We present a method for propagating linear constraints. Our technique exploits the fact that the interior point method converges on a central point of the polytope. A variable assigned to an extreme point is therefore assigned to this extreme point in all solutions. We show how linear relaxations and the interior point method can be combined to prune variable domains. We also describe a class o...

This paper presents a logarithmic barrier method for solving a semi-definite linear program. The descent direction is the classical Newton direction. We propose alternative ways to determine the step-size along the direction which are more efficient than classical linesearches.

In his Introductory Lectures on Convex Programming Nesterov has given an algorithm to nd the analytic centre x F for a given -self-concordant barrier F with bounded domain and a given interior point of this domain. The intended use of this algorithm is as an auxiliary phase in a primal short-step path-following method for solving convex programming problems. For the number of iterations in this...

We present an improved version of an infeasible interior-point method for linear optimization published in 2006. In the earlier version each iteration consisted of one so-called feasibility step and a few – at most three – centering steps. In this paper each iteration consists of only a feasibility step, whereas the iteration bound improves the earlier bound by a factor 2 √ 2. The improvements ...

Given a nonnegative, symmetric matrix of weights, H , we study the problem of finding an Hermitian, positive semidefinite matrix which is closest to a given Hermitian matrix, A, with respect to the weighting H . This extends the notion of exact matrix completion problems in that, Hi j = 0 corresponds to the element Ai j being unspecified (free), while Hi j large in absolute value corresponds to...

The inexact primal-dual interior point method which is discussed in this paper chooses a new iterate along an approximation to the Newton direction. The method is the Kojima, Megiddo, and Mizuno globally convergent infeasible interior point algorithm The inexact variation is shown to have the same convergence properties accepting a residual in both the primal and dual Newton step equation also ...

Ill-posed problems are numerically underdetermined. It is therefore often beneficial to impose known properties of the desired solution, such as nonnegativity, during the solution process. This paper proposes the use of an interior-point method in conjunction with truncated iteration for the solution of large-scale linear discrete ill-posed problems with box constraints. An estimate of the erro...

The formulation of interior point algorithms for semide nite programming has become an active research area, following the success of the methods for large{ scale linear programming. Many interior point methods for linear programming have now been extended to the more general semide nite case, but the initialization problem remained unsolved. In this paper we show that the initialization strate...

In digital communications, orthogonal pulse shapes are often used to represent message symbols for transmission through a channel. The design of such pulse shapes is formulated as a convex semidefinite programming problem, from which a globally optimal pulse shape can be efficiently found using interior point methods. The formulation is used to design filters which achieve the minimal bandwidth...

In this paper we prove relations between the eigenvalues of matrices that occur during the solution of linear programming problems with interior-point methods. We will present preconditioners for these matrices that preserve the relations and discuss the practical implications of our results when iterative linear solvers are used.