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. 531. ISSN 00255610 (Print) ; 14364646 (Online)
This is the latest version of this item.
Official URL: http://dx.doi.org/10.1007/s101070110458y
Abstract
For a proper cone K subset of R(n) and its dual cone K* the complementary slackness condition < x, s > = 0 defines an ndimensional 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 nonsymmetric 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 wellknown 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 halfline 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 
URI:  https://research.sabanciuniv.edu/id/eprint/17267 
Available Versions of this Item

Bilinear optimality constraints for the cone of positive
polynomials. (deposited 15 Dec 2008 16:10)

Bilinearity rank of the cone of positive polynomials and related cones. (deposited 03 Dec 2009 12:41)

Bilinearity rank of the cone of positive polynomials and related cones. (deposited 13 Oct 2010 10:17)
 Bilinear optimality constraints for the cone of positive polynomials. (deposited 20 Oct 2011 14:33) [Currently Displayed]

Bilinearity rank of the cone of positive polynomials and related cones. (deposited 13 Oct 2010 10:17)

Bilinearity rank of the cone of positive polynomials and related cones. (deposited 03 Dec 2009 12:41)