Bilinear optimality constraints for the cone of positive polynomials

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Rudolf, Gabor and Noyan, Nilay and Papp, David and Alizadeh, Farid (2011) Bilinear optimality constraints for the cone of positive polynomials. Mathematical Programming, 129 (1). pp. 5-31. ISSN 0025-5610 (Print) ; 1436-4646 (Online)

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For a proper cone K subset of R(n) and its dual cone K* the complementary slackness condition < x, s > = 0 defines an n-dimensional manifold C(K) in the space R(2n). When K is a symmetric cone, points in C(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(K). We examine several well-known cones, in particular the cone of positive polynomials P(2n+1) and its dual, and show that there are exactly four linearly independent bilinear identities which hold for all (x, s) is an element of C(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 Muntz polynomials.
Item Type: Article
Uncontrolled Keywords: Optimality conditions; Positive polynomials; Complementarity slackness; Bilinearity rank; Bilinear cones
Subjects: Q Science > Q Science (General)
Divisions: Faculty of Engineering and Natural Sciences
Faculty of Engineering and Natural Sciences > Academic programs > Manufacturing Systems Eng.
Depositing User: Nilay Noyan
Date Deposited: 20 Oct 2011 14:33
Last Modified: 30 Jul 2019 14:08

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