Newton-type method for bilevel programs with linear lower level problem and application to toll optimization

TL;DR

This paper develops a Newton-type method for solving bilevel programs with linear lower levels, using a KKT reformulation and penalization, and demonstrates its effectiveness on toll optimization in transportation networks.

Contribution

It introduces a novel Newton-type algorithm for bilevel problems with linear lower levels, incorporating a KKT reformulation and partial penalization, with practical application to toll setting.

Findings

Algorithm converges effectively in numerical experiments.

Method successfully applied to toll optimization in transportation networks.

Provides a new approach for bilevel problems with linear lower levels.

Abstract

We consider a bilevel program involving a linear lower level problem with left-hand-side perturbation. We then consider the Karush-Kuhn-Tucker reformulation of the problem and subsequently build a tractable optimization problem with linear constraints by means of a partial exact penalization. A semismooth system of equations is then generated from the later problem and a Newton-type method is developed to solve it. Finally, we illustrate the convergence and practical implementation of the algorithm on the optimal toll-setting problem in transportation networks.