Browsing by Author "Wang, Weichung"
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Item Adaptive Use of Iterative Methods in Interior Point Methods for Linear Programming(1998-10-15) Wang, Weichung; O'Leary, Dianne P.In this work we devise efficient algorithms for finding the search directions for interior point methods applied to linear programming problems. There are two innovations. The first is the use of updating of preconditioners computed for previous barrier parameters. The second is an adaptive automated procedure for determining whether to use a direct or iterative solver, whether to reinitialize or update the preconditioner, and how many updates to apply. These decisions are based on predictions of the cost of using the different solvers to determine the next search direction, given costs in determining earlier directions. These ideas are tested by applying a modified version of the OB1-R code of Lustig, Marsten, and Shanno to a variety of problems from the NETLIB and other collections. If a direct method is appropriate for the problem, then our procedure chooses it, but when an iterative procedure is helpful, substantial gains in efficiency can be obtained. (Also cross-referenced as UMIACS-TR-95-111)Item Adaptive Use of Iterative Methods in Predictor-Corrector Interior Point Methods for Linear Programming(1999-04-06) Wang, Weichung; O'Leary, Dianne P.In this work we devise efficient algorithms for finding the search directions for interior point methods applied to linear programming problems. There are two innovations. The first is the use of updating of preconditioners computed for previous barrier parameters. The second is an adaptive automated procedure for determining whether to use a direct or iterative solver, whether to reinitialize or update the preconditioner, and how many updates to apply. These decisions are based on predictions of the cost of using the different solvers to determine the next search direction, given costs in determining earlier directions. We summarize earlier results using a modified version of the OB1-R code of Lustig, Marsten, and Shanno, and we present results from a predictor-corrector code PCx modified to use adaptive iteration. If a direct method is appropriate for the problem, then our procedure chooses it, but when an iterative procedure is helpful, substantial gains in efficiency can be obtained. (Also cross-referenced as UMIACS-TR-99-21)Item Final Iterations in Interior Point Models -- Preconditioned Conjugate Gradients and Modified Search Directions(1998-10-15) Wang, WeichungIn this article we consider modified search directions in the endgame of interior point methods for linear programming. In this stage, the normal equations determining the search directions become ill-conditioned. The modified search directions are computered by solving perturbed systems in which the systems may be solved efficiently by the preconditioned conjugate gradient solver. We prove the convergence of the interior point methods using the modified search directions and show that each barrier problem is solved with a superlinear convergence rate. A variation of Cholesky factorization is presented for computing a better preconditioner when the normal equations are ill-conditioned. These ideas have been implemented successfully and the numerical results show that the algorithms enhance the performance of the preconditioned conjugate gradients-based interior point methods.