Levenberg-Marquardt method and partial exact penalty parameter selection in bilevel optimization

TL;DR

This paper introduces a Levenberg-Marquardt method for solving bilevel optimization problems reformulated into single-level problems, focusing on optimal parameter selection through experimental analysis.

Contribution

It develops a Levenberg-Marquardt algorithm tailored for bilevel problems with partial exact penalty parameter selection, supported by comprehensive experiments.

Findings

Effective parameter selection improves solution accuracy.

The method performs well on BOLIB test problems.

Partial calmness aids in reformulating bilevel problems.

Abstract

We consider the optimistic bilevel optimization problem, known to have a wide range of applications in engineering, that we transform into a single-level optimization problem by means of the lower-level optimal value function reformulation. Subsequently, based on the partial calmness concept, we build an equation system, which is parameterized by the corresponding partial exact penalization parameter. We then design and analyze a Levenberg-Marquardt method to solve this parametric system of equations. Considering the fact that the selection of the partial exact penalization parameter is a critical issue when numerically solving a bilevel optimization problem, we conduct a careful experimental study to this effect, in the context the Levenberg-Marquardt method, while using the Bilevel Optimization LIBrary (BOLIB) series of test problems.