Optimality conditions for a class of nonsmooth semidefinite bilevel optimization problems via convexifications applied to bilevel supply chain under uncertainty

Abstract

This paper addresses a nonsmooth semidefinite bilevel optimization problem, where both the upper- and lower-level problems include semidefinite constraints. We derive necessary optimality conditions of Fritz–John and Karush–Kuhn–Tucker types. Our approach utilizes partial exact penalization, and employs upper semi-regular convexificators and the optimal value reformulation. Additionally, we establish sufficient optimality conditions under generalized convexity assumptions. The nonsmooth nature of the problem, along with semidefinite constraints at both levels, introduces significant technical challenges, which are addressed through the proposed tools and methods. We illustrate our findings by considering an application to a bilevel supply chain problem focused on robust production planning under demand uncertainty.