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