A robust and efficient proposal for solving linear systems arising in interior-point methods for linear programming

• Published in: Computational optimization and applications Vol. 56; no. 3; pp. 573 - 597
• Main Authors: Gonzalez-Lima, María D., Oliveira, Aurelio R. L., Oliveira, Danilo E.
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01.12.2013; Springer Nature B.V
• Subjects: Algebra; Algorithms; Blocking; Computational mathematics; Convex and Discrete Geometry; Data envelopment analysis; Iterative methods; Lagrange multiplier; Linear algebra; Linear programming; Management Science; Mathematical models; Mathematics; Mathematics and Statistics; Methods; Operations Research; Operations Research/Decision Theory; Optimization; Statistics; Studies; Linear systems; Primal-dual interior point methods; Linear programming
• ISSN: 0926-6003; 1573-2894
• DOI: 10.1007/s10589-013-9572-5

We introduce an efficient and robust proposal for solving linear systems arising at each iteration of primal-dual interior-point methods for linear programming. Our proposal is based on the stable system presented by Gonzalez-Lima et al. (Comput. Opt. Appl. 44:213–247, 2009 ). Using similar techniques as those employed in the splitting preconditioner introduced by Oliveira and Sorensen (Linear Algebra Appl. 394:1–24, 2005 ) we are able to express the stable system matrix in block form such that the diagonal blocks are nonsingular diagonal matrices and the off-diagonal blocks are matrices close to zero when the iterates are close to the solution set of the linear programming problem. For degenerate problems a perturbation of the diagonal is added. We use a low-cost fixed iterative method to solve this system. Numerical experiments have shown that our approach leads to very accurate solutions for the linear programming problem.