Certifying convergence of Lasserre’s hierarchy via flat truncation
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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.
KeywordsFlat truncation Lasserre’s hierarchy Quadratic module Preordering Semidefinite program Sum of squares
Mathematics Subject Classification (2000)65K05 90C22
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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