Optimality conditions and finite convergence of Lasserre’s hierarchy

• Published in: Mathematical programming Vol. 146; no. 1-2; pp. 97 - 121
• Main Author: Nie, Jiawang
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.08.2014; Springer Nature B.V
• Subjects: Analysis; Boundaries; Calculus of Variations and Optimal Control; Optimization; Combinatorics; Convergence; Full Length Paper; Hierarchies; Mathematical analysis; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematical programming; Mathematics; Mathematics and Statistics; Mathematics of Computing; Nonlinear programming; Numerical Analysis; Optimization; Polynomials; Semidefinite programming; Studies; Theoretical; Optimality conditions; Semidefinite program; 65K05; Polynomial optimization; Sum of squares; 90C22; 90C26; Lasserre’s hierarchy
• ISSN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-013-0680-x

Lasserre’s hierarchy is a sequence of semidefinite relaxations for solving polynomial optimization problems globally. This paper studies the relationship between optimality conditions in nonlinear programming theory and finite convergence of Lasserre’s hierarchy. Our main results are: (i) Lasserre’s hierarchy has finite convergence when the constraint qualification, strict complementarity and second order sufficiency conditions hold at every global minimizer, under the standard archimedean condition; the proof uses a result of Marshall on boundary hessian conditions. (ii) These optimality conditions are all satisfied at every local minimizer if a finite set of polynomials, which are in the coefficients of input polynomials, do not vanish at the input data (i.e., they hold in a Zariski open set). This implies that, under archimedeanness, Lasserre’s hierarchy has finite convergence generically.