Computational Experience with a Preconditioner for Interior Point Methods for Linear Programming

In this paper, we discuss eecient implementation of a new class of preconditioners for linear systems arising from interior point methods. These new preconditioners give superior performancenear the solution of a linear programming problem where the linear systems are typically highly ill-conditioned. They rely upon the computation of an LU factorization of a subset of columns of the matrix of constraints. The implementation of these new techniques require some sophistication since the subset of selected columns is not known a priori. The conjugate gradient method using this new preconditioner compares favorably with the Cholesky factorization approach. The new approach is clearly superior for large scale problems where the Cholesky factorization produces intractable ll-in. Numerical experiments on several representative classes of linear programming problems are presented to demonstrate the promise of the new preconditioner. Abbreviated Title: Computational Experience with a Preconditioner. 1. Introduction. In 9] a new class of preconditioners for for the linear systems from interior point methods for linear programming is proposed and its theoretical properties are discussed. This class avoids computing the Schur complement matrix. Instead, these preconditioners rely upon an LU factorization of a subset of columns of the constraint matrix. In this work we present several techniques for an eecient implementation of these preconditioners. Among these, are techniques for the ee-cient utililization of the nonzero structure of the matrix to speed up the numerical factorization. We investigate the performance of some important large scale problem cases using this preconditioner with the conjugate gradient method. Performance of this iterative approach is compared with the performance of the direct Cholesky factorization technique. We use the following notation throughout this work. Lower case Greek letters denote scalars, lower case Latin letters denote vectors and upper case Latin letters denote matrices. Components of matrices and vectors are represented by the corresponding Greek letter with subscripts. The symbol 0 will denote the scalar zero, the zero column vector and the zero matrix, its dimension will be clear from context. The identity matrix will be denoted by I, a subscript will determine its dimension when it is not clear from context. The Euclidean norm is represented by k k which will also represent the 2-norm for matrices. The relation X = diag(x) means that X is a diagonal matrix whose the diagonal entries are the components of x. On the other hand, diag(A) means the column vector formed by the diagonal entries …