Advances in Pure Mathematics
Vol.08 No.08(2018), Article ID:86736,9 pages
10.4236/apm.2018.88045
Sufficiency and Wolfe Type Duality for Nonsmooth Multiobjective Programming Problems
Gang An1, Xiaoyan Gao2
1College of Science, Xi’an Shiyou University, Xi’an, China
2College of Science, Xi’an University of Science and Technology, Xi’an, China
Copyright © 2018 by authors and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).
http://creativecommons.org/licenses/by/4.0/
Received: July 30, 2018; Accepted: August 17, 2018; Published: August 20, 2018
ABSTRACT
In this paper, a class of nonsmooth multiobjective programming problems is considered. We introduce the new concept of invex of order type II for nondifferentiable locally Lipschitz functions using the tools of Clarke subdifferential. The new functions are used to derive the sufficient optimality condition for a class of nonsmooth multiobjective programming problems. Utilizing the sufficient optimality conditions, weak and strong duality theorems are established for Wolfe type duality model.
Keywords:
Multiobjective Programming, Optimality Condition, Locally Lipschitz Function, Wolfe Type Dual Problem
1. Introduction
The field of multiobjective programming, also called vector programming, has grown remarkably in different directions in the settings of optimality conditions and duality theory since the 1980s. It has been enriched by the applications of various types of generalizations of convexity theory, with and without differentiability assumptions. The Clarke subdifferential [1] (also called the Clarke generalized gradient) is an important tool to derive optimality conditions for nonsmooth optimization problems. Together with the Clarke’s subdifferential, many generalized convexity or invexity functions were generalized to locally Lipschitz functions. Based upon the generalized functions, several sufficient optimality conditions and duality results were established for the optimization problems. We can see for examples [2] - [8] . In [9] Upadhyay introduced some new generalizations of the concept of -invexity and established the necessary and sufficient optimality conditions for a class of nonsmooth semi-infinite minmax programming problems. In [10] the new concepts of type I were introduced. Sufficient optimality conditions and Mond-Weir duality results were obtained for nonsmooth multiobjective programming problems. Recently, many researchers have been interested in other types of solution concepts. One of them is higher order strict minimizer. In [11] and [12] some sufficient conditions and duality results were obtained for the new concept of strict minimizer of higher order for a multiobjective optimization problem.
In this paper, we consider the nonsmooth multiobjective programming including the locally Lipschitz functions. The new concepts of invex of order type II functions are introduced. Then, a sufficient optimality condition is obtained for the nondifferentiable multiobjective programming problem under the new functions and the Wolfe type duality results are obtained.
2. Preliminaries and Definitions
Let be the n-dimensional Euclidean space and let X be a nonempty open subset of . For , we denote:
Definition 2.1. [1] The function is said to be locally Lipschitz at , if there exist scalars and , such that
. (1)
where is the open ball of radius about x.
Definition 2.2. [1] The generalized directional derivative of a locally Lipschitz function f at x in the direction d, denoted by , is as follows:
. (2)
Definition 2.3. [1] The generalized gradient of at , denoted by , is defined as follows:
. (3)
where is the inner product in .
Consider the following nonsmooth multiobjective programming problem:
(MP)
where and are locally Lipschitz functions and X is a convex set in .
Let be the set of feasible solutions of (MP), and
.
Definition 2.4. A point is a strict minimizer of order for (MP) with respect to a nonlinear function , if for a constant , such that
. (4)
Throughout the paper, we suppose that ; ; .
Definition 2.5. is said to be invex of order type II at , if there exist and some vectors and such that for all the following inequalities hold:
(5)
(6)
Definition 2.6. is said to be (pseudo, quasi) invex of order type II at , if there exist and some vectors and such that for all the following inequalities hold:
(7)
(8)
3. Optimality Condition
In this section, we establish sufficient optimality conditions for a strict minimizer of (MP).
Theorem 3.1. Let . Suppose that
1) There exist , , such that
2) is invex of order type II at ,
3) .
Then is a strict minimizer of order for (MP).
Proof: Since , there exists , , such that
. (9)
whence
. (10)
Suppose that is not a strict minimizer of order for (MP). Then there exists and , such that
. (11)
By and hypothesis 3), we have
. (12)
Since and , and hypothesis 3), we get
. (13)
In view of the hypothesis 1), one finds from (12) and (13) that
. (14)
(15)
From and , we obtain
. (16)
. (17)
Also
(18)
which contradicts (10). Hence the result is true.
4. Wolfe Type Duality
In this section, we consider the Wolfe type dual for the primal problem (MP) and establish various duality theorems. Let e be the vector of whose components are all ones.
Let
be the set of all feasible solutions in problem (MD).
Theorem 4.1. (weak duality) Let and be feasible solutions for (MP) and (MD), respectively. Moreover, assume that
1) is invex of order type II at u,
2) .
Then the following can hold:
. (19)
Proof: Suppose contrary to the result that holds, then we have
(20)
which implies
(21)
Using , we have
(22)
By hypothesis 2), we have
(23)
with hypothesis 1) and 2), the above inequality yields
(24)
That is
(25)
From , which implies
(26)
On the other hand, by using the constraint conditions of (MD), there exist
and ,
such that
(27)
Also,
(28)
which contradicts (26). Then the result is true.
Theorem 4.2. (weak duality) Let and be feasible solutions for (MP) and (MD), respectively. Moreover, assume that
1) is (pseudo,quasi) invex of order type II at u,
2)
Then the following can hold:
. (29)
Proof: Suppose contrary to the result that holds, then we have
(30)
Also
(31)
Since , which yields
(32)
It follows from hypothesis 2) that
(33)
In the view of hypothesis 1), one finds from (33) that
(34)
For , we have
(35)
Since is a feasible solution for (MD), there exist and such that
(36)
whence
(37)
It follows from (35) that
(38)
For and , which yields
(39)
From hypothesis 1), it follows that
(40)
whence
(41)
which contradicts (33). Then the result is true.
The following definition is needed in the proof of the strong duality theorem.
Definition 4.1. A point is called a strict maximizer of order for (MD) with respect to a nonlinear function , if there exists a constant such that
. (42)
Theorem 4.3. (strong duality) Assume that is a strict minimizer of order with respect to for (MP), also there exist and , such that and . Furthermore, if all the hypothesis of Theorem 4.1 are satisfied for all feasible solutions of (MP) and (MD), then is a strict maximizer of order for (MD) with respect to .
Proof: The hypothesis implies that is a feasible solution of (MD). By Theorem 4.1, for any feasible of (MD), we have
(43)
That is
(44)
Using , which yields
(45)
whence
(46)
Thus is a strict maximizer of order for (MD) with respect to .
5. Conclusion
In this paper, we have defined a class of new generalized functions. By using the new functions, we have presented a sufficient optimality condition and Wolfe type duality results for a nondifferentiable multiobjective problem. The present results can be further generalized for other programming problems.
Acknowledgements
This work is supported by Scientific Research Program Funded by Shaanxi Provincial Education Department (Program No. 15JK1456); Natural Science Foundation of Shaanxi Province of China (Program No. 2017JM1041).
Conflicts of Interest
The authors declare no conflicts of interest regarding the publication of this paper.
Cite this paper
An, G. and Ga, X.Y. (2018) Sufficiency and Wolfe Type Duality for Nonsmooth Multiobjective Programming Problems. Advances in Pure Mathematics, 8, 755-763. https://doi.org/10.4236/apm.2018.88045
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