We describe a Jordan-algebraic version of results related to convexity of images of quadratic mappings as well as related results on exactness of symmetric relaxations of certain classes of nonconvex optimization problems. The exactness of relaxations is proved based on rank estimates. Our approach provides a unifying viewpoint on a large number of… (More)
We consider primal-dual algorithms for certain types of infinite-dimensional optimization problems. Our approach is based on the generalization of the technique of finite-dimensional Eu-clidean Jordan algebras to the case of infinite-dimensional JB-algebras of finite rank. This generalization enables us to develop polynomial-time primal-dual algorithms for… (More)
We describe an implementation of an infinite-dimensional primal-dual algorithm based on the Nesterov-Todd direction. Several applications to both continuous and discrete-time multi-criteria linear-quadratic control problems and linear-quadratic control problem with quadratic constraints are described. Numerical results show a very fast convergence… (More)
We describe a version of randomization technique within a general framework of Euclidean Jordan algebras. It is shown how to use this technique to evaluate the quality of symmetric relaxations for several non-convex optimization problems.
We explicitly calculate characteristic functions of cones of generalized polynomials corresponding to Chebyshev systems on intervals of the real line and the circle. Thus, in principle, we calculate homogeneous self-concordant barriers for this class of cones. This class includes almost all "cones of squares" considered in 5]. Our construction, however,… (More)
Based on previous explicit computations of universal barrier functions, we describe numerical experiments for solving certain classes of convex optimization problems. The comparison is given of the performance of the classical affine-scaling algorithm with the similar algorithm built upon the universal barrier function. 1. Introduction. By now there is a… (More)