Knud D. Andersen
Abstract
The main computational work in interior-point methods for linear programming (LP) is to solve a least-squares problem. The normal equations are often used, but if the LP constraint matrix contains a nearly dense column the normal-equations matrix will be nearly dense. Assuming that the nondense part of the constraint matrix is of full rank, the Schur complement can be used to handle dense columns. In this article we propose a modified Schur-complement method that relaxes this assumption. Encouraging numerical results are presented.
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Index Terms
- A modified Schur-complement method for handling dense columns in interior-point methods for linear programming
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Reviews
Maurice W. Benson
Concern with a particular sparse least squares problem arising in linear programming motivates this work. A direct normal equation approach results in a non-sparse linear system if a nearly dense column occurs in the problem. Nearly dense columns are collected into a separate part of the matrix, and methods for exploiting the resulting structure are explored. The author notes stability difficulties with a standard Schur-complement approach. He suggests an efficient modified Schur-complement method that works with a linear system that has the same solution as the original system, but with better conditioning. The modified method has the same sparsity structure for a Cholesky factorization subproblem as the original Schur-complement approach. The new method can, however, better compensate for rank (or near rank) deficiency. A good discussion of implementation details is included. Experimental results with a dozen test problems demonstrate the utility of the suggested approach. This report should be useful to researchers interested in the computational linear algebra aspects of the problem as well as those interested in the linear programming applications. Andersen does a good job of relating the algorithm development to the literature, with ample references for readers seeking more detail.
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