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Real Radicals and Finite Convergence of Polynomial Optimization Problems

  • Yoshiyuki Sekiguchi
Research Article
Part of the Advances in Mathematical Economics book series (MATHECON, volume 20)

Abstract

Polynomial optimization appears various areas of mathematics. Although it is a fully nonlinear nonconvex optimization problems, there are numerical algorithms to approximate the global optimal value by generating sequences of semidefinite programming relaxations. In this paper, we study how real radicals of ideals have roles in duality theory and finite convergence property. Especially, duality theory is considered in the case that the truncated quadratic module is not necessarily closed. We will also try to explain the results by giving concrete examples.

Keywords

Polynomial optimization Real radicals Sums of squares Moment problems 

References

  1. 1.
    Blekherman G, Parrilo P, Thomas R (2013) Semidefinite optimization and convex algebraic geometry. MOS-SIAM series on optimization, vol 13. SIAM, PhiladelphiaGoogle Scholar
  2. 2.
    Lasserre JB (2001) Global optimization with polynomials and the problem of moments. SIAM J Optim 11:796–817MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Lasserre JB (2010) Moments, positive polynomials and their applications. Imperial College Press, LondonzbMATHGoogle Scholar
  4. 4.
    Laurent M (2009) Sums of squares, moments and polynomial optimization, emerging applications of algebraic geometry. IMA volumes in mathematics and its applications, vol 149. Springer, New York, pp 157–270Google Scholar
  5. 5.
    Marshall M (2008) Positive polynomials and sums of squares. Mathematical surveys and monographs, vol 146. American Mathematical Society, ProvidenceGoogle Scholar
  6. 6.
    Nie J (2014) Optimality conditions and finite convergence of Lasserre’s hierarchy. Math Program 146:97–121MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Nie J (2013) Polynomial optimization with real varieties. SIAM J Optim 23:1634–1646MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Putinar M (1993) Positive polynomials on compact semi-algebraic sets. Indiana Univ Math J 42:969–984MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Schweighofer M (2005) Optimization of polynomials on compact semialgebraic sets. SIAM J Optim 15:805–825MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Sekiguchi Y, Takenawa T, Waki H (2013) Real ideal and the duality of semidefinite programming for polynomial optimization. Jpn J Ind Appl Math 30:321–330MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Todd MJ (2001) Semidefinite optimization. Acta Numer 10:515–560MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Singapore 2016

Authors and Affiliations

  1. 1.Graduate School of Marine Science and TechnologyTokyo University of Marine Science and TechnologyTokyoJapan

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