖=1𝜆_𝑖∇_𝑧𝑓𝜉,𝜂_𝑖+∇_𝑧𝑓𝜉,𝜂_𝑖+𝐴𝑤<−𝜌₁𝜃²(𝑧,𝑢),(91) which by sublinearity of 𝐹 becomes 𝐹𝑧,𝑢;^𝑘_𝑖=1𝜆_𝑖∇_𝑧𝑓𝜉,𝜂_𝑖+∇_𝑧𝑓𝜉,𝜂_𝑖𝜌+𝐴𝑤<−¹𝛼₁()𝜃𝑧,𝑢²(𝑧,𝑢).(92) By the feasibility of 𝜁=(𝑧,𝑧) to (P), 0≠𝜇∈𝑆^∗, and the inequality (81), we obtain Reℎ(𝜁),𝜇≤0≤Reℎ(𝜉),𝜇.(93) The above inequality together with the (𝐹,𝛼₂,𝜌₂,𝜃)-quasiconvexity of ℎ(𝜁) on 𝑄 with respect to the polyhedral cone 𝑆⊂𝐶^𝑝 implies 𝐹𝑧,𝑢;𝛼₂(𝑧,𝑢)𝜇^𝑇∇_𝑧ℎ(𝜉)+𝜇^𝐻∇_𝑧ℎ(𝜉)≤−𝜌₂𝜃²(𝑧,𝑢),(94) which by sublinearity of 𝐹 becomes 𝐹𝑧,𝑢;𝜇^𝑇∇_𝑧ℎ(𝜉)+𝜇^𝐻∇_𝑧ℎ(𝜉𝜌)≤−²𝛼₂()𝜃𝑧,𝑢²(𝑧,𝑢).(95) On adding (92) and (95) and using the sublinearity of 𝐹, we get 𝐹𝑧,𝑢;^𝑘_𝑖=1𝜆_𝑖∇_𝑧𝑓𝜉,𝜂_𝑖+∇_𝑧𝑓𝜉,𝜂_𝑖+𝐴𝑤+𝜇^𝑇∇_𝑧ℎ(𝜉)+𝜇^𝐻∇_𝑧ℎ(𝜉𝜌)<−¹𝛼₁()+𝜌𝑧,𝑢𝛼₂(²)𝜃𝑧,𝑢²(𝑧,𝑢).(96) From the assumption 𝜌₁/𝛼₁(𝑧,𝑢)+𝜌₂/𝛼₂(𝑧,𝑢)≥0, the above inequality yields 𝐹𝑧,𝑢;^𝑘_𝑖=1𝜆_𝑖∇_𝑧𝑓𝜉,𝜂_𝑖+∇_𝑧𝑓𝜉,𝜂_𝑖+𝐴𝑤+𝜇^𝑇∇_𝑧ℎ(𝜉)+𝜇^𝐻∇_𝑧ℎ(𝜉)<0,(97) which contradicts (80), hence the theorem.

Theorem 22 (weak duality). Let 𝜁=(𝑧,𝑧) and (𝑘,𝜆,𝜂,𝜉,𝜇,𝑤) be feasible solutions to (P) and (D2), respectively. Further, if Re[∑^𝑘_𝑖=1𝜆_𝑖[𝑓(𝜁,𝜂_𝑖)+𝑧^𝐻𝐴𝑤]] is (𝐹,𝛼₁,𝜌₁,𝜃)-quasiconvex with respect to 𝑅₊ on 𝑄,ℎ(𝜁) is (𝐹,𝛼₂,𝜌₂,𝜃)-pseudoconvex on 𝑄 with respect to the polyhedral cone 𝑆⊂𝐶^𝑝, and 𝜌₁/𝛼₁(𝑧,𝑢)+𝜌₂/𝛼₂(𝑧,𝑢)≥0, then 𝜂sup_∈𝑌Re𝑓𝜁,𝜂+𝑧^𝐻𝐴𝑧^1/2𝜂≥sup_∈𝑌Re𝑓𝜉,𝜂+𝑢^𝐻𝐴𝑢^1/2.(98)

Proof. The proof follows the same lines as in Theorem 21.

Theorem 23 (strong duality). Let 𝜁₀=(𝑧₀,𝑧₀) be an optimal solution to the problem (P) at which a constraint qualification is a satisfied. Then there exist (𝑘,𝜆,𝜂)∈𝐾(𝜁₀) and (𝜁₀,𝜇,𝑤)∈𝑋(𝑘,𝜆,𝜂) such that (𝑘,𝜆,𝜂,𝜁₀,𝜇,𝑤) is a feasible solution to the dual problem (D2). Further, if the hypotheses of Theorem 21 or Theorem 22 are satisfied, then (𝑘,𝜆,𝜂,𝜁₀,𝜇,𝑤) is optimal to (D2), and the two problems (P) and (D2) have the same optimal values.

Proof. The proof follows along the lines of Theorem 8 (Lai and Liu [13]).

Theorem 24 (strict converse duality). Let 𝜁̂ and (𝑘,𝜆,𝜂,𝜉̂,𝜇,𝑤) be optimal solutions to (P) and (D2), respectively, and the conditions of Theorem 23 are satisfied. Further, assume that the following conditions are satisfied: (i) Re∑𝑘_𝑖=1𝜆_𝑖[𝑓(𝜁̂,𝜂_𝑖)+𝑧^𝐻𝐴𝑤] is strict (𝐹,𝛼₁,𝜌₁,𝜃)-pseudoconvex with respect to 𝑅₊ on 𝑄 and ℎ(𝜁̂) is (𝐹,𝛼₂,𝜌₂,𝜃)-quasiconvex on 𝑄 with respect to the polyhedral cone 𝑆⊂𝐶^𝑝;(ii) 𝜌₁/𝛼₁(𝑧,𝑢)+𝜌₂/𝛼₂(𝑧,𝑢)≥0. Then 𝜁̂=𝜉̂; that is, 𝜉̂ is an optimal solution to (D2).

Proof. On the contrary, we assume that (𝑧,𝑧)=𝜁̂≠𝜉̂=(𝑢,𝑢).
On applying Theorem 23, we know that 𝜂sup_∈𝑌Re𝑓𝜁̂,𝜂+𝑧^𝐻𝐴𝑧^1/2𝜂=sup_∈𝑌Re𝑓𝜉̂,𝜂+𝑢^𝐻𝐴𝑢^1/2.(99) From the feasibility of 𝜁̂∈𝑄 to (P), 𝜇∈𝑆^∗, inequality (81) yields Reℎ𝜁̂,𝜇≤0≤Reℎ𝜉̂,𝜇.(100) Since ℎ(𝜁̂) is (𝐹,𝛼₂,𝜌₂,𝜃)-quasiconvex on 𝑄 with respect to the polyhedral cone 𝑆 in 𝐶^𝑝, the above inequality yields 𝐹𝑧,𝑢;𝛼₂(𝑧,𝑢)𝜇^𝑇∇_𝑧ℎ𝜉̂+𝜇^𝐻∇_𝑧ℎ𝜉̂≤−𝜌₂𝜃²(𝑧,𝑢),(101) which by sublinearity of 𝐹 implies 𝐹𝑧,𝑢;𝜇^𝑇∇_𝑧ℎ𝜉̂+𝜇^𝐻∇_𝑧ℎ𝜉̂𝜌≤−²𝛼₂(𝑢)𝜃𝑧,²(𝑧,𝑢).(102) By (80) and the sublinearity of 𝐹, we have 𝐹𝑘𝑧,𝑢;_𝑖=1𝜆_𝑖∇_𝑧𝑓𝜉̂,𝜂_𝑖+∇_𝑧𝑓𝜉̂,𝜂_𝑖𝑤𝜇+𝐴+𝐹𝑧,𝑢;^𝑇∇_𝑧ℎ𝜉̂+𝜇^𝐻∇_𝑧ℎ𝜉̂𝑘≥𝐹𝑧,𝑢;_𝑖=1𝜆_𝑖∇_𝑧𝑓𝜉̂,𝜂_𝑖+∇_𝑧𝑓𝜉̂,𝜂_𝑖𝑤+𝜇+𝐴_𝑇∇_𝑧ℎ𝜉̂+𝜇^𝐻∇_𝑧ℎ𝜉̂=0.(103) The above inequality, together with (102) and 𝜌₁/𝛼₁(𝑧,𝑢)+𝜌₂/𝛼₂(𝑧,𝑢)≥0, gives 𝐹𝑘𝑧,𝑢;_𝑖=1𝜆_𝑖∇_𝑧𝑓𝜉̂,𝜂_𝑖+∇_𝑧𝑓𝜉̂,𝜂_𝑖𝑤𝜇+𝐴≥−𝐹𝑧,𝑢;^𝑇∇_𝑧ℎ𝜉̂+𝜇^𝐻∇_𝑧ℎ𝜉̂≥𝜌₂𝛼₂(𝑢)𝜃𝑧,²(𝑢)𝜌𝑧,≥−¹𝛼₁(𝑢)𝜃𝑧,²(𝑧,𝑢).(104) That is, 𝐹𝑘𝑧,𝑢;_𝑖=1𝜆_𝑖∇_𝑧𝑓𝜉̂,𝜂_𝑖+∇_𝑧𝑓𝜉̂,𝜂_𝑖𝑤𝜌+𝐴≥−¹𝛼₁(𝑢)𝜃𝑧,²(𝑧,𝑢),(105) which by sublinearity of 𝐹 implies 𝐹𝑧,𝑢;𝛼₁(𝑢)𝑘𝑧,_𝑖=1𝜆_𝑖∇_𝑧𝑓𝜉̂,𝜂_𝑖+∇_𝑧𝑓𝜉̂,𝜂_𝑖+𝐴𝑤≥−𝜌₁𝜃²(𝑧,𝑢).(106) Since Re∑𝑘_𝑖=1𝜆_𝑖[𝑓(𝜁̂,𝜂_𝑖)+𝑧^𝐻𝐴𝑤] is strict (𝐹,𝛼₁,𝜌₁,𝜃)-pseudoconvex with respect to 𝑅₊ on 𝑄, the above inequality implies that 𝑘Re_𝑖=1𝜆_𝑖𝑓𝜁̂,𝜂_𝑖+𝑧^𝐻𝐴𝑤𝑘>Re_𝑖=1𝜆_𝑖𝑓𝜉̂,𝜂_𝑖+𝑢^𝐻𝐴𝑤.(107) From (82), (83), and the generalized Schwarz inequality, we get Re𝑧^𝐻𝐴𝑤≤𝑧^𝐻𝐴𝑧^1/2,Re𝑢^𝐻𝐴𝑤^1/2=𝑢^𝐻𝐴𝑢^1/2,(108) which on substituting in (107), we obtain 𝑘Re_𝑖=1𝜆_𝑖𝑓𝜁̂,𝜂_𝑖+𝑧^𝐻𝐴𝑧^1/2𝑘>Re_𝑖=1𝜆_𝑖𝑓𝜉̂,𝜂_𝑖+𝑢^𝐻𝐴𝑢^1/2.(109) Consequently, there exist certain 𝑖₀ which satisfy Re𝑓𝜁̂,𝜂_𝑖0+𝑧^𝐻𝐴𝑧^1/2>Re𝑓𝜉̂,𝜂_𝑖0+𝑢^𝐻𝐴𝑢^1/2.(110) Hence, 𝜂sup_∈𝑌Re𝑓𝜁̂,𝜂+𝑧^𝐻𝐴𝑧^1/2≥Re𝑓𝜁̂,𝜂_𝑖0+𝑧^𝐻𝐴𝑧^1/2>Re𝑓𝜉̂,𝜂_𝑖0+𝑢^𝐻𝐴𝑢^1/2𝜂=sup_∈𝑌Re𝑓𝜉̂,𝜂+𝑢^𝐻𝐴𝑢^1/2,(111) which contradicts (99), hence the theorem.

6. Conclusion

In this paper, we introduced generalized (𝐹,𝛼,𝜌,𝜃)-convex functions and established sufficient optimality conditions for a class of nondifferentiable minimax programming problems in complex space. These optimality conditions are then used to construct two types of dual model and finally we derived weak, strong, and strict converse duality theorems to show that there is no duality gap between the two dual problems with respect to the primal problem under some generalized convexities of complex functions in the complex programming problem. As a future task, the authors would like to extend these results to second and higher order cases and establish the relations between primal and its second and higher order dual problems.

Acknowledgments

The authors are thankful to the anonymous referees for their valuable comments which have improved the presentation of the paper. The research of the first author is financially supported by the University Grant Commission, New Delhi, India, through Grant No. F. no. 41-801/2012(SR).