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

TL;DR

This paper explores how semismoothness influences the implicit programming method for nonsmooth optimization problems, providing theoretical insights and computational techniques that improve convergence in complex MPECs and bilevel programs.

Contribution

It introduces a novel approach leveraging semismoothness and SCD mappings to enhance bundle methods for nonsmooth optimization, with proven convergence guarantees.

Findings

Successful application to a complex MPEC with variational inequality

Effective handling of nonsmooth upper-level objectives in bilevel programs

Convergence to stationary points demonstrated through computational experiments

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 results obtained allow us to supply the bundle algorithm with suitable, easily computable ``pseudogradients'', 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 and by an academic bilevel program with a nonsmooth upper-level objective.