Mathematical Programming

, Volume 146, Issue 1–2, pp 97–121 | Cite as

Optimality conditions and finite convergence of Lasserre’s hierarchy

  • Jiawang NieEmail author
Full Length Paper Series A


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.


Lasserre’s hierarchy Optimality conditions Polynomial optimization Semidefinite program Sum of squares 

Mathematics Subject Classification (2000)

65K05 90C22 90C26 



The author was partially supported by NSF grants DMS-0757212 and DMS-0844775. He would like very much to thank Murray Marshall for communications on the boundary hessian condition.


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© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2013

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California San DiegoLa JollaUSA

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