On the role of semismoothness in the implicit programming approach to selected nonsmooth optimization problems

Abstract

The paper deals with the implicit programming approach to a class of Mathematical Programs with Equilibrium Constraints (MPECs) and bilevel programs in the case when the corresponding reduced problems are solved using a bundle method of nonsmooth optimization. The obtained results allow us to supply the bundle algorithm with suitable, easily computable ``pseudosubgradients'', ensuring convergence to points satisfying a stationary condition. Both the theory and computational implementation heavily rely on the notion of SCD (subspace containing derivatives) mappings and the associated calculus. The approach is validated via a complex MPEC with equilibrium governed by a variational inequality of the 2nd kind, by a shape optimization problem with a nonsmooth objective and by an academic bilevel program with a nonsmooth upper-level objective.