Polyhedral Analysis of Symmetric Multilinear Polynomials over Box Constraints

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.


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