Second-order optimality conditions in set-valued optimization via asymptotic derivatives

Abstract

In this article we give new second-order optimality conditions in set-valued optimization. We use the second-order asymptotic tangent cones to define second-order asymptotic derivatives and employ them to give the optimality conditions. We extend the well-known Dubovitskii–Milutin approach to set-valued optimization to express the optimality conditions given as an empty intersection of certain cones in the objective space. We also use some duality arguments to give new multiplier rules. By following the more commonly adopted direct approach, we also give optimality conditions in terms of a disjunction of certain cones in the image space. Several particular cases are discussed.

Keywords:

• set-valued optimization
• second-order asymptotic derivatives
• second-order contingent derivatives
• second-order contingent sets
• contingent cones
• adjacent cones
• interiorly tangent cones
• Aubin property
• optimality conditions

AMS Subject Classification:

• 90C26
• 90C29
• 90C30

Acknowledgements

We express our sincere gratitude to the referees for their very careful reading and suggestions that brought substantial improvements to our work. This work was supported by a grant from the Simons Foundation (#210443 to Akhtar Khan).