Facets of a mixed-integer bilinear covering set with bounds on variables

TL;DR

This paper characterizes the convex hull of a mixed-integer bilinear covering set with bounds, providing a compact extended formulation and an efficient separation algorithm, advancing optimization techniques for such sets.

Contribution

It offers a closed-form convex hull description, an extended formulation with fewer constraints, and a linear-time separation algorithm for the bilinear covering set.

Findings

Extended formulation reduces the number of constraints

Separation algorithm efficiently finds facet inequalities

New inequalities improve optimization performance

Abstract

We derive a closed form description of the convex hull of mixed-integer bilinear covering set with bounds on the integer variables. This convex hull description is determined by considering some orthogonal disjunctive sets defined in a certain way. This description does not introduce any new variables, but consists of exponentially many inequalities. An extended formulation with a few extra variables and much smaller number of constraints is presented. We also derive a linear time separation algorithm for finding the facet defining inequalities of this convex hull. We study the effectiveness of the new inequalities and the extended formulation using some examples.