An Unregularized Third Order Newton Method

TL;DR

This paper introduces a third-order Newton method that uses third-order Taylor expansions and semidefinite programming to achieve local cubic convergence, with experimental comparisons to standard algorithms.

Contribution

The paper presents a novel third-order Newton method utilizing semidefinite programming and demonstrates its local cubic convergence property.

Findings

Achieves local cubic convergence.

Performs favorably in numerical experiments.

Uses semidefinite programming for subproblem solving.

Abstract

In this paper, we propose a third-order Newton's method which in each iteration solves a semidefinite program as a subproblem. Our approach is based on moving to the local minimum of the third-order Taylor expansion at each iteration, rather than that of the second order. We show that this scheme has local cubic convergence. We then provide numerical experiments comparing this scheme to some standard algorithms.