First- and second-order optimality conditions in optimistic bilevel set-valued programming

ABSTRACT

In this work we adapt the main results from Khanh and Tung [First and second-order optimality conditions without differentiability in multivalued vector optimization. Positivity. 2015;19:817–841] for a general set-valued optimization problem to an optimistic bilevel set-valued programming problem as an optimization problem with implicitly given set-valued constraints. Using the optimal value function, we convert our problem into a one level set-valued optimization problem with general inequality constraints and derive both necessary conditions and sufficient conditions of order one and two. The main tools we exploit are approximations of set-valued mappings. To illustrate the obtained results, some examples are given.

Keywords:

• Bilevel optimization
• first- and second-order approximations
• optimality conditions
• set-valued optimization

2000 Mathematics Subject Classifications:

• Primary: 90C29
• 90C26
• 90C70
• Secondary:49K99

Acknowledgments

Thanks are due to the anonymous referees for the careful reading and the improvements they bring to our paper.

Disclosure statement

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