Generating Cutting Inequalities Successively for Quadratic Optimization Problems in Binary Variables

TL;DR

This paper introduces a method for successively generating cutting inequalities in binary quadratic optimization, improving solution accuracy by iteratively refining the convex hull of optimal solutions using conic relaxations.

Contribution

It presents a novel successive cutting inequalities approach that tests multiple candidates in parallel, enhancing the ability to find exact solutions in binary quadratic problems.

Findings

Achieved exact optimal solutions for 70% of tested problems.

Demonstrated the effectiveness of the method on 60 quadratic binary problems.

Utilized parallel processing to validate multiple inequalities simultaneously.

Abstract

We propose a successive generation of cutting inequalities for binary quadratic optimization problems. Multiple cutting inequalities are successively generated for the convex hull of the set of the optimal solutions $\subset \{0, 1\}^n$, while the standard cutting inequalities are used for the convex hull of the feasible region. An arbitrary linear inequality with integer coefficients and the right-hand side value in integer is considered as a candidate for a valid inequality. The validity of the linear inequality is determined by solving a conic relaxation of a subproblem such as the doubly nonnegative relaxation, under the assumption that an upper bound for the unknown optimal value of the problem is available. Moreover, the candidates generated for the multiple cutting inequalities are tested simultaneously for their validity in parallel. Preliminary numerical results on 60 quadratic unconstrained binary optimization problems with a simple implementation of the successive cutting inequalities using an 8- or 32-core machine show that the exact optimal values are obtained for 70\% of the tested problems, demonstrating the strong potential of the proposed technique.