Higher-order efficiency conditions for constrained vector equilibrium problems

ABSTRACT

In this paper, we investigate some optimality conditions of higher-order for nonsmooth nonconvex vector equilibrium problems with constraints (or without constraints) in terms of the higher-order upper and lower Studniarski derivatives in Banach spaces. The calculus rule of all the data involved in the problem is taken into account. Using the notion of higher-order upper Studniarski derivative with the class of m-stable/m-steady functions, we first provide the higher-order necessary optimality conditions for the local weak efficient solution of vector equilibrium problem without constraints and then we present the higher-order necessary and sufficient optimality conditions for the local strict minimum of order m to such problem. We second obtain the necessary and sufficient optimality conditions for those efficient solutions of vector equilibrium problem with cone and equality constraints through Lagrange multiplier rules in finite-dimensional spaces. Final, the higher-order necessary and sufficient optimality conditions in terms of the higher-order upper and lower Studniarski derivatives for the local weak efficient solution of vector equilibrium problem with set, cone and equality constraints in Banach spaces are also established. Some examples are proposed to demonstrate our findings.

Keywords:

• Higher-order optimality conditions
• vector equilibrium problem with constraints
• local weak efficient solutions
• mth-order strict local minimums
• higher-order upper and lower Studniarski derivatives

AMS Classifications:

• 49J52
• 90C46
• 90C29
• 49K27

Acknowledgments

The authors are grateful to the anonymous referees and the associate editor for their valuable comments and suggestions, which helped to improve the quality of the paper in its present form.

Disclosure statement

No potential conflict of interest was reported by the author(s).