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)