New finite relaxation hierarchies for concavo-convex, disjoint bilinear programs, and facial disjunctions

TL;DR

This paper develops new finite relaxation hierarchies for concavo-convex programs, including disjoint bilinear programs, using geometric and algebraic techniques to achieve tight convex relaxations in a finite number of steps.

Contribution

It introduces a unified framework combining geometric barycentric coordinates with algebraic rational functions to tighten relaxations for CXP and FDP problems.

Findings

Achieves convex hull in m iterations for m inequalities.

Extends hierarchies to facial disjunctive programs.

Provides a unified approach for disjunctive programming relaxations.

Abstract

This paper introduces novel relaxation hierarchies for concavo-convex programs (CXP), a class of problems that includes disjoint bilinear programming (DBP) and concave minimization (CM) as special cases. At the core of these hierarchies is an algorithm based on double-description (DD) that computes the barycentric coordinates of a polyhedral cone as rational, non-negative functions representing multipliers associated with the cone's rays. These hierarchies combine geometric structure derived from barycentric coordinates with algebraic techniques via rational functions, achieving the convex hull in $m$ iterations, where $m$ is the number of inequalities that a subset of the variables must satisfy. Our framework offers the first unified approach to analyze and tighten relaxations from disjunctive programming (DP) and reformulation-linearization technique (RLT) for CXP. We also demonstrate that our methods extend to facial disjunctive programs (FDP), where solutions are constrained to lie on faces of a Cartesian product of polytopes, generalizing known hierarchies for 0-1 programs.