Mixed-integer bilevel representability

TL;DR

This paper explores the representability of sets via mixed-integer bilevel programs, showing equivalences with unions of polyhedra and analyzing the modeling power of integer variables in different levels.

Contribution

It characterizes the sets representable by mixed-integer bilevel programs and demonstrates the equivalence with unions of polyhedra, clarifying the modeling capabilities of integer variables.

Findings

Feasible regions with continuous bilevel constraints are finite unions of polyhedra.

Finite unions of polyhedra can be represented by these bilevel paradigms.

Integer constraints only in the upper level suffice for representing certain sets.

Abstract

We study the representability of sets that admit extended formulations using mixed-integer bilevel programs. We show that feasible regions modeled by continuous bilevel constraints (with no integer variables), complementarity constraints, and polyhedral reverse convex constraints are all finite unions of polyhedra. Conversely, any finite union of polyhedra can be represented using any one of these three paradigms. We then prove that the feasible region of bilevel problems with integer constraints exclusively in the upper level is a finite union of sets representable by mixed-integer programs and vice versa. Further, we prove that, up to topological closures, we do not get additional modeling power by allowing integer variables in the lower level as well. To establish the last statement, we prove that the family of sets that are finite unions of mixed-integer representable sets forms an algebra of sets (up to topological closures).