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Univariate Polynomial Optimization with Sum-of-Squares Interpolants

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Modeling and Optimization: Theory and Applications (MOPTA 2016)

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Abstract

One of the most common tools in polynomial optimization is the approximation of the cone of nonnegative polynomials with the cone of sum-of-squares polynomials. This leads to polynomial-time solvable approximations for many NP-hard optimization problems using semidefinite programming (SDP). While theoretically satisfactory, the translation of optimization problems involving sum-of-squares polynomials to SDPs is not always practical. First, in the common SDP formulation, the dual variables are semidefinite matrices whose condition numbers grow exponentially with the degree of the polynomials involved, which is detrimental for a floating-point implementation. Second, the SDP representation of sum-of-squares polynomials roughly squares the number of optimization variables, increasing the time and memory complexity of the solution algorithms by several orders of magnitude. In this paper we focus on the first, numerical, issue. We show that a reformulation of the sum-of-squares SDP using polynomial interpolants yields a substantial improvement over the standard formulation, and problems involving sum-of-squares interpolants of hundreds of degrees can be handled without difficulty by commonly used semidefinite programming solvers. Preliminary numerical results using semi-infinite optimization problems align with the theoretical predictions. In all problems considered, available memory is the only factor limiting the degrees of polynomials.

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Acknowledgements

The author is grateful to Sercan Yıldız for many constructive comments on the presentation of the material.

This material was based upon work partially supported by the National Science Foundation under Grant DMS-1127914 to the Statistical and Applied Mathematical Sciences Institute. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

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Authors and Affiliations

  1. North Carolina State University, Raleigh, NC, 27695, USA

    Dávid Papp

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  1. Dávid Papp
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Correspondence to Dávid Papp .

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Editors and Affiliations

  1. Industrial and Systems Engineering Department, Lehigh University, Bethlehem, Pennsylvania, USA

    Martin Takáč

  2. Industrial and Systems Engineering Department, Lehigh University, Bethlehem, Pennsylvania, USA

    Tamás Terlaky

Appendix

Appendix

Below are the tabulated numerical results from Sect. 4 that are too large to conveniently fit in the text.

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Papp, D. (2017). Univariate Polynomial Optimization with Sum-of-Squares Interpolants. In: Takáč, M., Terlaky, T. (eds) Modeling and Optimization: Theory and Applications. MOPTA 2016. Springer Proceedings in Mathematics & Statistics, vol 213. Springer, Cham. https://doi.org/10.1007/978-3-319-66616-7_9

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