ISRN Mathematical Analysis

VolumeΒ 2011, Article IDΒ 786978, 22 pages

http://dx.doi.org/10.5402/2011/786978

## Sufficient Conditions and Duality Theorems for Nondifferentiable Minimax Fractional Programming

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

Received 20 October 2010; Accepted 28 November 2010

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

Copyright Β© 2011 Shun-Chin Ho. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We consider nondifferentiable minimax fractional programming problems involving -(, )-invex functions with respect to and . Sufficient optimality conditions and duality results for a class of nondifferentiable minimax fractional programming problems are obtained undr -(, )-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 -(, )-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., [1β6]).

Many authors investigated the optimality conditions and duality theorems for minimax (fractional) programming problems. For details, one can consult [1β14]. 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 -invexity [5] and generalized -unvexity [13]. Recently, Antczak [15] defined a new class of functions, named --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 --invexity assumption on objective and constraint functions.

In this paper, we are inspired to extend the result of Lai et al. [10] to --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 --invex with respect to the same function and with respect to, not necessarily, the same function in Section 3. Employing these results, we construct three dual problems in Sections 4β6. Here we investigate weak, strong, and strict converse duality theorems under the framework of --invex with respect to the same function and with respect to, not necessarily, the same function .

#### 2. Some Notations and Preliminary Results

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

The following definition can be found in [15].

*Definition 2.1 (see [15]). *Let and be any real numbers. The differentiable function is said to be (strictly) --invex with respect to and at on a nonempty set if, there exist a function and a function such that, for all , the inequalities
hold.

is said to be --invex (strictly --invex) with respect to and on if it is --invex with respect to the same and at each .

It should be pointed out that exponentials appearing on the right-hand sides of the above inequalities are understood to be taken componentwise and .

We consider the following nondifferentiable minimax fractional programming problem:
where is a compact subset of , , , and are -functions, and are positive semidefinite matrices, , and for each in , where is the set of feasible solutions of problem (*P*); that is, . This is a nondifferentiable programming problem if either or is nonzero. If and are null matrices, then problem (*P*) is a minimax fractional programming problem.

For each define We let

Because and are contionuous differentiable and is compact subset of , we see that for each , , and for any , we have a postive constant

We will use the generalized Schwarz inequality the equality holds when , for some .

Hence if , we have

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

Theorem 2.2 ((necessary conditions) see, [10]). *Let be a -optimal solution and satisfying , , and is linearly independent. Then there exist , , , and such that
*

It should be noted that both the matrices and are positive definite at the solution in the above theorem. If one of and is zero, or both and are singular at , then, for , we define a set by

Here conditions (i)β(iii) are given as follows:(i)if and , then (ii)if and , then (iii)if and , then

If we take condition 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 --invexity. All theorems in this work will be proved only in the case when , (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 --invex functions with respect to and for given and ). The proofs of the other cases are easier than this one.

We would establish the sufficient conditions under the --invex function.

Theorem 3.1 (sufficient optimality conditions). *Let be a feasible solution of ( P). There exist a positive interger , , , , , , and to satisfy relations (2.7)~(2.11). Furthermore suppose that any one of conditions (a) and (b) holds: *(a)

*is --invex with respect to and at , and is --invex with respect to the same function and another function at on , not necessarily, equal to ,*(b)

*is --invex with respect to and at on ,*

*then is an optimal solution of (*

*P*).*Proof. *Suppose that is not an optimal solution of (*P*). Then there exists a -feasible solution such that
We note that
for , , and
Then, we obtain
It follows that
From relations (2.6), (2.11), (2.8), (2.10), and (3.5), we have
That is,
From relations (*P*) and (2.9), we obtain
If hypothesis (a) holds, from the --invexity with respect to and at of , we have
From the inequalities (3.8) and (3.9), we get
Now, multiplying equality (2.7) by , we know
From relations (3.10) and (3.11), we have
From the --invexity with respect to the same function and the function at of ,
From inequality (3.12) and the above inequality, we obtain
which contradicts (3.7), and proves that is an optimal solution to (*P*).

If hypothesis (b) holds, from the --invexity with respect to and at of , then
The above inequality along with (2.7) yields
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*):
where denotes the set of satisfying
If for a triplet the set is empty, then we define the supremum over it to be .

Let denote the set of all feasible points of (*D*). Moreover, we denote .

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

Theorem 4.1 (weak duality). *Let and be -feasible and -feasible, respectively. Suppose that any one of the following conditions and holds: *(a)* is --invex with respect to and at and is --invex with respect to the same function and another function at on , not necessarily, equal to , *(b)* is -invex with respect to and at on . **Then
*

*Proof. *Suppose on the contrary that
Then, we have an inequality
It follows that for , with , we have
with at least one strict inequality because . From relations (2.6), (4.4), (4.8), and (4.2), we obtain
That is,
From relations (*P*) and (4.3), we have
If hypothesis (a) holds, from --invexity with respect to and at of , we get
From the above inequality together with relation (4.10), we have
Multiplying (4.1) by , we obtain
From the above equality and inequality (4.13), we get
Using the --invexity of with respect to the same function and the function at and inequality (4.15), we get
which contradicts (4.11) and proves that .

If hypothesis (b) holds, from the --invexity with respect to and at of , then
By the above inequality and equality (4.1), we have
From relations (4.10) and (4.11), we obtain
which contradicts inequality (4.18). Thus, the proof is complete.

Theorem 4.2 (strong duality). *Let be an optimal solution of ( P), and let satisfy a constraint qualification for (P). Then there exist and such that is a feasible solution of (D). If in addition the hypothesis of Theorem 4.1 holds, then 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 and such that is feasible for (*D*), and
The optimality of this feasible solution for (*D*) follows from Theorem 4.1.

Theorem 4.3 (strict converse duality). *Let and 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)

*is strictly --invex with respect to and at and is -invex with respect to the same function and another function at on , not necessarily, equal to ,*(b)

*is strictly -invex with respect to and at on .*

*Then , that is, solves (*

*P*) and .*Proof. *We shall assume that and reach a contradiction. From Theorem 4.2, we know that there exist and such that is an optimal solution for (*D*) with the optimal value
Now like the proof of Theorem 4.1 by replacing by and by ,,,,,,,, we obtain
The above inequality contradicts
Therefore, we conclude that . Here, the proof of the theorem is complete.

*Remark 4.4. *In Theorem 4.3, if is a strictly --invex function with respect to and and is a --invex function with respect to the same function and the function , not necessarily, equal to , 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 as follows: where denotes the set of satisfying where

If for a triplet the set is empty, then we define the supremum over it to be .

Let denote the set of all feasible points of (*MWD*). Moreover, we denote .

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 and be -feasible and -feasible, respectively. Suppose that any one of the following conditions (a) and (b) holds: *(a)* is --invex, respect to and at and is --invex with respect to the same function and another function at on , not necessarily, equal to , *(b)* is --invex with respect to and at on . **Then
*

*Proof. *On the contrary, if possible, suppose that for each ,
From the above inequality and , , we obtain
By the above inequality, we know that
for all and .

Multiplying the above inequality by with , we have
From relations (2.6), (5.3), and (5.9), we get
By relations (*P*) and (5.2), we have
Now, if condition (a) holds, from --invexity with respect to and at of , we get
From the above inequality and inequality (5.11), we obtain
Multiplying (5.1) by , we have
By the above equality and inequality (5.13), we obtain
Using the --invexity with respect to the same function and the function at of and the above inequality, we have
which contradicts (5.10) and proves that .

If hypothesis (b) holds, from the --invexity with respect to and at of and the equality (5.1), then
From relations (5.10) and (5.11), we have
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 be an optimal solution of ( P) and let satisfy a constraint qualification for (P). Then there exist and such that is a feasible solution of (MWD). If in addition the hypothesis of Theorem 5.1 holds, then 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 and 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)

*is strictly --invex with respect to and at and is --invex with respect to the same function and another function at on , not necessarily, equal to ,*(b)

*is strictly --invex with respect to and at on .*

*Then , that is, solves (*

*P*) and .*Proof. *We suppose on the contrary that if , then
Since , for ,
It follows that
From and with , we have
From relations (2.6), (5.3), and inequality (5.22), we obtain that
By relations (*P*) and (5.2), we get
Now, if condition (a) holds, from --invexity with respect to and at of , then
From the above inequality and inequality (5.24), we know that
Multiplying (*MWD*) by , we have
Basing on the above inequality and (5.26), we get the inequality
Since is strictly --invex with respect to the same function and the function at and the above inequality, we obtain
which contradicts relations (5.23). Hence and is an optimal solution of (*P*).

If hypothesis (b) holds, from the strict --invexity with respect to and at of and equality (5.1), then
Relations (5.23) together with (5.24) yield
which contradicts (5.30). Hence is also an optimal solution of (*P*). The proof is complete.

*Remark 5.4. *In Theorem 5.3, if is a strictly --invex function with respect to and and is a --invex function with respect to the same function and the function , not necessarily, equal to , 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 be a -optimal solution, and let , be linearly independent. Then there exist ,, and such that
*

In this section, we present the Wolfe dual (*WD*) to the minimax program (*P*):
where denotes the set of satisfying
If for a triplet the set is empty, then we define the supremum over it to be .

Let denote the set of all feasible points of (*WD*). Moreover, we denote .

We assume throughout this section that and .

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

Theorem 6.2 (weak duality). *Let and be -feasible and -feasible, respectively, and assume that
*