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A bounded degree SOS hierarchy for polynomial optimization

Original Paper

Abstract

We consider a new hierarchy of semidefinite relaxations for the general polynomial optimization problem \((P):\,f^{*}=\min \{f(x):x\in K\}\) on a compact basic semi-algebraic set \(K\subset \mathbb {R}^n\). This hierarchy combines some advantages of the standard LP-relaxations associated with Krivine’s positivity certificate and some advantages of the standard SOS-hierarchy. In particular it has the following attractive features: (a) in contrast to the standard SOS-hierarchy, for each relaxation in the hierarchy, the size of the matrix associated with the semidefinite constraint is the same and fixed in advance by the user; (b) in contrast to the LP-hierarchy, finite convergence occurs at the first step of the hierarchy for an important class of convex problems; and (c) some important techniques related to the use of point evaluations for declaring a polynomial to be zero and to the use of rank-one matrices make an efficient implementation possible. Preliminary results on a sample of non convex problems are encouraging.

Keywords

Global optimization Polynomial optimization convex relaxations LP and semidefinite hierarchies 

Mathematics Subject Classification

90C26 90C22 

Notes

Acknowledgments

The work of the first author is partially supported by a PGMO grant from Fondation Mathématique Jacques Hadamard, and an ERC-ADG grant from the European Research Council (ERC): grant agreement 666981 TAMING.

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Copyright information

© EURO - The Association of European Operational Research Societies 2015

Authors and Affiliations

  • Jean B. Lasserre
    • 1
  • Kim-Chuan Toh
    • 2
  • Shouguang Yang
    • 2
  1. 1.LAAS-CNRS and Institute of MathematicsUniversity of ToulouseToulouseFrance
  2. 2.Department of MathematicsNational University of SingaporeSingaporeSingapore

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