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
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
In the paper we study some well-known cases of nonlinear programming problems, presenting them as instances of Inexact Linear Programming. The class of problems considered contains, in particular, semidefinite programming, second order cone programming and special cases of inexact semidefinite programming. Strong duality results for the nonlinear problems studied are obtained via the Lagrangian duality. … Read more
We study capacitated network flow problems with supplies and demands defined on a countably infinite collection of nodes having finite degree. This class of network flow models includes, for example, all infinite horizon deterministic dynamic programs with finite action sets since these are equivalent to the problem of finding a shortest infinite path in an … Read more
We describe a complete solution of the linear-quaratic control problem with the linear term in the objective function on a semiinfinite interval. This problem has important applications to calculation of Nesterov-Todd and other primal-dual directions in infinite-dimensional setting. CitationTechnical report, University of Notre Dame, December, 2003ArticleDownload View PDF
We explicitly calculate universal barrier functions for cones generated by (weakly) Chebyshev systems over finite sets. We show that universal barrier functions corresponding to Chebyshev systems on intervals are obtained as limits of universal barrier functions of their discretizations. The results are heavily rely upon classical work of M. Krein, A. Nudelman and I.J. Schoenberg … Read more
The aim of this paper is to discuss an asymptotic approximation model and its convergence for the minimax semi-infinite programming problem. An asymptotic surrogate constraints method for the minimax semi-infinite programming problem is presented making use of two general iscreteapproximation methods. Simultaneously, we discuss the consistenceand the epi-convergence of the asymptotic approximation problem. CitationSchool of … Read more
Cutting plane methods provide the means to solve large scale semidefinite programs (SDP) cheaply and quickly. They can also conceivably be employed for the purposes of re-optimization after branching, or the addition of cutting planes. We give a survey of various cutting plane approaches for SDP in this paper. These cutting plane approaches arise from … Read more
It is with the aim of solving scheduling problems with irregular cost functions that this paper focuses on the continuous assignment problem. It consists in partitioning a d dimensional region into subregions of prescribed volumes so that the total cost is minimized. The dual problem of the continuous assignment problem is an unconstrained maximisation of … Read more
We present a multigrid approach to the optimization of systems governed by differential equations. Such optimization problems have many applications, and are a broader class of problems than systems of equations. Using several model problems we give evidence (both theoretical and numerical) that a multigrid approach can often be successful in the setting of optimization. … Read more