Journal of Global Optimization

, Volume 61, Issue 4, pp 615–625 | Cite as

On an extension of Pólya’s Positivstellensatz

Article

Abstract

In this paper we provide a generalization of a Positivstellensatz by Pólya [Pólya in Naturforsch Ges Zürich 73:141–145 1928]. We show that if a homogeneous polynomial is positive over the intersection of the non-negative orthant and a given basic semialgebraic cone (excluding the origin), then there exists a “Pólya type” certificate for non-negativity. The proof of this result uses the original Positivstellensatz by Pólya, and a Positivstellensatz by Putinar and Vasilescu [Putinar and Vasilescu C R Acad Sci Ser I Math 328(7) 1999].

Keywords

Positivstellensatz Semialgebraic set Non-negativity certificate  Polynomial optimization 

Mathematics Subject Classification

11E25 14P05 14P10 90C30 

Notes

Acknowledgments

This research was started whilst the first author was at the Johann Bernoulli Institute for Mathematics and Computer Science, University of Groningen, The Netherlands. It was then continued whilst the author was at the Department of Statistics and Operations Research, University of Vienna, Austria. The authors would like to thank both of these institutes for the support that they provided. The second author wishes to thank to Slovenian research agency for support via program P1-0383 and project L74119 and to Creative Core FISNM-3330-13-500033 ‘Simulations’ project funded by the European Union. The authors also wish to thank the referees for their thorough reading and very constructive comments which improved the paper substantially.

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of Applied MathematicsUniversity of Twente EnschedeThe Netherlands
  2. 2.Faculty of Information Studies in Novo MestoNovo MestoSlovenia

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