Recursive McCormick Linearization of Multilinear Programs

TL;DR

This paper introduces a systematic method for identifying optimal Recursive McCormick Linearizations in multilinear programs, leading to stronger relaxations and improved bounds compared to existing solvers.

Contribution

It presents a novel, exact MIP formulation for finding minimal RMLs, analyzes its complexity, and demonstrates superior performance on benchmark problems.

Findings

The proposed algorithms outperform existing RML strategies in solvers.

The problem of identifying minimal RMLs is NP-hard, with some fixed-parameter tractable cases.

Structural properties enable the design of optimal RMLs with best possible relaxations.

Abstract

Linear programming (LP) relaxations are widely employed in exact solution methods for multilinear programs (MLP). One example is the family of Recursive McCormick Linearization (RML) strategies, where bilinear products are substituted for artificial variables, which deliver a relaxation of the original problem when introduced together with concave and convex envelopes. In this article, we introduce the first systematic approach for identifying RMLs, in which we focus on the identification of linear relaxation with a small number of artificial variables and with strong LP bounds. We present a novel mechanism for representing all the possible RMLs, which we use to design an exact mixed-integer programming (MIP) formulation for the identification of minimum-size RMLs; we show that this problem is NP-hard in general, whereas a special case is fixed-parameter tractable. Moreover, we explore structural properties of our formulation to derive an exact MIP model that identifies RMLs of a given size with the best possible relaxation bound is optimal. Our numerical results on a collection of benchmarks indicate that our algorithms outperform the RML strategy implemented in state-of-the-art global optimization solvers.