For linear optimization (LO) problems, we consider a curvature integral first introduced by Sonnevend et al. (1991). Our main result states that in order to establish an upper bound for the total Sonnevend curvature of the central path, it is sufficient to consider only the case when n = 2m. This also implies that the … Read more
In this paper we characterize the slack matrices of cones and polytopes among all nonnegative matrices. This leads to an algorithm for deciding whether a given matrix is a slack matrix. The underlying decision problem is equivalent to the polyhedral verifi cation problem whose complexity is unknown. CitationApril 2013ArticleDownload View PDF
Several variations of index selection rules for simplex type algorithms for linear programming, like the Last-In-First-Out or the Most-Often-Selected-Variable are rules not only theoretically finite, but also provide significant flexibility in choosing a pivot element. Based on an implementation of the primal simplex and the monotonic build-up (MBU) simplex method, the practical benefit of the … Read more
Nonnegative matrix factorization (NMF) has been shown recently to be tractable under the separability assumption, under which all the columns of the input data matrix belong to the convex cone generated by only a few of these columns. Bittorf, Recht, R\’e and Tropp (`Factoring nonnegative matrices with linear programs’, NIPS 2012) proposed a linear programming … Read more
This paper introduces three novel techniques for updating the invertible representation of the basis matrix when solving practical sparse linear programming (LP) problems using a high performance implementation of the dual revised simplex method, being of particular value when suboptimization is used. Two are variants of the product form update and the other permits multiple … Read more
Our main goal is to compute or estimate the calmness modulus of the argmin mapping of linear semi-infinite optimization problems under canonical perturbations, i.e., perturbations of the objective function together with continuous perturbations of the right-hand side of the constraint system (with respect to an index ranging in a compact Hausdorff space). Specifically, we provide … Read more
Although nonnegative matrix factorization (NMF) is NP-hard in general, it has been shown very recently that it is tractable under the assumption that the input nonnegative data matrix is separable (that is, there exists a cone spanned by a small subset of the columns containing all columns). Since then, several algorithms have been designed to … Read more
The computational time required by interior-point methods is often dominated by the solution of linear systems of equations. An efficient specialized interior-point algorithm for primal block-angular problems has been used to solve these systems by combining Cholesky factorizations for the block constraints and a conjugate gradient based on a power series preconditioner for the linking … Read more
The aim of this paper is to demonstrate a new approach to solve the linearized (n-1) security constrained optimal power flow (SCOPF) problem by a structure exploiting interior point solver. Firstly, we present a reformulation of the linearized SCOPF model, in which most matrices that need to be factorized are constant. Hence, most factorizations and … Read more
This paper presents a new warmstarting technique in the context of a primal-dual column generation method applied to solve a particular class of combinatorial optimization problems. The technique relies on calculating an initial point and on solving auxiliary linear optimization problems to determine the step direction needed to fully restore primal and dual feasibilities after … Read more