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.
JEL Classification: C61
Mathematics Subject Classification (2010): 90C46, 13J30
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Maxima is a free computer algebra system.
References
Blekherman G, Parrilo P, Thomas R (2013) Semidefinite optimization and convex algebraic geometry. MOS-SIAM series on optimization, vol 13. SIAM, Philadelphia
Lasserre JB (2001) Global optimization with polynomials and the problem of moments. SIAM J Optim 11:796–817
Lasserre JB (2010) Moments, positive polynomials and their applications. Imperial College Press, London
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 157–270
Marshall M (2008) Positive polynomials and sums of squares. Mathematical surveys and monographs, vol 146. American Mathematical Society, Providence
Nie J (2014) Optimality conditions and finite convergence of Lasserre’s hierarchy. Math Program 146:97–121
Nie J (2013) Polynomial optimization with real varieties. SIAM J Optim 23:1634–1646
Putinar M (1993) Positive polynomials on compact semi-algebraic sets. Indiana Univ Math J 42:969–984
Schweighofer M (2005) Optimization of polynomials on compact semialgebraic sets. SIAM J Optim 15:805–825
Sekiguchi Y, Takenawa T, Waki H (2013) Real ideal and the duality of semidefinite programming for polynomial optimization. Jpn J Ind Appl Math 30:321–330
Todd MJ (2001) Semidefinite optimization. Acta Numer 10:515–560
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer Science+Business Media Singapore
About this chapter
Cite this chapter
Sekiguchi, Y. (2016). Real Radicals and Finite Convergence of Polynomial Optimization Problems. In: Kusuoka, S., Maruyama, T. (eds) Advances in Mathematical Economics Volume 20. Advances in Mathematical Economics, vol 20. Springer, Singapore. https://doi.org/10.1007/978-981-10-0476-6_4
Download citation
DOI: https://doi.org/10.1007/978-981-10-0476-6_4
Received:
Revised:
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-10-0475-9
Online ISBN: 978-981-10-0476-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)