Proper efficiency and duality for a new class of nonconvex multitime multiobjective variational problems

Ariana Pitea,Tadeusz Antczak

doi:10.1186/1029-242x-2014-333

Abstract

In this paper, a new class of generalized of nonconvex multitime multiobjective variational problems is considered. We prove the sufficient optimality conditions for efficiency and proper efficiency in the considered multitime multiobjective variational problems with univex functionals. Further, for such vector variational problems, various duality results in the sense of Mond-Weir and in the sense of Wolfe are established under univexity. The results established in the paper extend and generalize results existing in the literature for such vector variational problems.MSC:65K10, 90C29, 90C30.

Highlights

Multiobjective variational problems are very prominent amongst constrained optimization models because of their occurrences in a variety of popular contexts, notably, economic planning, advertising investment, production and inventory, epidemic, control of a rocket, etc.; for an excellent survey, see [ ] Chinchuluun and Pardalos.Several classes of functions have been defined for the purpose of weakening the limitations of convexity in mathematical programming, and for multiobjective variational problems
Several authors have contributed in this direction: [ ] Aghezzaf and Khazafi, [ ] Ahmad and Sharma, [ ] Arana-Jiménez et al, [ ] Bector and Husain, [ ] Bhatia and Mehra, [ ] Hachimi and Aghezzaf, [ ] Mishra and Mukherjee, [ – ] Nahak and Nanda, and others. One class of such multiobjective optimization problems is the class of vector PDI&PDEconstrained optimization problems in which partial differential inequalities or/and equations represent a multitude of natural phenomena of some applications in science and engineering
We study a new class of nonconvex multitime multiobjective variational problems of minimizing a vector-valued functional of curvilinear integral type

Summary

Introduction

Multiobjective variational problems are very prominent amongst constrained optimization models because of their occurrences in a variety of popular contexts, notably, economic planning, advertising investment, production and inventory, epidemic, control of a rocket, etc.; for an excellent survey, see [ ] Chinchuluun and Pardalos. Antczak [ ] used the introduced η-approximation approach for nonlinear multiobjective programming problems with univex functions to obtain new sufficient optimality conditions for such a class of nonconvex vector optimization problems. Mishra et al [ ] established some sufficiency results for multiobjective programming problems using Lagrange multiplier conditions, and under various types of generalized V -univexity type-I requirements, they proved weak, strong and converse duality theorems. In [ ], Khazafi and Rueda established sufficient optimality conditions and mixed type duality results under generalized V -univexity type I conditions for multiobjective variational programming problems. We study a new class of nonconvex multitime multiobjective variational problems of minimizing a vector-valued functional of curvilinear integral type. We establish the sufficient optimality conditions for a proper efficiency in the multitime multiobjective variational problem under univexity assumptions imposed on the functionals constituting such vector variational problems.

We get

Using the following relation

Using the relation