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
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
VSDP is a software package that is designed for the computation of verified results in conic programming. The current version of VSDP supports the constraint cone consisting of the product of semidefinite cones, second-order cones and the nonnegative orthant. It provides functions for computing rigorous error bounds of the true optimal value, verified enclosures of … 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
In this paper we will discuss two variants of an inexact feasible interior point algorithm for convex quadratic programming. We will consider two different neighbourhoods: a (small) one induced by the use of the Euclidean norm which yields a short-step algorithm and a symmetric one induced by the use of the infinity norm which yields … Read more
In this paper we argue for aggregated models in decomposition schemes for two-stage stochastic programming problems. We observe that analogous schemes proved effective for single-stage risk-averse problems, and for general linear programming problems. A major drawback of the aggregated approach for two-stage problems is that an aggregated master problem can not contain all the information … 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
This paper describes a new approach for computing nonnegative matrix factorizations (NMFs) with linear programming. The key idea is a data-driven model for the factorization, in which the most salient features in the data are used to express the remaining features. More precisely, given a data matrix X, the algorithm identifies a matrix C that … Read more
In this paper, we consider the solution of the standard linear programming (LP). A remarkable result in LP claims that all optimal solutions form an optimal face of the underlying polyhedron. In practice, many real-world problems have infinitely many optimal solutions and pursuing the optimal face, not just an optimal vertex, is quite desirable. The … Read more
Implementations of the Simplex method differ only in very specific aspects such as the pivot rule. Similarly, most relaxation methods for mixed-integer programming differ only in the type of cuts and the exploration of the search tree. Implementing instances of those frameworks would therefore be more efficient if linear and mixed-integer programming solvers let users … Read more