Asymptotic regularity for Lipschitzian nonlinear optimization problems with applications to complementarity-constrained and bilevel programming

TL;DR

This paper extends asymptotic regularity concepts to nonsmooth Lipschitzian optimization problems, compares new constraint qualifications, and applies the theory to complementarity and bilevel programming, including algorithm development and numerical validation.

Contribution

It introduces novel asymptotic regularity notions for nonsmooth problems and demonstrates their applications to complementarity and bilevel optimization, including algorithm design.

Findings

Recovered recent results in complementarity-constrained optimization

Justified a stationarity system for bilevel problems

Proposed a combined penalty and DC-programming algorithm

Abstract

Asymptotic stationarity and regularity conditions turned out to be quite useful to study the qualitative properties of numerical solution methods for standard nonlinear and complementarity-constrained programs. In this paper, we first extend these notions to nonlinear optimization problems with nonsmooth but Lipschitzian data functions in order to find reasonable notions of asymptotic stationarity and regularity in terms of Clarke's and Mordukhovich's subdifferential construction. Particularly, we compare the associated novel asymptotic constraint qualifications with already existing ones. The second part of the paper presents two applications of the obtained theory. On the one hand, we specify our findings for complementarity-constrained optimization problems and recover recent results from the literature which demonstrates the power of the approach. Furthermore, we hint at potential extensions to or- and vanishing-constrained optimization. On the other hand, we demonstrate the usefulness of asymptotic regularity in the context of bilevel optimization. More precisely, we justify a well-known stationarity system for affinely constrained bilevel optimization problems in a novel way. Afterwards, we suggest a solution algorithm for this class of bilevel optimization problems which combines a penalty method with ideas from DC-programming. After a brief convergence analysis, we present results of some numerical experiments.