Journal of Optimization Theory and Applications

, Volume 163, Issue 3, pp 707–718 | Cite as

On Polynomial Optimization Over Non-compact Semi-algebraic Sets

  • V. JeyakumarEmail author
  • J. B. Lasserre
  • G. Li


The optimal value of a polynomial optimization over a compact semi-algebraic set can be approximated as closely as desired by solving a hierarchy of semidefinite programs and the convergence is finite generically under the mild assumption that a quadratic module generated by the constraints is Archimedean. We consider a class of polynomial optimization problems with non-compact semi-algebraic feasible sets, for which an associated quadratic module, that is generated in terms of both the objective function and the constraints, is Archimedean. For such problems, we show that the corresponding hierarchy converges and the convergence is finite generically. Moreover, we prove that the Archimedean condition (as well as a sufficient coercivity condition) can be checked numerically by solving a similar hierarchy of semidefinite programs. In other words, under reasonable assumptions, the now standard hierarchy of semidefinite programming relaxations extends to the non-compact case via a suitable modification.


Polynomial optimization Non-compact semi-algebraic sets  Semidefinite programming relaxations Positivstellensatzë 

Mathematics Subject Classification (2000)

90C30 90C26 14P10 90C46 


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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of Applied MathematicsUniversity of New South WalesSydney Australia
  2. 2.LAAS-CNRS and Institute of MathematicsToulouseFrance

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