Optimality conditions and duality for nonsmooth vector equilibrium problems with constraints

Abstract

We consider optimality conditions and duality for a general nonsmooth setvalued vector equilibrium problem with inequality constraints. We focus on the Q-solution, which contains most other concepts, and the firm solution, which is hardly expressed as a Q-solution. To face high-level nonsmoothness, we employ several notions of contingent variations as generalized derivatives. As relaxed convexity assumptions, some types of arcwise connectedness conditions are imposed. Both necessary and sufficient optimality conditions of orders m ≥ 1 are investigated for global Q-solutions and local firm solutions, with consequences for Henig-and Benson-proper solutions. For duality, the Wolfe and Mond-Weir schemes are dealt with, using first-order contingent variations. We discuss weak, strong, direct and converse duality. As illustrative applications, we choose three optimization-related models: a vector minimization problem with inequality constraints, a cone saddle point problem and a variational inequality. Our results are new or improve several existing ones in the literature.

Second-order sensitivity analysis for parametric equilibrium problems in set-valued optimization

“…For possible developments of this paper, since several theoretical models in optimization can be expressed as special cases of equilibrium problems, such as constrained set-valued optimization problems, cone saddle point problems, variational inequalities (see [18]), we can learn about applications of the obtained results in the paper to these particular cases. Furthermore, finding answers for open questions in Remark 3.12 may be a promising study.…”

Higher-order optimality conditions for strict and weak efficient solutions in set-valued optimization

“…(iii) In general, the preinvexness is incomparable with the near subconvexlikeness, see Remark 2.1(iii) in [11].…”

“…(x 0 , (y 0 , z 0 )) for all m ≥ 2. If there exist m ≥ 1, c * ∈ C * \{0} and d * ∈ D * such that (7) and (11) hold for all (y, z) ∈ E D m (F, G)(x 0 , (y 0 , z 0 ))( \{0}), then (x 0 , y 0 ) is a local efficient solution of (SOP).…”

Optimality and Duality of Approximate Quasi Weakly Efficient Solution for Nonsmooth Vector Optimization Problems

“…It is known that duality assertions allow to study a minimization problem through a maximization problem and to know what one can expect in the best case. ere are many dual models in the literature, for instance, Lagrange dual (see [15,16]), Mond-Weir duality (see [17][18][19]), Wolfe duality (see [17,20]), conjugate duality (see [21]), and symmetric duality (see [22]). In this paper, we will introduce an approximate Mond-Weir dual model for the problem (VOP) and examine duality theorems between it and primal problem involving approximate quasi weakly e cient solution.…”