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

Journal of Inequalities and Applications, Sep 2014

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.

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

Journal of Inequalities and Applications Proper efficiency and duality for a new class of nonconvex multitime multiobjective variational problems Ariana Pitea 0 2 Tadeusz Antczak 1 0 Faculty of Applied Sciences, University 'Politehnica' of Bucharest , Splaiul Independent ̧ei, No. 313, Bucharest, 060042 , Romania 1 Faculty of Mathematics and Computer Science, University of Łódz ́ , Banacha 22, Łódz ́, 90-238 , Poland 2 Faculty of Applied Sciences, University 'Politehnica' of Bucharest , Splaiul Independent ̧ei, No. 313, Bucharest, 060042 , Romania 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 multitime variational problem; univex function; proper efficient solution; optimality conditions; duality 1 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. Several classes of functions have been defined for the purpose of weakening the limitations of convexity in mathematical programming, and also 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, 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. The areas of research which strongly motivate the PDI&PDE-constrained optimization include: shape optimization in fluid mechanics and medicine, optimal control of processes, structural optimization, material inversion - in geophysics, data assimilation in regional weather prediction modeling, etc. PDI&PDE-constrained optimization problems are generally infinite dimensional in nature, large and complex, [] Chinchuluun The basic optimization problems of path-independent curvilinear integrals with PDE constraints or with isoperimetric constraints, expressed by the multiple integrals or pathindependent curvilinear integrals, were stated for the first time by Udrişte and Ţevy in []. Later, optimality and duality results for PDI&PDE-constrained optimization problems were established by Pitea et al. in [] and []. Recently, nonconvex optimization problems with the so-called class of univex functions have been the object of increasing interest, both theoretical and applicative, and there exists nowadays a wide literature. This class of generalized convex functions was introduced in nonlinear scalar optimization problems by Bector et al. [] as a generalization of the definition of an invex function introduced by Hanson []. Later, 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. In [], Popa and Popa defined the concept of ρ-univexity as a generalization univexity and ρ-invexity. 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. In this paper, we study a new class of nonconvex multitime multiobjective variational problems of minimizing a vector-valued functional of curvilinear integral type. In order to prove the main results in the paper, we introduce the definition of univexity for a vectorial functional of curvilinear integral type. Thus, 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. Further, we define the multiobjective variational dual problems in the (...truncated)


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Ariana Pitea, Tadeusz Antczak. Proper efficiency and duality for a new class of nonconvex multitime multiobjective variational problems, Journal of Inequalities and Applications, 2014, pp. 333, 2014,