Takashi Tsuchiya 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 similar algorithm based upon the universal barrier function CitationTo appear in “Computational Optimization and Applications”ArticleDownload View PDF
The problem of maximizing the sum of linear functional and several weighted logarithmic determinant (logdet) functions under semidefinite constraints is a generalization of the semidefinite programming (SDP) and has a number of applications in statistics and datamining, and other areas of informatics and mathematical sciences. In this paper, we extend the framework of standard primal-dual … Read more
The main goals of this paper are to: i) relate two iteration-complexity bounds associated with the Mizuno-Todd-Ye predictor-corrector algorithm for linear programming (LP), and; ii) study the geometrical structure of the central path in the context of LP. The first forementioned iteration-complexity bound is expressed in terms of an integral introduced by Sonnevend, Stoer and … Read 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 (typically, within 3-4 iterations) to optimal solutions CitationPreprint, May, 2004, University of Notre DameArticleDownload … Read more
Density estimation is a classical and important problem in statistics. The aim of this paper is to develop a new computational approach to density estimation based on semidefinite programming (SDP), a new technology developed in optimization in the last decade. We express a density as the product of a nonnegative polynomial and a base density … Read 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 Euclidean 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 “infinite-dimensional second-order cone programs.” We consider as an example a … Read more
Solving systems of linear equations with “normal” matrices of the form $A D^2 A^T$ is a key ingredient in the computation of search directions for interior-point algorithms. In this article, we establish that a well-known basis preconditioner for such systems of linear equations produces scaled matrices with uniformly bounded condition numbers as $D$ varies over … Read more
In this paper, we consider a continuous version of the convex network flow problem which involves the integral of the Euclidean norm of the flow and its square in the objective function. A discretized version of this problem can be cast as a second-order cone program, for which efficient primal-dual interior-point algorithms have been developed … Read more
In this paper we present a new iteration-complexity bound for the Mizuno-Todd-Ye predictor-corrector (MTY P-C) primal-dual interior-point algorithm for linear programming. The analysis of the paper is based on the important notion of crossover events introduced by Vavasis and Ye. For a standard form linear program $\min\{c^Tx : Ax=b, \, x \ge 0\}$ with decision … Read more