Bilinear optimality constraints for the cone of positive polynomials

This is the latest version of this item.

Official URL: http://dx.doi.org/10.1007/s10107-011-0458-y

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.