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It is well-known that the convex and concave envelope of a multilinearpolynomial over a box are polyhedral functions. Exponential-sized extended andprojected formulations for these envelopes are also known. We consider theconvexification question for multilinear polynomials that are symmetric withrespect to permutations of variables. Such a permutation-invariant structurenaturally implies a quadratic-sized extended formulation for the envelopesthrough the use of disjunctive programming. The optimization and separationproblems are answered directly without using this extension. The problemsymmetry allows the optimization and separation problems to be answereddirectly without using any extension. It also implies that permuting thecoefficients of a core set of facets generates all the facets. We provide somenecessary conditions and some sufficient conditions for a valid inequality tobe a core facet. These conditions are applied to obtain envelopes for twoclasses: symmetric supermodular functions and multilinear monomials withreflection symmetry, thereby yielding alternate proofs to the literature.Furthermore, we use constructs from the reformulation-linearization-techniqueto completely characterize the set of points lying on each facet.

Polyhedral Analysis of Symmetric Multilinear Polynomials over Box Constraints