Extended formulations for convex hulls of some bilinear functions

Abstract

We consider the problem of characterizing the convex hull of the graph of a bilinear function f on the n-dimensional unit cube [0, 1] n . Extended formulations for this convex hull are obtained by taking subsets of the facets of the Boolean Quadric Polytope (BQP). Extending existing results, we propose a systematic study of properties of f that guarantee that certain classes of BQP facets are sufficient for an extended formulation. We use a modification of Zuckerberg's geometric method for proving convex hull characterizations [Geometric proofs for convex hull defining formulations, Operations Research Letters 44 (2016), 625-629] to prove some initial results in this direction. In particular, we provide small-sized extended formulations for bilinear functions whose corresponding graph is either a cycle with arbitrary edge weights or a clique or an almost clique with unit edge weights.

Set characterizations and convex extensions for geometric convex-hull proofs

“…On the one hand, these extensions allow to prove convex-hull descriptions for graphs of Boolean functions over 0/1-polytopes. On the other hand, they can be applied to bilinear functions over arbitrary polytopes, generalizing the result from [11] for bilinear functions over unit-boxes. In summary, we show that Zuckerberg's method is a valuable tool for conducting convex-hull proofs.…”

“…In this section, we revisit Zuckerberg's method for convex-hull proofs for combinatorial decision or optimization problems (see [1,22]). We start by briefly summarizing it, based on the condensed version of the method that was derived in [11]. Then we introduce the concept of set characterizations to significantly simplify the derivation of the set construction algorithms which form the core of Zuckerberg-type convex-hull proofs.…”

“…Theorem 1 ([11,Theorem 4], Zuckerberg's convex-hull characterization) Let F ⊆ {0, 1} n and h ∈ [0, 1] n . Then we have h ∈ conv(F) iff there are sets S 1 , .…”

“…Zuckerberg has stated his method in terms of abstract measure spaces over which set-theoretic expressions have to be derived. In the recent work [11], the authors give a significantly simplified version of his approach by passing over to a concrete measure space: a real interval equipped with the Lebesgue measure. Their actual aim in this article are proofs on the facial structure of the graphs of bilinear functions.…”

“…Afterwards, they show that their simplification of Zuckerberg's method allows for elegant proofs of their statements. In [12], the authors continue the work of [11] and give further convexhull results on special graph classes. The authors of [6] have adopted their simplified approach of Zuckerberg's method in order to give convex-hull proofs for special cases of the Boolean quadric polytope (see [18]) with multiple-choice constraints.…”