In this paper we present the convergence analysis of the inexact infeasible path-following(IIPF) interior point algorithm. In this algorithm the preconditioned conjugate gradient method is used to solve the reduced KKT system (the augmented system). The augmented system is preconditioned by using a block triangular matrix. The KKT system is solved approximately. Therefore, it becomes … Read more
In this paper we present a cutting plane algorithm for the Set Covering problem. Cutting planes are generated by a “general” (i.e. not based on the “template paradigm”) separation algorithm based on the following idea: i) identify a suitably small subproblem defined by a subset of the constraints of the formulation; ii) run an exact … Read more
This paper describes an active-set algorithm for nonlinear programming that solves a parametric linear programming subproblem at each iteration to generate an estimate of the active set. A step is then computed by solving an equality constrained quadratic program based on this active-set estimate. This approach respresents an extension of the standard sequential linear-quadratic programming … Read more
A regularized Newton-like method for solving nonnegative least-squares problems is proposed and analysed in this paper. A preconditioner for KKT systems arising in the method is introduced and spectral properties of the preconditioned matrix are analysed. A bound on the condition number of the preconditioned matrix is provided. The bound does not depend on the … Read more
The idea of a finite collection of closed sets having “strongly regular intersection” at a given point is crucial in variational analysis. We show that this central theoretical tool also has striking algorithmic consequences. Specifically, we consider the case of two sets, one of which we assume to be suitably “regular” (special cases being convex … Read more
The transportation paradox is related to the classical transportation problem. For certain instances of this problem an increase in the amount of goods to be transported may lead to a decrease in the optimal total transportation cost. Even though the paradox has been known since the early days of linear programming, it has got very … Read more
We recall the use of squared slacks used to transform inequality constraints into equalities and several reasons why their introduction may be harmful in many algorithmic frameworks routinely used in nonlinear programming. Numerical examples performed with the sequential quadratic programming method illustrate those reasons. CitationCahier du GERAD G-2007-62, Aug. 2007ArticleDownload View PDF
Traditionally, optimality and uniqueness of an (n,N,t) spherical code is proved using linear programming bounds. However, this approach does not apply to the parameter (4,10,1/6). We use semidefinite programming bounds instead to show that the Petersen code (which are the vertices of the 4-dimensional second hypersimplex or the midpoints of the edges of the regular … Read more
We consider the classical problem of estimating a density on $[0,1]$ via some maximum entropy criterion. For solving this convex optimization problem with algorithms using first-order or second-order methods, at each iteration one has to compute (or at least approximate) moments of some measure with a density on $[0,1]$, to obtain gradient and Hessian data. … Read more
We study cooperative games with supermodular costs. We show that supermodular costs arise in a variety of situations: in particular, we show that the problem of minimizing a linear function over a supermodular polyhedron–a problem that often arises in combinatorial optimization–has supermodular optimal costs. In addition, we examine the computational complexity of the least core … Read more