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.

中文翻译：

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一种用于线性和凸二次规划乘法器的内点邻近方法的新预处理方法

在本文中，我们讨论线性和二次规划问题的有效数值解，通常是大规模的。出于这个目的，我们设计了一种不可行的内点方法，与乘法器的近端方法相结合，从而产生了原始对偶正则化内点方法。由于内存和 CPU 时间的限制，这种方法的应用产生了一系列病态的线性系统，这些线性系统不能总是通过分解方法来解决。我们提出了一种新颖的预处理策略，该策略基于线性情况下正规方程矩阵的适当稀疏化，并且还构成了块对角预处理器的基础，以加速线性系统的 MINRES 求解一般二次规划问题。