Projection onto simplicial cones by Picard's method

TL;DR

This paper introduces a Picard's method-based approach for projecting onto simplicial cones, demonstrating linear convergence and comparing its performance with semi-smooth Newton's method through numerical experiments.

Contribution

The paper develops a Picard's iterative method for projecting onto simplicial cones and analyzes its convergence properties, providing a new computational technique for this problem.

Findings

Picard's method converges linearly to the solution.

Numerical experiments compare Picard's and semi-smooth Newton's methods.

The proposed method is effective for projecting onto simplicial cones.

Abstract

By using Moreau's decomposition theorem for projecting onto cones, the problem of projecting onto a simplicial cone is reduced to finding the unique solution of a nonsmooth system of equations. It is shown that Picard's method applied to the system of equations associated to the problem of projecting onto a simplicial cone generates a sequence that converges linearly to the solution of the system. Numerical experiments are presented making the comparison between Picard's and semi-smooth Newton's methods to solve the nonsmooth system associated with the problem of projecting a point onto a simplicial cone.