S. Pougkakiotis and J. Gondzio
Abstract
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.
Key words: Proximal method of multipliers, Interior point methods, Semidefinite programming.