  In this paper we combine an infeasible Interior Point Method (IPM) with the
Proximal Method of Multipliers (PMM). The resulting algorithm (IP-PMM) is
interpreted as a primal-dual regularized IPM, suitable for solving linearly
constrained convex quadratic programming problems. We apply few iterations of
the interior point method to each sub-problem of the proximal method of
multipliers. Once a satisfactory solution of the PMM sub-problem is found, we
update the PMM parameters, form a new IPM neighbourhood and repeat this
process. Given this framework, we prove polynomial complexity of the algorithm,
under standard assumptions. To our knowledge, this is the first polynomial
complexity result for a primal-dual regularized IPM. The algorithm is guided by
the use of a single penalty parameter; that of the logarithmic barrier. In
other words, we show that IP-PMM inherits the polynomial complexity of IPMs, as
well as the strict convexity of the PMM sub-problems. The updates of the
penalty parameter are controlled by IPM, and hence are well-tuned, and do not
depend on the problem solved. Furthermore, we study the behavior of the method
when it is applied to an infeasible problem, and identify a necessary condition
for infeasibility. The latter is used to construct an infeasibility detection
mechanism. Subsequently, we provide a robust implementation of the presented
algorithm and test it over a set of small to large scale linear and convex
quadratic programming problems. The numerical results demonstrate the benefits
of using regularization in IPMs as well as the reliability of the method.
