Geometric and computational hardness of bilevel programming

TL;DR

This paper demonstrates the intrinsic computational and geometric complexity of bilevel programming, showing its equivalence to general minimization problems and its high complexity class, emphasizing the need for regularity assumptions.

Contribution

It establishes the equivalence of unconstrained smooth bilevel programming to general lower semicontinuous minimization and analyzes the complexity of polynomial bilevel problems, revealing their high computational hardness.

Findings

Unconstrained smooth bilevel programming is as hard as general lower semicontinuous minimization.

Any semi-algebraic extended-real-valued function can be represented as a polynomial bilevel program's value function.

Decision problems in polynomial bilevel programming are $ ext{Σ}_2^p$-hard, above NP complexity.

Abstract

We first show a simple but striking result in bilevel optimization: unconstrained $C^\infty$ smooth bilevel programming is as hard as general extended-real-valued lower semicontinuous minimization. We then proceed to a worst-case analysis of box-constrained bilevel polynomial optimization. We show in particular that any extended-real-valued semi-algebraic function, possibly non-continuous, can be expressed as the value function of a polynomial bilevel program. Secondly, from a computational complexity perspective, the decision version of polynomial bilevel programming is one level above NP in the polynomial hierarchy ($\Sigma^p_2$-hard). Both types of difficulties are uncommon in non-linear programs for which objective functions are typically continuous and belong to the class NP. These results highlight the irremediable hardness attached to general bilevel optimization and the necessity of imposing some form of regularity on the lower level.