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Doubly nonnegative relaxations for quadratic and polynomial optimization problems with binary and box constraints

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Abstract

We propose a doubly nonnegative (DNN) relaxation for polynomial optimization problems (POPs) with binary and box constraints. This work is an extension of the work by Kim, Kojima and Toh in 2016 from quadratic optimization problems to POPs. The dense and sparse DNN relaxations are reduced to a simple conic optimization problem (COP) to which an accelerated bisection and projection (BP) algorithm is applied. The COP involves a single equality constraint in a matrix variable which is restricted to the intersection of the positive semidefinite cone and a polyhedral cone representing the algebraic properties of binary and box constraints as well as the sparsity structure of the objective polynomial. Our DNN relaxation serves as a variant of the Lasserre moment-SOS hierarchy of relaxations for binary and box constrained POPs when the size of the cones is gradually increased to compute tighter lower bounds for their optimal values. A significant part of the paper is devoted to deriving and analyzing a class of polyhedral cones to strengthen the DNN relaxation, and the design of efficient computation of the metric projections onto these cones in the accelerated BP algorithm. Numerical results on various POPs are available in Ito et al. (ACM Trans Math Softw 45, Article 34, 2019).

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Notes

  1. Linear inequality constraints are converted to linear ones by adding slacks.

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Acknowledgements

The authors would like to thank Prof. Satoru Iwata for the information on a closed relation of the class of polyhedral cone discussed in Sect. 3 to the isotonic regression problem, and Dr. Naoki Ito for improving the Matlab codes for Algorithm 2.1.

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Authors and Affiliations

  1. Department of Mathematics, Ewha W. University, 52 Ewhayeodae-gil, Seodaemun-gu, Seoul, 120-750, Korea

    Sunyoung Kim

  2. Department of Industrial and Systems Engineering, Chuo University, Tokyo, 192-0393, Japan

    Masakazu Kojima

  3. Department of Mathematics, and Institute of Operations Research and Analytics, National University of Singapore, 10 Lower Kent Ridge Road, Singapore, 119076, Singapore

    Kim-Chuan Toh

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  1. Sunyoung Kim
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  2. Masakazu Kojima
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  3. Kim-Chuan Toh
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Correspondence to Kim-Chuan Toh.

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S.K.: The research was supported by National Research Foundation of Korea (Grant Nos. 2017-R1A2B2005119, 2014-R1A2A1A11049618). M.K.: This research was supported by Grant-in-Aid for Scientific Research (A) 26242027 and the Japan Science and Technology Agency (JST), the Core Research of Evolutionary Science and Technology (CREST) research project. K.-C.T.: This research was supported by Ministry of Education - Singapore (Grant No. R-146-000-257-112).

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Kim, S., Kojima, M. & Toh, KC. Doubly nonnegative relaxations for quadratic and polynomial optimization problems with binary and box constraints. Math. Program. 193, 761–787 (2022). https://doi.org/10.1007/s10107-020-01594-y

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  • DOI: https://doi.org/10.1007/s10107-020-01594-y

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