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A new approximation hierarchy for polynomial conic optimization

  • Peter J. C. Dickinson
  • Janez PovhEmail author
Article

Abstract

In this paper we consider polynomial conic optimization problems, where the feasible set is defined by constraints in the form of given polynomial vectors belonging to given nonempty closed convex cones, and we assume that all the feasible solutions are non-negative. This family of problems captures in particular polynomial optimization problems (POPs), polynomial semi-definite polynomial optimization problems (PSDPs) and polynomial second-order cone-optimization problems (PSOCPs). We propose a new general hierarchy of linear conic optimization relaxations inspired by an extension of Pólya’s Positivstellensatz for homogeneous polynomials being positive over a basic semi-algebraic cone contained in the non-negative orthant, introduced in Dickinson and Povh (J Glob Optim 61(4):615–625, 2015). We prove that based on some classic assumptions, these relaxations converge monotonically to the optimal value of the original problem. Adding a redundant polynomial positive semi-definite constraint to the original problem drastically improves the bounds produced by our method. We provide an extensive list of numerical examples that clearly indicate the advantages and disadvantages of our hierarchy. In particular, in comparison to the classic approach of sum-of-squares, our new method provides reasonable bounds on the optimal value for POPs, and strong bounds for PSDPs and PSOCPs, even outperforming the sum-of-squares approach in these latter two cases.

Keywords

Polynomial conic optimization Polynomial semi-definite programming Polynomial second-order cone programming Approximation hierarchy Linear programming Semi-definite programming 

Mathematics Subject Classification

11E25 13P25 14P10 90C05 90C22 90C30 

Notes

Acknowledgements

The research for this paper was started while P.J.C. Dickinson was at the Johann Bernoulli Institute for Mathematics and Computer Science, University of Groningen, The Netherlands. It was then continued while this author was at the Department of Statistics and Operations Research, University of Vienna, Austria, and then again after he joined the Department of Applied Mathematics, University of Twente, The Netherlands. This author would like to gratefully acknowledge support from The Netherlands Organisation for Scientific Research (NWO) through Grant No. 613.009.021. The second author started this research when he was affiliated to the Faculty of Information Studies in Novo Mesto, Slovenia, and continued the work after moving to the University of Ljubljana, Slovenia. He wishes to thank the Slovenian Research Agency for support via the program P1-0383 and projects J1-8132, N1-0057, N1-0071. Both authors would also like to the thank the anonymous referees for their useful comments with regards to this paper.

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Authors and Affiliations

  1. 1.University of GroningenGroningenThe Netherlands
  2. 2.University of ViennaViennaAustria
  3. 3.Department of Applied MathematicsUniversity of TwenteEnschedeThe Netherlands
  4. 4.Faculty of Mechanical EngineeringUniversity of LjubljanaLjubljanaSlovenia

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