  In this paper we generalize the Interior Point-Proximal Method of Multipliers
(IP-PMM) presented in [An Interior Point-Proximal Method of Multipliers for
Convex Quadratic Programming, Computational Optimization and Applications, 78,
307--351 (2021)] for the solution of linear positive Semi-Definite Programming
(SDP) problems, allowing inexactness in the solution of the associated Newton
systems. In particular, we combine an infeasible Interior Point Method (IPM)
with the Proximal Method of Multipliers (PMM) and interpret the algorithm
(IP-PMM) as a primal-dual regularized IPM, suitable for solving SDP problems.
We apply some iterations of an IPM to each sub-problem of the PMM until a
satisfactory solution is found. We then update the PMM parameters, form a new
IPM neighbourhood, and repeat this process. Given this framework, we prove
polynomial complexity of the algorithm, under mild assumptions, and without
requiring exact computations for the Newton directions. We furthermore provide
a necessary condition for lack of strong duality, which can be used as a basis
for constructing detection mechanisms for identifying pathological cases within
IP-PMM.
