A new approach for finding a basis for the splitting preconditioner for linear systems from interior point methods

• Published in: Computational optimization and applications 2017-05, Vol.67 (1), p.111-127
• Main Authors: Suñagua, Porfirio, Oliveira, Aurelio R. L.
• Format: Article
• Language: English
• Subjects: Computation; Convex and Discrete Geometry; Heuristic; Linear programming; Linear systems; Management Science; Mathematical analysis; Mathematics; Mathematics and Statistics; Matrices (mathematics); Matrix methods; Methods; Operations Research; Operations Research/Decision Theory; Optimization; Predictor-corrector methods; Splitting; Statistics; Studies
• ISSN: 0926-6003; 1573-2894
• DOI: 10.1007/s10589-016-9887-0

The class of splitting preconditioners for the iterative solution of linear systems arising from Mehrotra’s predictor-corrector method for large scale linear programming problems needs to find a basis through a sophisticated process based on the application of a rectangular LU factorization. This class of splitting preconditioners works better near a solution of the linear programming problem when the matrices are highly ill-conditioned. In this study, we develop and implement a new approach to find a basis for the splitting preconditioner, based on standard rectangular LU factorization with partial permutation of the scaled transpose linear programming constraint matrix. In most cases, this basis is better conditioned than the existing one. In addition, we include a penalty parameter in Mehrotra’s predictor-corrector method in order to reduce ill-conditioning of the normal equations matrix. Computational experiments show a reduction in the average number of iterations of the preconditioned conjugate gradient method. Also, the increased efficiency and robustness of the new approach become evident by the performance profile.