A class of valid inequalities for multilinear 0–1 optimization problems

Abstract

This paper investigates the polytope associated with the classical standard linearization technique for the unconstrained optimization of multilinear polynomials in 0–1 variables. A new class of valid inequalities, called 2-links, is introduced to strengthen the LP relaxation of the standard linearization. The addition of the 2-links to the standard linearization inequalities provides a complete description of the convex hull of integer solutions for the case of functions consisting of at most two nonlinear monomials. For the general case, various computational experiments show that the 2-links improve both the standard linearization bound and the computational performance of exact branch & cut methods. The improvements are especially significant for a class of instances inspired from the image restoration problem in computer vision. The magnitude of this effect is rather surprising in that the 2-links are in relatively small number (quadratic in the number of terms of the objective function).