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Foundations of Computational Mathematics

, Volume 19, Issue 6, pp 1223–1263 | Cite as

On the Exactness of Lasserre Relaxations and Pure States Over Real Closed Fields

  • Tom-Lukas Kriel
  • Markus SchweighoferEmail author
Article

Abstract

Consider a finite system of non-strict polynomial inequalities with solution set \(S\subseteq \mathbb R^n\). Its Lasserre relaxation of degree d is a certain natural linear matrix inequality in the original variables and one additional variable for each nonlinear monomial of degree at most d. It defines a spectrahedron that projects down to a convex semialgebraic set containing S. In the best case, the projection equals the convex hull of S. We show that this is very often the case for sufficiently high d if S is compact and “bulges outwards” on the boundary of its convex hull. Now let additionally a polynomial objective function f be given, i.e., consider a polynomial optimization problem. Its Lasserre relaxation of degree d is now a semidefinite program. In the best case, the optimal values of the polynomial optimization problem and its relaxation agree. We prove that this often happens if S is compact and d exceeds some bound that depends on the description of S and certain characteristics of f like the mutual distance of its global minimizers on S.

Keywords

Moment relaxation Lasserre relaxation Pure state Basic closed semialgebraic set Positive polynomial Sum of squares Polynomial optimization Semidefinite programming Linear matrix inequality Spectrahedron Semidefinitely representable set 

Mathematics Subject Classification

Primary: 13J30 14P10 46L30 52A20 Secondary: 12D15 52A20 52A41 90C22 90C26 

Notes

Acknowledgements

Both authors were supported by the DFG Grant SCHW 1723/1-1.

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Copyright information

© SFoCM 2018

Authors and Affiliations

  1. 1.Fachbereich Mathematik und StatistikUniversität KonstanzKonstanzGermany

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