Polyhedral Analysis of Symmetric Multilinear Polynomials over Box Constraints

TL;DR

This paper investigates the convexification of symmetric multilinear polynomials over box constraints, providing polyhedral descriptions, facet characterizations, and new proofs for specific classes, leveraging symmetry to simplify formulations.

Contribution

It introduces a symmetry-based approach to characterize convex envelopes of multilinear polynomials, including necessary and sufficient conditions for facets and alternative proofs for known classes.

Findings

Symmetry reduces the complexity of convexification.

Facet generation can be achieved through coefficient permutation.

Complete facet characterization for symmetric supermodular functions.

Abstract

It is well-known that the convex and concave envelope of a multilinear polynomial over a box are polyhedral functions. Exponential-sized extended and projected formulations for these envelopes are also known. We consider the convexification question for multilinear polynomials that are symmetric with respect to permutations of variables. Such a permutation-invariant structure naturally implies a quadratic-sized extended formulation for the envelopes through the use of disjunctive programming. The optimization and separation problems are answered directly without using this extension. The problem symmetry allows the optimization and separation problems to be answered directly without using any extension. It also implies that permuting the coefficients of a core set of facets generates all the facets. We provide some necessary conditions and some sufficient conditions for a valid inequality to be a core facet. These conditions are applied to obtain envelopes for two classes: symmetric supermodular functions and multilinear monomials with reflection symmetry, thereby yielding alternate proofs to the literature. Furthermore, we use constructs from the reformulation-linearization-technique to completely characterize the set of points lying on each facet.