# Dynamic Non-Diagonal Regularization in Interior Point Methods for Linear and Convex Quadratic Programming

## TL;DR

This paper introduces a dynamic non-diagonal regularization technique for interior point methods that improves spectral properties and sparsity of the Newton system, enhancing efficiency in solving linear and quadratic programs.

## Contribution

It proposes a novel implicit non-diagonal regularization approach with a dynamic tuning rule, reducing the need for trial-and-error parameter selection in interior point methods.

## Key findings

- Improved spectral properties of the Newton system.
- Enhanced sparsity leading to more efficient factorizations.
- Effective in solving small and medium-scale linear and quadratic programs.

## Abstract

In this paper, we present a dynamic non-diagonal regularization for interior point methods. The non-diagonal aspect of this regularization is implicit, since all the off-diagonal elements of the regularization matrices are cancelled out by those elements present in the Newton system, which do not contribute important information in the computation of the Newton direction. Such a regularization has multiple goals. The obvious one is to improve the spectral properties of the Newton system solved at each iteration of the interior point method. On the other hand, the regularization matrices introduce sparsity to the aforementioned linear system, allowing for more efficient factorizations. We also propose a rule for tuning the regularization dynamically based on the properties of the problem, such that sufficiently large eigenvalues of the non-regularized system are perturbed insignificantly. This alleviates the need of finding specific regularization values through experimentation, which is the most common approach in literature. We provide perturbation bounds for the eigenvalues of the non-regularized system matrix and then discuss the spectral properties of the regularized matrix. Finally, we demonstrate the efficiency of the method applied to solve standard small and medium-scale linear and convex quadratic programming test problems.