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A semi-smooth Newton method for general projection equations applied to the nearest correlation matrix problem

Nicolas F. Armijo, Yunier Bello-Cruz, Gabriel Haeser

Abstract

In this paper, we extend and investigate the properties of the semi-smooth Newton method when applied to a general projection equation in finite dimensional spaces. We first present results concerning Clarke's generalized Jacobian of the projection onto a closed and convex cone. We then describe the iterative process for the general cone case and establish two convergence theorems. We apply these results to the constrained quadratic conic programming problem, emphasizing its connection to the projection equation. To illustrate the performance of our method, we conduct numerical experiments focusing on semidefinite least squares, in particular the nearest correlation matrix problem. In the latter scenario, we benchmark our outcomes against previous literature, presenting performance profiles and tabulated results for clarity and comparison.

A semi-smooth Newton method for general projection equations applied to the nearest correlation matrix problem

Abstract

In this paper, we extend and investigate the properties of the semi-smooth Newton method when applied to a general projection equation in finite dimensional spaces. We first present results concerning Clarke's generalized Jacobian of the projection onto a closed and convex cone. We then describe the iterative process for the general cone case and establish two convergence theorems. We apply these results to the constrained quadratic conic programming problem, emphasizing its connection to the projection equation. To illustrate the performance of our method, we conduct numerical experiments focusing on semidefinite least squares, in particular the nearest correlation matrix problem. In the latter scenario, we benchmark our outcomes against previous literature, presenting performance profiles and tabulated results for clarity and comparison.
Paper Structure (8 sections, 17 theorems, 61 equations, 2 figures, 1 table)

This paper contains 8 sections, 17 theorems, 61 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Notations and preliminaries
  3. On the projection mapping onto a closed and convex cone
  4. A semi-smooth Newton method for general projection equations
  5. Application to quadratic conic programming
  6. The nearest correlation matrix problem
  7. Numerical experiments
  8. Concluding remarks

Key Result

Theorem 1.1

Let $F\colon \mathbb{X} \rightarrow \mathbb{X}$ be a Lipschitz mapping. Then, we have that is, $F(y)-F(x) = U(z)(y-x)$ where $U(z) \in \partial_C F(z)$ and $z$ is a convex combination of $x$ and $y$.

Figures (2)

  • Figure 1: Performance profile for Experiment 5.5 with $\alpha = 0.1$.
  • Figure 2: Performance profiles for Experiment 5.8.

Theorems & Definitions (30)

  • Theorem 1.1: Mean Value Theorem Clarke:1990, Proposition 2.6.5, Page 79
  • Lemma 1.1: Banach's Lemma horn_johnson:2012, Page 351
  • Lemma 1.2: Weyl's inequality horn_johnson:2012, Theorem 4.3.1, Page 239
  • Theorem 1.2: Contraction mapping principle Ortega:1987, Thm. 8.2.2, page 153
  • Theorem 2.1
  • proof
  • Lemma 2.1
  • proof
  • Theorem 3.1: Sufficient condition for existence and uniqueness of a solution
  • proof
  • ...and 20 more