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Convergence Analysis Of Inexact InfeasibleInteriorPointAlgorithms For Solving Linear Programming Problems
 SIAM J. Optim
, 2000
"... . In this paper we present a convergence analysis for some inexact (polynomial) variants of the infeasibleinteriorpointalgorithm of Kojima, Megiddo and Mizuno. For this analysis we assume that the iterates are bounded. The new variants allow the use of search directions that are calculated from t ..."
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. In this paper we present a convergence analysis for some inexact (polynomial) variants of the infeasibleinteriorpointalgorithm of Kojima, Megiddo and Mizuno. For this analysis we assume that the iterates are bounded. The new variants allow the use of search directions that are calculated from the defining linear system with only moderate accuracy, e.g. via the use of Krylov subspace methods like CG or QMR. Furthermore, some numerical results for the proposed methods are given. Key words. Linear programming, infeasibleinteriorpoint method, inexact search direction AMS subject classifications. 90C05, 65K05, 90C06 1. Introduction. Considering the fact that the primaldual algorithm of Kojima, Megiddo and Mizuno [3] (henceforth called the KMMAlgorithm) does not use a predictorcorrector approach, it is surprising that it is an efficient algorithm for solving linear programming problems in practice. Of course, this efficiency does not only follow from the use of the Newton search...
Convergence of Infeasible InteriorPoint Algorithms from Arbitrary Starting Points
 SIAM Journal on Optimization
, 1993
"... An important advantage of infeasible interiorpoint methods compared to feasible interiorpoint methods is their ability to be warmstarted from approximate solutions. It is therefore important for the convergence theory of these algorithms not to depend on being able to alter the starting point. In ..."
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Cited by 7 (3 self)
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An important advantage of infeasible interiorpoint methods compared to feasible interiorpoint methods is their ability to be warmstarted from approximate solutions. It is therefore important for the convergence theory of these algorithms not to depend on being able to alter the starting point. In two recent papers, Yin Zhang and Stephen Wright prove convergence results for some infeasible interiorpoint methods. Unfortunately, their analysis places a restriction on the starting point. It is easy to meet the restriction by altering the starting point, but this may take the point farther away from the solution, removing the advantage of warmstarting the algorithms. In this paper we extend Zhang and Wright's results to apply to arbitrary strictly positive starting points. We then present an algorithm for solving the BoxConstrained Linear Complementarity problem and prove its convergence. 1 Introduction Quite often, in using an iterative method to solve a problem, it is possible to u...
An Inexact TrustRegion FeasiblePoint Algorithm for Nonlinear Systems of Equalities and Inequalities
 Department of Computational and Applied Mathematics, Rice University
, 1995
"... In this work we define a trustregion feasiblepoint algorithm for approximating solutions of the nonlinear system of equalities and inequalities F(x, y) = 0, y ≥ 0, where F: R^n × R^m → R^p is continuously differentiable. This formulation is quite general; the KarushKuhn ..."
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Cited by 4 (0 self)
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In this work we define a trustregion feasiblepoint algorithm for approximating solutions of the nonlinear system of equalities and inequalities F(x, y) = 0, y &ge; 0, where F: R^n &times; R^m &rarr; R^p is continuously differentiable. This formulation is quite general; the KarushKuhnTucker conditions of a general nonlinear programming problem are an obvious example, and a set of equalities and inequalities can be transformed, using slack variables, into such form. We will be concerned with the possibility that n, m, and p may be large and that the Jacobian matrix may be sparse and rank deficient. Exploiting the convex structure of the local model trustregion subproblem, we propose a globally convergent inexact trustregion feasiblepoint algorithm to minimize an arbitrary norm of the residual, say F(x, y)a, subject to the nonnegativity constraints. This algorithm uses a trustregion globalization strategy to determine a descent direction as an inexact solution of the local model trustregion subproblem and then, it uses linesearch techniques to obtain an acceptable steplength. We demonstrate that, under rather weak hypotheses, any accumulation point of the iteration sequence is a constrained stationary point for f = Fa, and that the sequence of constrained residuals converges to zero.
An Investigation of the Interior Point Algorithms for the Linear Transportation Problem
 SIAM J. Sci. Computing
, 1993
"... . Recently, Resende and Veiga [31] have proposed an efficient implementation of the Dual Affine (DA) interiorpoint algorithm for the solution of linear transportation models with integer costs and righthand side coefficients. This procedure incorporates a Preconditioned Conjugate Gradient (PCG) me ..."
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. Recently, Resende and Veiga [31] have proposed an efficient implementation of the Dual Affine (DA) interiorpoint algorithm for the solution of linear transportation models with integer costs and righthand side coefficients. This procedure incorporates a Preconditioned Conjugate Gradient (PCG) method for solving the linear system that is required in each iteration of the DA algorithm. In this paper, we introduce an Incomplete QR Decomposition (IQRD) preconditioning for the PCG algorithm. Computational experience shows that the IQRD preconditioning is quite appropriate in this instance and is more efficient than the preconditioning introduced by Resende and Veiga. We also show that the Primal Dual (PD) and the Predictor Corrector (PC) interior point algorithms can also be implemented by using the same type of technique. A comparison among these three algorithms is also included and indicates that the PD an PC algorithms are more appropriate for the solution of transportation problems...