Bilinear optimality constraints for the cone of positive polynomials |
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Authors: | Gábor Rudolf Nilay Noyan Dávid Papp Farid Alizadeh |
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Institution: | 1.Virginia Commonwealth University,Richmond,USA;2.Faculty of Engineering and Natural Sciences,Sabanci University,Orhanli, Tuzla, Istanbul,Turkey;3.Department of Industrial Engineering and Management Sciences,Northwestern University,Evanston,USA;4.RUTCOR and School of Business,Rutgers, State University of New Jersey,Piscataway,USA |
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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. |
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