Solving Linear Bilevel Problems Using Big-Ms: Not All That Glitters Is Gold

TL;DR

This paper critically examines the common use of big-M constants in transforming linear bilevel problems into single-level MILPs, highlighting potential pitfalls and the need for better selection methods.

Contribution

It demonstrates through a counterexample that trial-and-error big-M tuning can lead to suboptimal solutions, emphasizing the need for improved approaches.

Findings

Trial-and-error big-M selection can cause suboptimal solutions

Counterexample shows limitations of current methods

Highlights need for better big-M determination techniques

Abstract

The most common procedure to solve a linear bilevel problem in the PES community is, by far, to transform it into an equivalent single-level problem by replacing the lower level with its KKT optimality conditions. Then, the complementarity conditions are reformulated using additional binary variables and large enough constants (big-Ms) to cast the single-level problem as a mixed-integer linear program that can be solved using optimization software. In most cases, such large constants are tuned by trial and error. We show, through a counterexample, that this widely used trial-and-error approach may lead to highly suboptimal solutions. Then, further research is required to properly select big-M values to solve linear bilevel problems.