An Interior Point-Proximal Method of Multipliers for Positive Semi-Definite Programming

TL;DR

This paper introduces a generalized interior point-proximal method of multipliers for solving positive semi-definite programming problems, allowing inexact solutions and proving polynomial complexity.

Contribution

It extends the IP-PMM framework to SDP problems with inexact Newton solutions and establishes polynomial complexity without requiring exact computations.

Findings

Algorithm has polynomial complexity under mild assumptions.

Allows inexact solutions in Newton system computations.

Provides a condition for detecting lack of strong duality.

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.