Mixed-integer bilevel representability

We study the representability of sets by 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 po...

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Veröffentlicht in:Mathematical programming 2021-01, Vol.185 (1-2), p.163-197
Hauptverfasser: Basu, Amitabh, Ryan, Christopher Thomas, Sankaranarayanan, Sriram
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Ryan, Christopher Thomas
Sankaranarayanan, Sriram
description We study the representability of sets by 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 variables 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 (up to topological closures) forms an algebra of sets; i.e., this family is closed under finite unions, intersections and complementation.
doi_str_mv 10.1007/s10107-019-01424-w
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subjects Applied mathematics
Calculus of Variations and Optimal Control
Optimization
Closures
Combinatorics
Constraint modelling
Formulations
Full Length Paper
Integer programming
Integers
Intersections
Linear programming
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematical programming
Mathematics
Mathematics and Statistics
Mathematics of Computing
Mixed integer
Numerical Analysis
Optimization
Polyhedra
Theoretical
Topology
Unions
title Mixed-integer bilevel representability
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