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In this work, in the context of Linear and Quadratic Programming, weinterpret Primal Dual Regularized Interior Point Methods (PDR-IPMs) in theframework of the Proximal Point Method. The resulting Proximal Stabilized IPM(PS-IPM) is strongly supported by theoretical results concerning convergenceand the rate of convergence, and can handle degenerate problems. Moreover, inthe second part of this work, we analyse the interactions between theregularization parameters and the computational foot-print of the linearalgebra routines used to solve the Newton linear systems. In particular, whenthese systems are solved using an iterative Krylov method, we propose generalpurpose preconditioners which, exploiting the regularization and a newrearrangement of the Schur complement, remain attractive for a series ofsubsequent IPM iterations. Therefore they need to be recomputed only in afraction of the total IPM iterations. The resulting regularized second ordermethods, for which low-frequency-updates of the preconditioners are allowed,pave the path for an alternative third way in-between first and second ordermethods.

Proximal stabilized Interior Point Methods for quadratic programming and low-frequency-updates preconditioning techniques