Abstract. Every Newton step in an interior-point method for optimization requires a solution of a symmetric indefinite system of linear equations. Most of today's codes apply direct solution methods to perform this task. The use of logarithmic barriers in interior point methods causes unavoidable ill-conditioning of linear systems and, hence, iterative methods fail to provide sufficient accuracy unless appropriately preconditioned. Two types of preconditioners which use some form of incomplete Cholesky factorization for indefinite systems are proposed in this paper. Although they involve significantly sparser factorizations than those used in direct approaches they still capture most of the numerical properties of the preconditioned system. The spectral analysis of the preconditioned matrix is performed: for convex optimization problems all the eigenvalues of this matrix are strictly positive. Numerical results are given for a set of public domain large linearly constrained convex quadratic programming problems with sizes reaching tens of thousands of variables. The analysis of these results reveals that the solution times for such problems on a modern PC are measured in minutes when direct methods are used and drop to seconds when iterative methods with appropriate preconditioners are used.

Abstract. Issues of indefinite preconditioning of reduced Newton systems arising in optimization with interior point methods are addressed in this paper. Constraint preconditioners have shown much promise in this context. However, there are situations in which an unfavorable sparsity pattern of Jacobian matrix may adversely affect the preconditioner and make its inverse representation unacceptably dense hence too expensive to be used in practice. A remedy to such situations is proposed in this paper. An approximate constraint preconditioner is considered in which sparse approximation of the Jacobian is used instead of the complete matrix. Spectral analysis of the preconditioned matrix is performed and bounds on its non-unit eigenvalues are provided. Preliminary computational results are encouraging.

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A numerical study of the efficiency of the generalized conjugate residual methods (GCR) is performed using three different preconditioners all based upon an incomplete LU factorization. The GCR behavior is evaluated in connection with the solution of large, sparse unsymmetric systems of equations, arising from the finite element integration of the diffusion-convection equation for 2-dimensional (2-D) and 3-D problems with different Peclet and Courant numbers. The order of the test matrices ranges from 450 to 1700. Results from a set of numerical experiments are presented and comparisons with preconditioned GCR methods and with direct method are carried out.

In this paper we present the results obtained in solving consistent sparse systems of n nonlinear equations F(x) = 0; by a Quasi-Newton method combined with a p block iterative row-projection linear solver of Cimmino-type, 1 p n: Under weak regularity conditions for F; it is proved that this Inexact Quasi-Newton method has a local, linear convergence in the energy norm induced by the preconditioned matrix HA; where A is an initial guess of the Jacobian matrix, and it may converge superlinearly too. is the Moore-Penrose pseudo inverse of the m i n block, A i is the preconditioner. A simple partitioning of the Jacobian matrix was used for solving a set of nonlinear test problems with sizes ranging from 1024 to 131072 on the CRAY T3E under the MPI environment.