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Mathematical Programming

, Volume 129, Issue 1, pp 5–31 | Cite as

Bilinear optimality constraints for the cone of positive polynomials

  • Gábor Rudolf
  • Nilay Noyan
  • Dávid Papp
  • Farid Alizadeh
Full Length Paper Series B

Abstract

For a proper cone \({{\mathcal K}\subset\mathbb{R}^n}\) and its dual cone \({{\mathcal K}^*}\) the complementary slackness condition \({\langle{\rm {\bf x}},{\rm {\bf s}}\rangle=0}\) defines an n-dimensional manifold \({C({\mathcal K})}\) in the space \({{\mathbb R}^{2n}}\) . When \({{\mathcal K}}\) is a symmetric cone, points in \({C({\mathcal K})}\) must satisfy at least n linearly independent bilinear identities. This fact proves to be useful when optimizing over such cones, therefore it is natural to look for similar bilinear relations for non-symmetric cones. In this paper we define the bilinearity rank of a cone, which is the number of linearly independent bilinear identities valid for points in \({C({\mathcal K})}\) . We examine several well-known cones, in particular the cone of positive polynomials \({{\mathcal P}_{2n+1}}\) and its dual, and show that there are exactly four linearly independent bilinear identities which hold for all \({({\rm {\bf x}},{\rm {\bf s}})\in C({\mathcal P}_{2n+1})}\), regardless of the dimension of the cones. For nonnegative polynomials over an interval or half-line there are only two linearly independent bilinear identities. These results are extended to trigonometric and exponential polynomials. We prove similar results for Müntz polynomials.

Keywords

Optimality conditions Positive polynomials Complementarity slackness Bilinearity rank Bilinear cones 

Mathematics Subject Classification (2000)

90C46 90C22 

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Copyright information

© Springer and Mathematical Optimization Society 2011

Authors and Affiliations

  • Gábor Rudolf
    • 1
  • Nilay Noyan
    • 2
  • Dávid Papp
    • 3
  • Farid Alizadeh
    • 4
  1. 1.Virginia Commonwealth UniversityRichmondUSA
  2. 2.Faculty of Engineering and Natural SciencesSabanci UniversityOrhanli, Tuzla, IstanbulTurkey
  3. 3.Department of Industrial Engineering and Management SciencesNorthwestern UniversityEvanstonUSA
  4. 4.RUTCOR and School of BusinessRutgers, State University of New JerseyPiscatawayUSA

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