A semi-smooth Newton method for the nonlinear conic problem with generalized simplicial cones

TL;DR

This paper introduces a semi-smooth Newton method for nonlinear conic programming with generalized simplicial cones, establishing local quadratic convergence and demonstrating effectiveness through numerical experiments.

Contribution

It generalizes Robinson's normal equations to a conic setting, develops a semi-smooth Newton method, and proves strong semi-smoothness of the projection operator.

Findings

The method achieves local quadratic convergence.

Numerical experiments show competitive performance against smoothing Newton methods.

Application to low-rank matrix completion demonstrates practical utility.

Abstract

In this work we develop and analyze a semi-smooth Newton method for the general nonlinear conic programming problem. In particular, we study the problem with a generalized simplicial cone, i.e., the image of a symmetric cone under a linear mapping. We generalize Robinson's normal equations to a conic setting, yielding what we call the conic projection equations. The resulting system is equivalent to the KKT conditions associated with the nonlinear conic programming problem. A semi-smooth Newton iteration is proposed for solving it, and local quadratic convergence is established. We study properties of generalized simplicial cones and prove strong semi-smoothness of the projection operator onto them. Numerical experiments compare the method against a recent smoothing Newton approach on the circular cone programming problem, and we also apply it to the low-rank matrix completion problem.