A new preconditioning approach for an interior point‐proximal method of multipliers for linear and convex quadratic programming

Abstract

In this article, we address the efficient numerical solution of linear and quadratic programming problems, often of large scale. With this aim, we devise an infeasible interior point method, blended with the proximal method of multipliers, which in turn results in a primal-dual regularized interior point method. Application of this method gives rise to a sequence of increasingly ill-conditioned linear systems which cannot always be solved by factorization methods, due to memory and CPU time restrictions. We propose a novel preconditioning strategy which is based on a suitable sparsification of the normal equations matrix in the linear case, and also constitutes the foundation of a block-diagonal preconditioner to accelerate MINRES for linear systems arising from the solution of general quadratic programming problems. Numerical results for a range of test problems demonstrate the robustness of the proposed preconditioning strategy, together with its ability to solve linear systems of very large dimension.

Low-rank Updates of Preconditioners for Sequences of Symmetric Linear Systems

“…We finally mention the work in [23] where the author use this correction to update an Inexact Constraint Preconditioner, by damping the largest eigenvalues of the preconditioned matrices.…”

An Interior Point-Proximal Method of Multipliers for Linear Positive Semi-Definite Programming

“…Under this framework, it was proved that IP-PMM converges in a polynomial number of iterations, under mild assumptions, and an infeasibility detection mechanism was established. An important feature of this method is that it provides a reliable tuning for the penalty parameters of the PMM; indeed, the reliability of the algorithm is established numerically in a wide variety of convex problems in [7,8,10,19]. In particular, the IP-PMMs proposed in [7,8,10] use preconditioned iterative methods for the solution of the resulting linear systems, and are very robust despite the use of inexact Newton directions.…”

“…An important feature of this method is that it provides a reliable tuning for the penalty parameters of the PMM; indeed, the reliability of the algorithm is established numerically in a wide variety of convex problems in [7,8,10,19]. In particular, the IP-PMMs proposed in [7,8,10] use preconditioned iterative methods for the solution of the resulting linear systems, and are very robust despite the use of inexact Newton directions. In what follows, we develop and analyze an IP-PMM for linear SDP problems, which furthermore allows for inexactness in the solution of the linear systems that have to be solved at every iteration.…”

“…A particularly important benefit of using regularization, is that the resulting Newton systems can be preconditioned effectively (e.g. see the developments in [7,10]), allowing for more efficient implementations, with significantly lowered memory requirements. We note that the paper is focused on the theoretical aspects of the method, and an efficient, scalable, and reliable implementation would require a separate study.…”

“…We note that the paper is focused on the theoretical aspects of the method, and an efficient, scalable, and reliable implementation would require a separate study. Nevertheless, the practical effectiveness of IP-PMM (both in terms of efficiency, scalability, and robustness) has already been demonstrated for linear, convex quadratic [7,10,19], and nonlinear convex problems [8].…”