Journal of Optimization

Volume 2013 (2013), Article ID 297015, 12 pages

http://dx.doi.org/10.1155/2013/297015

Research Article

Nondifferentiable Minimax Programming Problems in Complex Spaces Involving Generalized Convex Functions

Anurag Jayswal, Ashish Kumar Prasad, and Krishna Kummari

Department of Applied Mathematics, Indian School of Mines, Dhanbad, Jharkhand 826 004, India

Received 17 June 2013; Accepted 7 November 2013

Academic Editor: Sheng-Jie Li

Copyright © 2013 Anurag Jayswal et al. 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 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 -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 -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 -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:

  = -dimensional vector space of complex (real) numbers,  = the set of complex (real) matrices,  = the nonnegative orthant of ,  = the conjugate transpose of ,  the inner product of in .

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 is polyhedral cone if there is and such that ; that is, is generated by a finite number of vectors (the columns of ).

Equivalently, 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 and -points such that

where , are closed half-spaces involving the point .

Definition 2 (see [5]). If , then constitute the dual (polar) of .

If is analytic in a neighbourhood of , then , , is the gradient of function at . Similarly, if the complex function is analytic in variables and , we define the gradients by

In this paper, we consider the following complex programming problem:

where is a compact subset in , is a positive semidefinite Hermitian matrix, is a polyhedral cone in is continuous, and, for each , and are analytic in , where 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 ; that is, .

Special Cases. (i) If problem is a real programming problem with two variables nondifferentiable minimax problem, it may be expressed as

where is compact subset of and are continuously differentiable functions at , and is a positive semidefinite symmetric matrix. This problem was studied by Ahmad et al. [14, 15].

(ii) If vanishes in , then problem reduces to the problem considered by Mond and Craven [16]; that is,

(iii) If , then becomes a differentiable complex minimax programming problem studied by Datta and Bhatia [3]; that is,

Definition 3. A functional is said to be sublinear in its third variable, if, for any , the following conditions are satisfied:

(i), (ii),

for any in and . From (ii), it is clear that .

Let be sublinear on the third variable, with , if and . Let and be analytic functions and 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 of analytic function is said to be -convex (strict -convex) with respect to on the manifold , if, for any , , one has

Definition 5. The real part of analytic function is said to be -quasiconvex (strict -quasiconvex) with respect to on the manifold , if, for any , , one has

Definition 6. The real part of analytic function is said to be -pseudoconvex (strict -pseudoconvex) with respect to on the manifold , if, for any , , one has

Definition 7. The mapping is said to be -convex (strict -convex) with respect to the polyhedral cone on the manifold if, for any and , , one has

Definition 8. The mapping is said to be -quasiconvex (strict -quasiconvex) with respect to the polyhedral cone on the manifold if, for any and , , one has

Definition 9. The mapping is said to be -pseudoconvex (strict -pseudoconvex) with respect to the polyhedral cone on the manifold if for any and , , one has

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 of analytic function is said to be -pseudoconvex with respect to on the manifold , if, for any , one has

Remark 11. If we take , then the above definitions reduce to that given by Lai et al. [9]. In addition, if we take , then we obtain the definitions given by Mishra and Rueda [17].

Let and ; then Schwarz inequality can be written as

The equality holds if or for .

Definition 12 (see [12]). The problem is said to satisfy the constraint qualification at a point , if, for any nonzero ,

In the next section, we recall some notations and discuss necessary and sufficient optimality conditions for problem on the basis of Lai and Liu [18] and Lai and Huang [12].

3. Necessary and Sufficient Conditions

Let , be a continuous function defined on , where is a specified compact subset in problem . Then the supremum will be attained to its maximum in , and the set

is then also a compact set in . In particular, if is an optimal solution of problem , there exist a positive integer and finite points , , with such that the Lagrangian function

satisfies the Kuhn-Tucker type condition at . That is,

Equivalent form of expression (15) at is

For the integer , corresponding a vector and , with , we define a set as follows:

where the set is the intersection of closed half-spaces having the point on their boundaries.

Theorem 13 (necessary optimality conditions). Let be an optimal solution to . Suppose that the constraint qualification is satisfied for at and . Then there exist , and a positive integer with the following properties:

(i), ,(ii) , , ,

such that satisfies the following conditions:

Theorem 14 (sufficient optimality conditions). Let be a feasible solution to . Suppose that there exists a positive integer , , with and satisfying conditions (19)–(22). Further, if is -convex with respect to on is -convex on with respect to the polyhedral cone , and , then is an optimal solution to .

Proof. We prove this theorem by contradiction. Suppose that there is a feasible solution such that

Since , , we have

Thus, from the above three inequalities, we obtain

Using (21) and generalized Schwarz inequality, we get

and inequality (22) yields

Using (26) and (27) in (25), we have

Since and , we have

Since is -convex with respect to on , we have

From (29) and (30), we conclude that

which due to sublinearity of can be written as

On the other hand, from the feasibility of to , we have , or for , which along with (20) yields

Since is -convex on with respect to the polyhedral cone , we have

From (33) and (34), it follows that

which due to sublinearity of can be written as

On adding (32) and (36) and using sublinearity of , we get

The above inequality, together with the assumption , gives

which contradicts (19), hence the theorem.

Theorem 15 (sufficient optimality conditions). Let be a feasible solution to . Suppose that there exists a positive integer , , with and satisfying conditions (19)–(22). Further, if is -pseudoconvex with respect to on is -quasiconvex on with respect to the polyhedral cone , and , then is an optimal solution to .

Proof. Proceeding as in Theorem 14, we have

which, by -pseudoconvexity of with respect to on , yields

Using the sublinearity of , the above inequality can be written as

On the other hand, from the feasibility of to , we have , or for , which along with (20) yields

Since is -quasiconvex on with respect to the polyhedral cone , the above inequality yields

which due to sublinearity of can be written as

On adding (41) and (44) and using sublinearity of , we get

The above inequality, together with the assumption , gives

which contradicts (19), hence the theorem.

Theorem 16 (sufficient optimality conditions). Let be a feasible solution to . Suppose that there exists a positive integer , , with and satisfying conditions (19)–(22). Further, if is -quasiconvex with respect to on is strict -pseudoconvex on with respect to the polyhedral cone , and , then is an optimal solution to .

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:

for .

Now, we formulate a parametric dual problem with respect to the complex minimax programming problem as follows:

where denotes the set of all to satisfy the following conditions:

If, for a triplet , the set , then we define the supremum over to be for nonexecption in the formulation of .

Theorem 17 (weak duality). Let and be feasible solutions to and , respectively. Further, if is -pseudoconvex with respect to on is -quasiconvex on with respect to the polyhedral cone , and , Then

Proof. Suppose, on the contrary, that

By compactness of in , there exist an integer and finite points , , with such that (49) holds. From (49) and (55), we have

From (51) and the generalized Schwarz inequality, we have

Using (52) and (57) in (56), we get

Since is -pseudoconvex with respect to on , the above inequality implies that

which due to sublinearity of can be written as

By the feasibility of to , we have , or , for , which along with (50) yields

The above inequality, together with the -quasiconvexity of on with respect to the polyhedral cone , implies

which due to sublinearity of can be written as

On adding (60) and (63) and using sublinearity of , we get

From the assumption , the above inequality yields

which contradicts (48), hence the theorem.

Theorem 18 (weak duality). Let and be feasible solutions to and , respectively. Further, if is -quasiconvex with respect to on is -pseudoconvex on with respect to the polyhedral cone , and , then

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

Theorem 19 (strong duality). Let be an optimal solution to the problem at which a constraint qualification is satisfied. Then there exist and such that is a feasible solution to the dual problem . If the hypotheses of Theorem 17 or 18 are satisfied, then is optimal to , and the two problems and 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 and be optimal solutions to and , respectively, and assume that the assumptions of Theorem 19 are satisfied. Further, assume that the following conditions are satisfied:

(i) is strict -pseudoconvex with respect to on and is -quasiconvex on with respect to the polyhedral cone ;(ii) .

Then ; that is, is optimal solution to .

Proof. On the contrary, suppose that .

On applying Theorem 19, we know that

From the feasibility of to , and (50), we have

Since is -quasiconvex on with respect to the polyhedral cone , the above inequality yields

which by sublinearity of implies

By (48) and the sublinearity of , we have

The above inequality, together with (70) and , gives

That is,

which by sublinearity of implies

Since is strict -pseudoconvex with respect to on , the above inequality implies that

From (51), (52), and the generalized Schwarz inequality, we have

which on substituting in (75) and by using (49), we obtain

Consequently, there exist certain which satisfy

Hence,

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 to the complex programming problem :

where denotes the set of all to satisfy the following conditions:

If, for a triplet , the set , then we define the supremum over to be for nonexception in the formulation of .

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

Theorem 21 (weak duality). Let and be feasible solutions to and , respectively. Further, if is -pseudoconvex with respect to on is -quasiconvex on with respect to the polyhedral cone , and , then

Proof. On the contrary, we suppose that

Since , , we have

Then the above three inequalities give

From (82), (83), (88), and the generalized Schwarz inequality, we have

As , and , we have

Since is -pseudoconvex with respect to on , the above inequality implies that

which by sublinearity of becomes

By the feasibility of to , , and the inequality (81), we obtain

The above inequality together with the -quasiconvexity of on with respect to the polyhedral cone implies

which by sublinearity of becomes

On adding (92) and (95) and using the sublinearity of , we get

From the assumption , the above inequality yields

which contradicts (80), hence the theorem.

Theorem 22 (weak duality). Let and be feasible solutions to and , respectively. Further, if is -quasiconvex with respect to on is -pseudoconvex on with respect to the polyhedral cone , and , then

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

Theorem 23 (strong duality). Let be an optimal solution to the problem at which a constraint qualification is a satisfied. Then there exist and such that is a feasible solution to the dual problem . Further, if the hypotheses of Theorem 21 or Theorem 22 are satisfied, then is optimal to , and the two problems and 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 and be optimal solutions to and , respectively, and the conditions of Theorem 23 are satisfied. Further, assume that the following conditions are satisfied:

(i) is strict -pseudoconvex with respect to on and is -quasiconvex on with respect to the polyhedral cone ;(ii) .

Then ; that is, is an optimal solution to .

Proof. On the contrary, we assume that .

On applying Theorem 23, we know that

From the feasibility of to , , inequality (81) yields

Since is -quasiconvex on with respect to the polyhedral cone in , the above inequality yields

which by sublinearity of implies

By (80) and the sublinearity of , we have

The above inequality, together with (102) and , gives

That is,

which by sublinearity of implies

Since is strict -pseudoconvex with respect to on , the above inequality implies that

From (82), (83), and the generalized Schwarz inequality, we get

which on substituting in (107), we obtain

Consequently, there exist certain which satisfy

Hence,

which contradicts (99), hence the theorem.

6. Conclusion

In this paper, we introduced generalized -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).

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