Optimality Conditions in Nondifferentiable G-Invex Multiobjective Programming

Abstract

We consider a class of nondifferentiable multiobjective programs with inequality and equality constraints in which each component of the objective function contains a term involving the support function of a compact convex set. We introduce G-Karush-Kuhn-Tucker conditions and G-Fritz John conditions for our nondifferentiable multiobjective programs. By using suitable G-invex functions, we establish G-Karush-Kuhn-Tucker necessary and sufficient optimality conditions, and G-Fritz John necessary and sufficient optimality conditions of our nondifferentiable multiobjective programs. Our optimality conditions generalize and improve the results in Antczak (2009) to the nondifferentiable case.

1. Introduction and Preliminaries

A number of different forms of invexity have appeared. In [1], Martin defined Kuhn-Tucker invexity and weak duality invexity. In [2], Ben-Israel and Mond presented some new results for invex functions. Hanson [3] introduced the concepts of invex functions, and Type I, Type II functions were introduced by Hanson and Mond [4]. Craven and Glover [5] established Kuhn-Tucker type optimality conditions for cone invex programs, and Jeyakumar and Mond [6] introduced the class of the so-called V-invex functions to proved some optimality for a class of differentiable vector optimization problems than under invexity assumption.Egudo [7] established some duality results for differentiable multiobjective programming problems with invex functions. Kaul et al. [8] considered Wolfe-type and Mond-Weir-type duals and generalized the duality results of Weir [9] under weaker invexity assumptions.

Based on the paper by Mond and Schechter [10], Yang et al. [11] studied a class of nondifferentiable multiobjective programs. They replaced the objective function by the support function of a compact convex set, constructed a more general dual model for a class of nondifferentiable multiobjective programs, and established only weak duality theorems for efficient solutions under suitable weak convexity conditions. Subsequently, Kim et al. [12] established necessary and sufficient optimality conditions and duality results for weakly efficient solutions of nondifferentiable multiobjective fractional programming problems.

Recently, Antczak [13, 14] studied the optimality and duality for G-multi-objective programming problems. They defined a new class of differentiable nonconvex vector valued functions, namely, the vector G-invex (G-incave) functions with respect to . They used vector G-invexity to develop optimality conditions for differentiable multiobjective programming problems with both inequality and equality constraints. Considering the concept of a (weak) Pareto solution, they established the so-called G-Karush-Kuhn-Tucker necessary optimality conditions for differentiable vector optimization problems under the Kuhn-Tucker constraint qualification.

In this paper, we obtain an extension of the results in [13],which were established in the differentiable to the nondifferentiable case. We proposed a class of nondifferentiable multiobjective programming problems in which each component of the objective function contains a term involving the support function of a compact convex set. We obtain G-Karush-Tucker necessary and sufficient conditions and G-Fritz John necessary and sufficient conditions for weak Pareto solution. Necessary optimal theorems are presented by using alternative theorem [15] and Mangasarian-Fromovitz constraint qualification [16]. In addition, we give sufficient optimal theorems under suitable G-invexity conditions.

We provide some definitions and some results that we shall use in the sequel. Throughout the paper, the following convention will be used.

For any we write

(1.1)

Throughout the paper, we will use the same notation for row and column vectors when the interpretation is obvious. We say that a vector is negative if and strictly negative if .

Definition 1.1.

A function is said to be strictly increasing if and only if

(1.2)

Let be a vector-valued differentiable function defined on a nonempty open set , and , the range of , that is, the image of under .

Definition 1.2 (see [11]).

Let C be a compact convex set in . The support function is defined by

(1.3)

The support function , being convex and everywhere finite, has a subdifferential, that is, there exists such that

(1.4)

Equivalently,

(1.5)

The subdifferential of at is given by

(1.6)

Now, in the natural way, we generalize the definition of a real-valued G-invex function. Let be a vector-valued differentiable function defined on a nonempty open set , and the range of , that is, the image of under .

Definition 1.3.

Let be a vector-valued differentiable function defined on a nonempty set and . If there exist a differentiable vector-valued function such that any of its component is a strictly increasing function on its domain and a vector-valued function such that, for all and for any

(1.7)

then is said to be a (strictly) vector -invex function at on (with respect to ) (or shortly, -invex function at on ). If (1.7) is satisfied for each , then is vector -invex on with respect to .

Lemma 1.4 (see [13]).

In order to define an analogous class of (strictly) vector -incave functions with respect to , the direction of the inequality in the definition of -invex function should be changed to the opposite one.

We consider the following multiobjective programming problem.

(1.8)

where , are differentiable functions on a nonempty open set . Moreover, are differentiable real-valued strictly increasing functions, are differentiable real-valued strictly increasing functions, and , are differentiable real-valued strictly increasing functions. Let be the set of all feasible solutions for problem (NMP), and . Further, we denote by the set of inequality constraint functions active at and by the objective functions indices set, for which the corresponding Lagrange multiplier is not equal . For such optimization problems, minimization means in general obtaining weak Pareto optimal solutions in the following sense.

Definition 1.5.

A feasible point is said to be a weak Pareto solution (a weakly efficient solution, a weak minimum) of (NMP) if there exists no other such that

(1.9)

Definition 1.6 (see [17]).

Let be a given set in ordered by or by . Specifically, we call the minimal element of defined by a minimal vector, and that defined by a weak minimal vector. Formally speaking, a vector is called a minimal vector in if there exists no vector in such that ; it is called a weak minimal vector if there exists no vector in such that .

By using the result of Antczak [13] and the definition of a weak minimal vector, we obtain the following proposition.

Proposition 1.7.

Let be feasible solution in a multiobjective programming problem and let , be a continuous real-valued strictly increasing function defined on . Further, we denote and . Then, is a weak Pareto solution in the set of all feasible solutions for a multiobjective programming problem if and only if the corresponding vector is a weak minimal vector in the set .

Proof.

Let be a weak Pareto solution. Then there does not exist such that

(1.10)

By the strict increase of involving the support function, we have

(1.11)

Therefore, is a weak minimal vector in the set W. The converse part is proved similarly.

Lemma 1.8 (see [13]).

In the case when , for any , we obtain a definition of a vector-valued invex function.

2. Optimality Conditions

In this section, we establish G-Fritz John and G-Karush-Kuhn-Tucker necessary and sufficient conditions for a weak Pareto optimal point of (NMP).

Theorem 2.1 (G-Fritz John Necessary Optimality Conditions).

Suppose that are differentiable real-valued strictly increasing functions defined on are differentiable real-valued strictly increasing functions defined on , are differentiable real-valued strictly increasing functions defined on , and let . Let be a weak Pareto optimal point in problem (NMP). Then there exist , , and such that

(2.1)

Proof.

Let . Since is convex and compact,

(2.2)

is finite. Also, ,

(2.3)

Since is a weak Pareto optimal point in (NMP)

(2.4)

has no solution .By [15, Corollary ], there exist , and , not all zero, such that for any ,

(2.5)

Let = . Then . Assume to the contrary that . By separation theorem, there exists such that , that is,

(2.6)

This contradicts (2.5).

Letting , we get

(2.7)

Since , we obtain the desired result.

Theorem 2.2 (G-Karush-Kuhn-Tucker Necessary Optimality Conditions).

Suppose  that are differentiable real-valued strictly increasing functions defined on are differentiable real-valued strictly increasing functions defined on , are differentiable real-valued strictly increasing functions defined on , and , are linearly independent, and let . Moreover, we assume that there exists such that , and . If is a weak Pareto optimal point in problem (NMP), then there exist , , and such that

(2.8)

Proof.

Since is a weak Pareto optimal point of (NMP), by Theorem 2.1, there exist , , and such that

(2.9)

Assume that there exists such that , and . Then . Assume to the contrary that . Then . If , then . Since , are linearly independent, has a trivial solution , this contradicts to the fact that . So . Define . Since , we have and so . This is a contradiction. Hence . Indeed, it is sufficient only to show that there exist , and such that . we set

(2.10)

It is not difficult to see that the G-Karush-Kuhn-Tucker necessary optimality conditions are satisfied with Lagrange multipliers, there exist ; and given by (2.10).

We denote by and the sets of equality constraints indices for which a corresponding Lagrange multiplier is positive and negative, respectively, that is, and .

Theorem 2.3 (G-Fritz John Sufficient Optimality Conditions).

Let satisfy the G-Fritz John optimality conditions as follow:

(2.11)

(2.12)

(2.13)

(2.14)

Further, assume that is vector -invex with respect to at , is strictly -invex with respect to at , -invex with respect to at , and -incave with respect to at . Moreover, suppose that for and for . Then is a weak Pareto optimal point in problem (NMP).

Proof.

Suppose that is not a weak Pareto optimal point in problem (NMP). Then there exists such that . Since

(2.15)

Thus we get

(2.16)

By assumption, is -invex with respect to at on . Then by Definition 1.3, for any ,

(2.17)

Hence by (2.16) and (2.17), we obtain

(2.18)

Since satisfy the G-Fritz John conditions, by ,

(2.19)

Since is strictly -invex with respect to at on ,

(2.20)

Thus, by ,

(2.21)

Then, (2.12) implies

(2.22)

By assumption, , is -invex with respect to at on , and , is -incave with respect to at on . Then, by Definition 1.3, we have,

(2.23)

Thus, for any ,

(2.24)

Since and , then the inequality above implies

(2.25)

Adding both sides of inequalities (2.19), (2.22), (2.25), and by (2.14),

(2.26)

which contradicts (2.11). Hence, is a weak Pareto optimal for (NMP).

Theorem 2.4 (G-Karush-Kuhn-Tucker Sufficient Optimality Conditions).

Let   satisfy the G-Karush-Kuhn-Tucker conditions as follow:

(2.27)

(2.28)

(2.29)

(2.30)

Further, assume that is vector -invex with respect to at , is strictly -invex with respect to at , -invex with respect to at , and -incave with respect to at . Moreover, suppose that for and for . Then is a weak Pareto optimal point in problem (NMP).

Proof.

Suppose that is not a weak Pareto optimal point in problem (NMP). Then there exists such that . Since

(2.31)

Thus we get

(2.32)

By assumption, is -invex with respect to at on . Then by Definition 1.3, for any ,

(2.33)

Hence by (2.32) and (2.33), we obtain

(2.34)

Since satisfy the G-Karush-Kuhn-Tucker conditions, by ,

(2.35)

Since is strictly -invex with respect to at on ,

(2.36)

Thus, by ,

(2.37)

Then, (2.28),(2.30) imply

(2.38)

By assumption, , is -invex with respect to at on , and , is -incave with respect to at on . Then, by Definition 1.3, we have,

(2.39)

Thus, for any ,

(2.40)

Since and , then the inequality above implies

(2.41)

Adding both sides of inequalities (2.35), (2.38) and (2.41),

(2.42)

which contradicts (2.27). Hence, is a weak Pareto optimal for (NMP).