Bilinear optimality constraints for the cone of positive polynomials

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

Kernel-based interior-point methods for monotone linear complementarity problems over symmetric cones

We present an interior-point method for monotone linear complementarity problems over symmetric cones (SCLCP) that is based on barrier functions which are defined by a large class of univariate functions, called eligible kernel functions. This class is fairly general and includes the classical logarithmic function, the self-regular functions, as well as many non-self-regular functions as special cases. We provide a unified analysis of the method and give a general scheme on how to calculate the iteration bounds for the entire class. We also calculate the iteration bounds of both large-step and short-step versions of the method for ten frequently used eligible kernel functions. For some of them we match the best known iteration bound for large-step methods, while for short-step methods the best iteration bound is matched for all cases. The paper generalizes results of Lesaja and Roos (SIAM J. Optim. 20(6):3014–3039, 2010) from P ?(?)-LCP over the non-negative orthant to monotone LCPs over symmetric cones.Software Computer TechnologyElectrical Engineering, Mathematics and Computer Scienc

Smoothing algorithms for complementarity problems over symmetric cones

Complementarity problem, Symmetric cone, Euclidean Jordan algebra, Smoothing algorithm, Merit function method,

An entropy-like proximal algorithm and the exponential multiplier method for convex symmetric cone programming

Symmetric cone optimization, Proximal-like method, Entropy-like distance, Exponential multiplier method,