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A sublevel moment-SOS hierarchy for polynomial optimization

Author

Listed:
  • Tong Chen

    (LAAS-CNRS)

  • Jean-Bernard Lasserre

    (LAAS-CNRS
    Université Toulouse 3 Paul Sabatier)

  • Victor Magron

    (LAAS-CNRS
    Université Toulouse 3 Paul Sabatier)

  • Edouard Pauwels

    (Université Toulouse 3 Paul Sabatier
    Université de Toulouse)

Abstract

We introduce a sublevel Moment-SOS hierarchy where each SDP relaxation can be viewed as an intermediate (or interpolation) between the d-th and $$(d+1)$$ ( d + 1 ) -th order SDP relaxations of the Moment-SOS hierarchy (dense or sparse version). With the flexible choice of determining the size (level) and number (depth) of subsets in the SDP relaxation, one is able to obtain different improvements compared to the d-th order relaxation, based on the machine memory capacity. In particular, we provide numerical experiments for $$d=1$$ d = 1 and various types of problems both in combinatorial optimization (Max-Cut, Mixed Integer Programming) and deep learning (robustness certification, Lipschitz constant of neural networks), where the standard Lasserre’s relaxation (or its sparse variant) is computationally intractable. In our numerical results, the lower bounds from the sublevel relaxations improve the bound from Shor’s relaxation (first order Lasserre’s relaxation) and are significantly closer to the optimal value or to the best-known lower/upper bounds.

Suggested Citation

  • Tong Chen & Jean-Bernard Lasserre & Victor Magron & Edouard Pauwels, 2022. "A sublevel moment-SOS hierarchy for polynomial optimization," Computational Optimization and Applications, Springer, vol. 81(1), pages 31-66, January.
    • Handle: RePEc:spr:coopap:v:81:y:2022:i:1:d:10.1007_s10589-021-00325-z
    DOI: 10.1007/s10589-021-00325-z
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    References listed on IDEAS

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    1. Jean B. Lasserre & Kim-Chuan Toh & Shouguang Yang, 2017. "A bounded degree SOS hierarchy for polynomial optimization," EURO Journal on Computational Optimization, Springer;EURO - The Association of European Operational Research Societies, vol. 5(1), pages 87-117, March.
    2. Monique Laurent, 2003. "A Comparison of the Sherali-Adams, Lovász-Schrijver, and Lasserre Relaxations for 0--1 Programming," Mathematics of Operations Research, INFORMS, vol. 28(3), pages 470-496, August.
    3. Jie Wang & Victor Magron, 2021. "Exploiting term sparsity in noncommutative polynomial optimization," Computational Optimization and Applications, Springer, vol. 80(2), pages 483-521, November.
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