A structured modified Newton approach for solving systems of nonlinear equations arising in interior-point methods for quadratic programming

TL;DR

This paper introduces a structured modified Newton method for solving nonlinear systems in interior-point algorithms for quadratic programming, reducing computational cost while maintaining solution quality through low-rank Jacobian updates.

Contribution

The paper proposes a novel structured low-rank update scheme for Jacobians in modified Newton methods, preserving sparsity and improving efficiency in interior-point quadratic programming.

Findings

The method reduces computational effort compared to standard Newton iterations.

Numerical experiments show competitive performance with theoretical guarantees.

Heuristics further enhance practical efficiency beyond initial theoretical results.

Abstract

The focus in this work is on interior-point methods for inequality-constrained quadratic programs, and particularly on the system of nonlinear equations to be solved for each value of the barrier parameter. Newton iterations give high quality solutions, but we are interested in modified Newton systems that are computationally less expensive at the expense of lower quality solutions. We propose a structured modified Newton approach where each modified Jacobian is composed of a previous Jacobian, plus one low-rank update matrix per succeeding iteration. Each update matrix is, for a given rank, chosen such that the distance to the Jacobian at the current iterate is minimized, in both 2-norm and Frobenius norm. The approach is structured in the sense that it preserves the nonzero pattern of the Jacobian. The choice of update matrix is supported by results in an ideal theoretical setting. We also produce numerical results with a basic interior-point implementation to investigate the practical performance within and beyond the theoretical framework. In order to improve performance beyond the theoretical framework, we also motivate and construct two heuristics to be added to the method.