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On the impact of running intersection inequalities for globally solving polynomial optimization problems

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Abstract

We consider global optimization of nonconvex problems whose factorable reformulations contain a collection of multilinear equations of the form \(z_e = \prod _{v \in e} {z_v}\), \(e \in E\), where E denotes a set of subsets of cardinality at least two of a ground set. Important special cases include multilinear and polynomial optimization problems. The multilinear polytope is the convex hull of the set of binary points z satisfying the system of multilinear equations given above. Recently Del Pia and Khajavirad introduced running intersection inequalities, a family of facet-defining inequalities for the multilinear polytope. In this paper we address the separation problem for this class of inequalities. We first prove that separating flower inequalities, a subclass of running intersection inequalities, is NP-hard. Subsequently, for multilinear polytopes of fixed degree, we devise an efficient polynomial-time algorithm for separating running intersection inequalities and embed the proposed cutting-plane generation scheme at every node of the branch-and-reduce global solver BARON. To evaluate the effectiveness of the proposed method we consider two test sets: randomly generated multilinear and polynomial optimization problems of degree three and four, and computer vision instances from an image restoration problem Results show that running intersection cuts significantly improve the performance of BARON and lead to an average CPU time reduction of 50% for the random test set and of 63% for the image restoration test set.

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Authors and Affiliations

  1. Department of Industrial and Systems Engineering and Wisconsin Institute for Discovery, University of Wisconsin-Madison, Madison, USA

    Alberto Del Pia

  2. Department of Management Science and Information Systems, Rutgers School of Business, Rutgers-State University of New Jersey, New Brunswick, USA

    Aida Khajavirad

  3. Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, USA

    Nikolaos V. Sahinidis

Authors
  1. Alberto Del Pia
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  2. Aida Khajavirad
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  3. Nikolaos V. Sahinidis
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Correspondence to Aida Khajavirad.

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Alberto Del Pia and Aida Khajavirad were partially supported by National Science Foundation award CMMI-1634768.

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Del Pia, A., Khajavirad, A. & Sahinidis, N.V. On the impact of running intersection inequalities for globally solving polynomial optimization problems. Math. Prog. Comp. 12, 165–191 (2020). https://doi.org/10.1007/s12532-019-00169-z

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