Improving the preconditioning of linear systems from interior point methods

Abstract

The solution of linear optimization problems through interior point methods involves the solution of linear systems. These systems often have high dimensions and high sparsity degree, especially in real applications. Typically algebraic operations are performed to reduce the systems in two simpler formulations: one of them is known as the augmented system, and the other one, referred as normal equation systems, has a smaller dimension matrix which is symmetric positive definite.The solution of linear systems is the interior point methods step that requires most of the processing time. Consequently, the choice of the solution methods are extremely important in order to have an efficient implementation. Usually, direct methods are applied for solving these systems as, for example, Bunch-Parllett factorization or Cholesky factorization. However, in large scale problems, the use of direct methods becomes discouraging by limitations of time and memory. In such cases, iterative approaches are more attractive.The success of iterative method approaches depends on good preconditioners once the coefficient matrix becomes very ill-conditioned, especially close to an optimal solution. An alternative to treat the problem of ill conditioning is to use hybrid approaches with two phases: phase I uses a preconditioner for the normal equation systems built with incomplete factorizations information, called controlled Cholesky factorization; phase II, used in the final iterations, adopts the splitting preconditioner, which was developed specifically for such ill conditioned systems.This project proposes a new ordering criterion for the columns of the splitting preconditioner that preserves the sparse structure of the original coefficient matrix aiming improve the computational time and memory.