# An Interior Point-Proximal Method of Multipliers for Convex Quadratic Programming

## TL;DR

This paper introduces a novel polynomial-time algorithm combining Interior Point Methods with Proximal Method of Multipliers for convex quadratic programming, providing theoretical guarantees and practical robustness.

## Contribution

It presents the first polynomial complexity proof for a primal-dual regularized IPM and develops an efficient, robust algorithm with infeasibility detection capabilities.

## Key findings

- Algorithm achieves polynomial complexity under standard assumptions.
- Numerical tests confirm the method's robustness and efficiency.
- Infeasibility detection mechanism effectively identifies infeasible problems.

## Abstract

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.