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On decomposability of Multilinear sets

Abstract

We consider the Multilinear set \({\mathcal {S}}\) defined as the set of binary points (xy) satisfying a collection of multilinear equations of the form \(y_I = \prod _{i \in I} x_i\), \(I \in {\mathcal {I}}\), where \({\mathcal {I}}\) denotes a family of subsets of \(\{1,\ldots , n\}\) of cardinality at least two. Such sets appear in factorable reformulations of many types of nonconvex optimization problems, including binary polynomial optimization. A great simplification in studying the facial structure of the convex hull of the Multilinear set is possible when \({\mathcal {S}}\) is decomposable into simpler Multilinear sets \({\mathcal {S}}_j\), \(j \in J\); namely, the convex hull of \({\mathcal {S}}\) can be obtained by convexifying each \({\mathcal {S}}_j\), separately. In this paper, we study the decomposability properties of Multilinear sets. Utilizing an equivalent hypergraph representation for Multilinear sets, we derive necessary and sufficient conditions under which \({\mathcal {S}}\) is decomposable into \({\mathcal {S}}_j\), \(j \in J\), based on the structure of pair-wise intersection hypergraphs. Our characterizations unify and extend the existing decomposability results for the Boolean quadric polytope. Finally, we propose a polynomial-time algorithm to optimally decompose a Multilinear set into simpler subsets. Our proposed algorithm can be easily incorporated in branch-and-cut based global solvers as a preprocessing step for cut generation.

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Acknowledgements

The authors would like to thank two anonymous referees for comments and suggestions that improved the quality of this manuscript.

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Correspondence to Alberto Del Pia.

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This research was supported in part by National Science Foundation award CMMI-1634768.

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Del Pia, A., Khajavirad, A. On decomposability of Multilinear sets. Math. Program. 170, 387–415 (2018). https://doi.org/10.1007/s10107-017-1158-z

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  • DOI: https://doi.org/10.1007/s10107-017-1158-z

Keywords

  • Multilinear functions
  • Convex hull
  • Decomposition
  • Zero–one polynomial optimization
  • Factorable relaxations
  • Polynomial-time algorithm

Mathematics Subject Classification

  • 90C11
  • 90C26
  • 90C57
  • 05C65