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Certifying convergence of Lasserre’s hierarchy via flat truncation

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Abstract

Consider the optimization problem of minimizing a polynomial function subject to polynomial constraints. A typical approach for solving it globally is applying Lasserre’s hierarchy of semidefinite relaxations, based on either Putinar’s or Schmüdgen’s Positivstellensatz. A practical question in applications is: how to certify its convergence and get minimizers? In this paper, we propose flat truncation as a certificate for this purpose. Assume the set of global minimizers is nonempty and finite. Our main results are: (1) Putinar type Lasserre’s hierarchy has finite convergence if and only if flat truncation holds, under some generic assumptions; the same conclusion holds for the Schmüdgen type one under weaker assumptions. (2) Flat truncation is asymptotically satisfied for Putinar type Lasserre’s hierarchy if the archimedean condition holds; the same conclusion holds for the Schmüdgen type one if the feasible set is compact. (3) We show that flat truncation can be used as a certificate to check exactness of standard SOS relaxations and Jacobian SDP relaxations.

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Notes

  1. If \(f_k = f_{min}\) and (1.1) has a minimizer, say \(x^*\), then (1.4) always has an optimizer that is flat, e.g., \([x^*]_{2k}\). Thus, it makes more sense to ask whether every optimizer has a flat truncatioin.

  2. Its entries are listed in the graded lexicographical ordering.

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Acknowledgments

The author was partially supported by NSF grants DMS-0757212 and DMS-0844775, and he would like very much to thank the referees for fruitful suggestions on improving the paper.

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  1. Department of Mathematics, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA, 92093, USA

    Jiawang Nie

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  1. Jiawang Nie
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Nie, J. Certifying convergence of Lasserre’s hierarchy via flat truncation. Math. Program. 142, 485–510 (2013). https://doi.org/10.1007/s10107-012-0589-9

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  • DOI: https://doi.org/10.1007/s10107-012-0589-9

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