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A sublevel moment-SOS hierarchy for polynomial optimization

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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 \((d+1)\)-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 \(d=1\) 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.

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  1. http://web.stanford.edu/~yyye/yyye/Gset/

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Acknowledgements

This work has benefited from the Tremplin ERC Stg Grant ANR-18-ERC2-0004-01 (T-COPS project), the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie Actions, grant agreement 813211 (POEMA) as well as from the AI Interdisciplinary Institute ANITI funding, through the French “Investing for the Future PIA3” program under the Grant agreement \(\hbox {n}^{\circ }\)ANR-19-PI3A-0004. The third author was supported by the FMJH Program PGMO (EPICS project) and EDF, Thales, Orange et Criteo. The fourth author acknowledge the support of Air Force Office of Scientific Research, Air Force Material Command, USAF, under grant numbers FA9550-19-1-7026, FA9550-18-1-0226, and ANR MasDol.

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Authors and Affiliations

  1. LAAS-CNRS, 7 avenue du Colonel Roche, BP 54200, 31031, Toulouse, Cédex 4, France

    Tong Chen, Jean-Bernard Lasserre & Victor Magron

  2. IMT, Université Toulouse 3 Paul Sabatier, Toulouse, France

    Jean-Bernard Lasserre, Victor Magron & Edouard Pauwels

  3. IRIT, CNRS, Université de Toulouse, Toulouse, France

    Edouard Pauwels

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  1. Tong Chen
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  2. Jean-Bernard Lasserre
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  3. Victor Magron
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  4. Edouard Pauwels
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Correspondence to Tong Chen.

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Chen, T., Lasserre, JB., Magron, V. et al. A sublevel moment-SOS hierarchy for polynomial optimization. Comput Optim Appl 81, 31–66 (2022). https://doi.org/10.1007/s10589-021-00325-z

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