NDLI: Bilinear optimality constraints for the cone of positive polynomials

Content Provider	Springer Nature Link
Author	yan, Nilay Rudolf, Gábor Alizadeh, Farid Papp, Dávid
Copyright Year	2011
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.
Ending Page	31
Page Count	27
Starting Page	5
File Format	PDF
ISSN	00255610
e-ISSN	14364646
Journal	Mathematical Programming
Issue Number	1
Volume Number	129
Language	English
Publisher	Springer-Verlag
Publisher Date	2011-05-22
Publisher Place	Berlin, Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Optimality conditions, duality Semidefinite programming Positive polynomials Theoretical, Mathematical and Computational Physics Bilinearity rank Bilinear cones Optimality conditions Complementarity slackness Mathematical Methods in Physics Calculus of Variations and Optimal Control; Optimization Mathematics of Computing Numerical Analysis Combinatorics
Content Type	Text
Resource Type	Article
Subject	Mathematics Software

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