Bilinear optimality constraints for the cone of positive polynomials

For a proper cone and its dual cone the complementary slackness condition defines an n -dimensional manifold in the space . When is a symmetric cone, points in must satisfy at least n linearly independent bilinear identities. This fact proves to be useful when optimizing over such cones, therefore i...

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Vydáno v:Mathematical programming Ročník 129; číslo 1; s. 5 - 31
Hlavní autoři: Rudolf, Gábor, Noyan, Nilay, Papp, Dávid, Alizadeh, Farid
Médium: Journal Article
Jazyk:angličtina
Vydáno: Berlin/Heidelberg Springer-Verlag 01.09.2011
Springer Nature B.V
Témata:
Algorithms
Calculus of Variations and Optimal Control; Optimization
Combinatorics
Errors
Full Length Paper
Intervals
Linear programming
Manifolds
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematical programming
Mathematics
Mathematics and Statistics
Mathematics of Computing
Numerical Analysis
Optimization
Polynomials
Studies
Syntax
Theoretical
Variables
Optimality conditions
Complementarity slackness
Positive polynomials
90C46
90C22
Bilinearity rank
Bilinear cones
ISSN:0025-5610, 1436-4646
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Shrnutí:For a proper cone and its dual cone the complementary slackness condition defines an n -dimensional manifold in the space . When is a symmetric cone, points in 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 . We examine several well-known cones, in particular the cone of positive polynomials and its dual, and show that there are exactly four linearly independent bilinear identities which hold for all , 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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ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-011-0458-y