Certifying convergence of Lasserre’s hierarchy via flat truncation

• Published in: Mathematical programming Vol. 142; no. 1-2; pp. 485 - 510
• Main Author: Nie, Jiawang
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.12.2013; Springer Nature B.V
• Subjects: Analysis; Asymptotic properties; Calculus of Variations and Optimal Control; Optimization; Combinatorics; Full Length Paper; Hierarchies; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematical programming; Mathematics; Mathematics and Statistics; Mathematics of Computing; Numerical Analysis; Optimization; Polynomials; Studies; Theoretical; Semidefinite program; 65K05; Sum of squares; 90C22; Preordering; Quadratic module; Flat truncation; Lasserre’s hierarchy
• ISSN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-012-0589-9

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.