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In this paper we show the global convergence of the anne scaling methods without assuming any condition on degeneracy. The behavior of the method near degenerate faces is analyzed in detail on the basis of the equivalence between the affine scaling methods for homogeneous LP problems and Kar-markar's method. It is shown that the step-size ~-, where the(More)
Systems of linear equations with " normal " matrices of the form AD 2 A T is a key ingredient in the computation of search directions for interior-point algorithms. In this article, we establish that a well-known basis preconditioner for such systems of linear equations produces scaled matrices with uniformly bounded condition numbers as D varies over the(More)
In this paper we present a new iteration-complexity bound for the Mizuno–Todd–Ye predictor-corrector (MTY P-C) primal-dual interior-point algorithm for linear programming. The analysis of the paper is based on the important notion of crossover events introduced by Vavasis and Ye. For a standard form linear program min{c T x : Ax = b, x ≥ 0} with decision(More)
This paper is a continuation of our previous paper in which we studied a polynomial primal-dual path-following algorithm for SOCP using an analogue of the HRVW/KSH/M direction for SDP. We develop an improved and simplied complexity analysis which can be also applied to the algorithm using the NT direction. Specically, we show that the long-step algorithm(More)
In this paper we study primal-dual path-following algorithms for the second-order cone programming (SOCP) based on a family of directions that is a natural extension of the Monteiro-Zhang (MZ) family for semideenite programming. We show that the polynomial iteration-complexity bounds of two well-known algorithms for linear programming, namely the short-step(More)
In this paper we investigate the global convergence property of the affine scaling method under the assumption of dual nondegeneracy. The behavior of the method near degenerate vertices is analyzed in detail on the basis of the equivalence between the affine scaling methods for homogeneous LP problems and Karmarkar's method. It is shown that the step-size(More)
This paper establishes the polynomial convergence of a new class of (feasible) primal-dual interior-point path following algorithms for semideenite programming (SDP) whose search directions are obtained by applying Newton method to the symmetric central path equation (P T XP) 1=2 (P ?1 SP ?T)(P T XP) 1=2 ? I = 0; where P is a nonsingular matrix.(More)
The layered-step interior-point algorithm was introduced by Vavasis and Ye. The algorithm accelerates the path following interior-point algorithm and its arithmetic complexity depends only on the coefficient matrix A. The main drawback of the algorithm is the use of an unknown big constant x, in computing the search direction and to initiate the algorithm.(More)
In this paper we present a variant of Vavasis and Ye's layered-step path following primal-dual interior-point algorithm for linear programming. Our algorithm is a predictor-corrector type algorithm which uses from time to time the least layered squares (LLS) direction in place of the affine scaling direction. It has the same iteration-complexity bound of(More)