Interval-Valued Optimization Problems Involving (𝛼,𝜌)-Right Upper-Dini-Derivative Functions

Abstract

We consider an interval-valued multiobjective problem. Some necessary and sufficient optimality conditions for weak efficient solutions are established under new generalized convexities with the tool-right upper-Dini-derivative, which is an extension of directional derivative. Also some duality results are proved for Wolfe and Mond-Weir duals.

1. Introduction

Recently, Yuan and Liu [1] considered some new generalized convexity concepts using right upper-Dini-derivative, which is an extension of directional derivative. Thus some optimality and duality results are established for a nondifferentiable multiobjective programming problem. For various approaches relative to generalized convexity, we refer to [2–9].

In many real-life situations data suffer from inexactness. The interval-valued optimization problems are closely related to optimization problems with inexact data. Recently, Wu [10–12] derived optimality conditions and duality results for a multiobjective programming problem with interval-valued objective functions. See also [13] and their references.

In this paper we consider an interval multiobjective optimization problem. Some new optimality conditions and duality results are stated under new generalized convexities with the tool-right upper-Dini-derivative. The paper is organized as follows. In Section 2 some definitions, notations, and some basic arithmetic of interval calculus are given. In Section 3, we state necessary optimality conditions and in Section 4 we present sufficient optimality conditions. The duality results are stated in Sections 5 and 6. The last section gives some conclusions.

2. Notations and Preliminaries

Let be the -dimensional Euclidean space and let be its nonnegative orthant. For and we consider the following conventions: Let be an arcwise connected set in Avriel and Zang [14] and Bhatia and Mehra [15] and a real-valued function defined on . Let , and be the arc connecting and in .

Definition 1. The right derivative (or right differential) of with respect to at is defined as

Yuan and Liu [1] give some new generalized convexity with the upper-Dini-derivative concept.

Definition 2. The right upper-Dini-derivative relative to is defined by

Definition 3. is locally arcwise connected at if for any and there exists a positive number , with and a continuous arc s.t. for any . The set is if is at any .

Definition 4 (see [1]). Let be a LAC set and let be a real function defined on . The function is said to be -right upper-Dini-derivative locally arcwise connected with respect to at , if there exist real functions and such that If is -right upper-Dini-derivative locally arcwise connected (with respect to ) at for any , then is called -right upper-Dini-derivative locally arcwise connected (with respect to ) on .

Definition 5 (see [1]). A -dimensional vector-valued function is called -right upper-Dini-derivative arcwise connected (with respect to ) at , if the th component of is -right upper-Dini-derivative arcwise connected (with respect to ) at for , where and. If is -right upper-Dini-derivative arcwise connected (with respect to ) at any , then is called -right upper-Dini-derivative arcwise connected (with respect to ) on .

Definition 6 (see [3]). A -dimensional vector-valued function is called convex-like (with respect to ) on if for all and any , there exists such that .

Definition 7 (see [1]). A -dimensional vector-valued function is called -generalized (strong) pseudoright upper-Dini-derivative arcwise connected (with respect to ) at , if there exists vector-valued function such that , for ; is called -generalized (weak) quasi-right upper-Dini-derivative arcwise connected (with respect to ) at , if there exists vector-valued function such that , for , where .

Lemma 8 (see [16]). Let be a nonempty set and let be a convex-like vector-valued function on . Then either has a solution , or there exists such that the system holds for all , but both are never true at the same time.

Let CBI() be the class of all closed and bounded intervals in . Thus if CBI, we have where and mean lower and upper bounds of . If , then is a real number. Also, let . Then, by definition we have

For a real number , we have

Using [17, 18], we consider some preliminary results about interval arithmetic calculus.

Definition 9. Let and CBI. We say that is less than and write if , .

Definition 10. Let CBI. We say that is less than or equal to and write if and .

Let be a nonempty subset of . A function CBI is called an interval-valued function. In this case, with , , .

We consider the following multiobjective interval-valued optimization problem: with , , , where , for and , . Let be the set of all feasible points of . We put .

Definition 11. Let . We say that is a weak efficient solution of if there exists no such that .

3. Necessary Optimality Conditions

In this section, we establish Fritz John and Karush-Kuhn-Tucker necessary optimality conditions for problem .

Theorem 12 (Fritz John necessary condition). Assume that is an efficient solution for . If, , and are convex-like on with respect to the variable and   is upper semicontinuous at for , then there exist , , such that

Proof. We prove firstly that the following system of inequalities has no solution for . We proceed by contradicting. If there exists , a solution of this system, we get that for each there exist and such that for and for each there exists such that Since , for , is semicontinuous at , then is semicontinuous at . Finally we get for any , where . These inequalities contradict that is a weak efficient solution for . Hence the systems (9) and (10) have no solution for . Now we can apply Lemma 8. Since , , and are convex-like on , we obtain that (9) and (10) hold and the proof is complete.

Theorem 13 (Karush-Kuhn-Tucker necessary optimality condition). Let , , and be convex-like on with respect to the variables and , for , be upper semicontinuous at . If there exists such that and is a weak efficient solution for the problem , then there exist , , and satisfying (9), (10), and .

Proof. We suppose . Then, by (9) we obtain Since for , by (10), and there exists such that , by (15), it results , which contradicts (15) and theorem is proved.

4. Sufficient Optimality Conditions

In this section we give some Karush-Kuhn-Tucker type sufficient optimality conditions under generalized convexity with upper-Dini-derivative concept.

Theorem 14. Let be a feasible solution of . Assume that there exist and , , such that and are and -right upper-Dini-derivative locally arcwise connected at with respect to , respectively. Also one assumes and hold for all feasible. Then is a weak efficient solution for .

Proof. We suppose to the contrary that is not a weak efficient solution for . Then there exists such that . Now, since , , and , we get
Since and are and -right upper-Dini-derivative locally arcwise connected at , by (18) we obtain
Now, by and (17) we get a contradiction. Thus the theorem is proved.

The next sufficient optimality condition is given in the case of generalized pseudo- and quasi-right upper-Dini-derivative arcwise connected type, where the proof is on the line of the above theorem.

Theorem 15. Let be a feasible solution of . Assume that there exist and , , such that is generalized pseudoright upper-Dini-derivative locally arcwise connected at with respect to and is -generalized quasi-right upper-Dini-derivative locally arcwise connected at with respect to , respectively. Also one assumes that there exist and such that for any . Then is a weak efficient solution for .

5. Wolfe Duality

Relative to we consider the following Wolfe type dual problem: where and such that . Let denote the set of all feasible solutions of and let be the projection of the set on . Now we present weak, strong, and strict converse duality theorems relative to and . The proofs, which will be skipped here, follow the classic lines of multiobjective optimization [4, 7] and interval optimization problems [11].

Theorem 16 (weak duality). Let and be a feasible solution for and , respectively. One supposes that and are and -right upper-Dini-derivative arcwise connected at on with respect to , respectively. Moreover, one assumes and , for all . Then the following cannot hold: .

Theorem 17 (strong duality). Let be a weak efficient solution of , at which the assumptions of Karush-Kuhn-Tucker necessary optimality conditions are satisfied. Then there exist and such that and the objective values of and are equal. Further, if the hypotheses of weak duality theorem hold for all feasible solutions for , then is an efficient solution of .

Theorem 18 (converse duality). Let be a weak efficient solution of . One assumes that    and are and -right upper Dini derivative arcwise connected at on with respect to , respectively. If and , for all , then is a weak efficient solution of .

6. Mond-Weir Duality

In this section we consider the following interval multiobjective dual problem, which is Mond-Weir dual type [6]: where , with . Let denote the set of all feasible solutions of and as the projection of the set on .

As in Section 5, we can establish some duality results. Here, we simply state them in the next theorems.

Theorem 19 (weak duality). Let and be a feasible solution for and , respectively. Assume that and are and -right upper-Dini-derivative arcwise connected at on with respect to , respectively. Also one supposes that and , for all . Then the following cannot hold: .

Theorem 20 (strong duality). Let be a weak efficient solution of the interval multiobjective programming problem , at which the assumptions of Karush-Kuhn-Tucker necessary optimality conditions are satisfied. Then there exist , , and , such that is a feasible solution for and the objective values of and are equal. Moreover, if the weak duality result between and holds, then is a weak efficient solution for .

Theorem 21 (converse duality). Let be a weak efficient solution of . Suppose that and are and -right upper-Dini-derivative arcwise connected at on with respect to , respectively. Further, if and , for all , then is a weak efficient solution of .

7. Conclusions

In this paper we considered an interval multiobjective optimization problem. Some new optimality conditions and duality results were studied under the generalized convexity considered by Yuan and Liu [1]. Necessary optimality conditions and sufficient optimality conditions were derived. Duality results were established. These results can be extended to a class of univex generalized convexity of Mishra type [4] with the tool-right upper-Dini-derivative. Further it is possible to establish duality results relative to a mixed dual of Xu type [9].

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.