Semismooth Newton-type method for bilevel optimization: Global convergence and extensive numerical experiments

TL;DR

This paper introduces a semismooth Newton method for bilevel optimization that guarantees global convergence and demonstrates excellent performance on numerous complex problems without requiring explicit derivatives of the lower-level value function.

Contribution

It develops a novel semismooth Newton framework for bilevel problems that avoids direct computation of the lower-level value function and ensures convergence properties.

Findings

Method exhibits global convergence and local superlinear convergence.

Achieves high accuracy with few penalty parameters.

Performs well on 124 benchmark nonlinear bilevel problems.

Abstract

We consider the standard optimistic bilevel optimization problem, in particular upper- and lower-level constraints can be coupled. By means of the lower-level value function, the problem is transformed into a single-level optimization problem with a penalization of the value function constraint. For treating the latter problem, we develop a framework that does not rely on the direct computation of the lower-level value function or its derivatives. For each penalty parameter, the framework leads to a semismooth system of equations. This allows us to extend the semismooth Newton method to bilevel optimization. Besides global convergence properties of the method, we focus on achieving local superlinear convergence to a solution of the semismooth system. To this end, we formulate an appropriate CD-regularity assumption and derive suffcient conditions so that it is fulfilled. Moreover, we develop conditions to guarantee that a solution of the semismooth system is a local solution of the bilevel optimization problem. Extensive numerical experiments on $124$ examples of nonlinear bilevel optimization problems from the literature show that this approach exhibits a remarkable performance, where only a few penalty parameters need to be considered.