Theoretical and numerical comparison of the Karush-Kuhn-Tucker and value function reformulations in bilevel optimization

TL;DR

This paper compares the KKT and value function reformulations in bilevel optimization, analyzing their theoretical properties and numerical performance, and finds the value function approach often outperforms KKT in practice.

Contribution

It introduces a fair comparison framework for these reformulations and provides the first systematic analysis of their relative effectiveness.

Findings

Value function reformulation often outperforms KKT in numerical experiments.

Neither reformulation dominates theoretically across all problem classes.

The comparison framework can guide the choice of reformulation in bilevel optimization.

Abstract

The Karush-Kuhn-Tucker and value function (lower-level value function, to be precise) reformulations are the most common single-level transformations of the bilevel optimization problem. So far, these reformulations have either been studied independently or as a joint optimization problem in an attempt to take advantage of the best properties from each model. To the best of our knowledge, these reformulations have not yet been compared in the existing literature. This paper is a first attempt towards establishing whether one of these reformulations is best at solving a given class of the optimistic bilevel optimization problem. We design a comparison framework, which seems fair, considering the theoretical properties of these reformulations. This work reveals that although none of the models seems to particularly dominate the other from the theoretical point of view, the value function reformulation seems to numerically outperform the Karush-Kuhn-Tucker reformulation on a Newton-type algorithm. The computational experiments here are mostly based on test problems from the Bilevel Optimization LIBrary (BOLIB).