Compact Linearization for Binary Quadratic Problems Comprising Linear Constraints

TL;DR

This paper extends a compact linearization technique to more general binary quadratic problems with various linear constraints, providing stronger relaxations and an automatic computation method for practical solver use.

Contribution

It generalizes the compact linearization approach to broader problem classes and introduces an automatic way to compute these linearizations for improved solver performance.

Findings

Linear programming relaxations are as strong as classical methods in certain cases.

The approach can be automated for use in general-purpose solvers.

Applicable to problems with linear and quadratic constraints, enhancing solution efficiency.

Abstract

In this paper, the compact linearization approach originally proposed for binary quadratic programs with assignment constraints is generalized to such programs with arbitrary linear equations and inequalities that have positive coefficients and right hand sides. Quadratic constraints may exist in addition, and the technique may as well be applied if these impose the only nonlinearities, i.e., the objective function is linear. We present special cases of linear constraints (along with prominent combinatorial optimization problems where these occur) such that the associated compact linearization yields a linear programming relaxation that is provably as least as strong as the one obtained with a classical linearization method. Moreover, we show how to compute a compact linearization automatically which might be used, e.g., by general-purpose mixed-integer programming solvers.