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On semismooth$^*$ path-following method and uniformity of strong metric subregularity at/around the reference point

Tomáš Roubal, Jan Valdman

Abstract

This paper investigates a path-following method inspired by the semismooth$^*$ approach for solving algebraic inclusions, with a primary emphasis on the role of uniform subregularity. Uniform subregularity is crucial for ensuring the robustness and stability of path-following methods, as it provides a framework to uniformly control the distance between the input and the solution set across a continuous path. We explore the problem of finding a mapping $ x: \mathbb{R} \longrightarrow \mathbb{R}^n $ that satisfies $ 0 \in F(t, x(t)) $ for each $ t \in [0, T] $, where $ F $ is a set-valued mapping from $ \mathbb{R} \times \mathbb{R}^n $ to $ \mathbb{R}^n $. The paper discusses two approaches: the first considers mappings with uniform semismooth$^*$ properties along continuous paths, leading to a consistent grid error throughout the interval, while the second examines mappings exhibiting pointwise semismooth$^*$ properties at individual points along the path. The uniform strong subregularity framework is integrated into these approaches to strengthen the stability of solution trajectories and improve algorithmic convergence.

On semismooth$^*$ path-following method and uniformity of strong metric subregularity at/around the reference point

Abstract

This paper investigates a path-following method inspired by the semismooth approach for solving algebraic inclusions, with a primary emphasis on the role of uniform subregularity. Uniform subregularity is crucial for ensuring the robustness and stability of path-following methods, as it provides a framework to uniformly control the distance between the input and the solution set across a continuous path. We explore the problem of finding a mapping that satisfies for each , where is a set-valued mapping from to . The paper discusses two approaches: the first considers mappings with uniform semismooth properties along continuous paths, leading to a consistent grid error throughout the interval, while the second examines mappings exhibiting pointwise semismooth properties at individual points along the path. The uniform strong subregularity framework is integrated into these approaches to strengthen the stability of solution trajectories and improve algorithmic convergence.

Paper Structure

This paper contains 8 sections, 24 theorems, 103 equations, 1 figure, 4 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Uniform semismoothness$^*$
  4. Uniform strong subregularity
  5. Uniform semismooth* path-following method
  6. Point-wise semismooth* path-following method
  7. Numerical Implementation of Semismooth$^*$ Path-Following Methods
  8. Conclusion

Key Result

Theorem 1.1

Assume that $F$ is semismooth* at $(\bar{x}, 0) \in \text{gph } F$ and assume that there are $\ell, \kappa > 0$ such that for every $x \notin F^{-1}(0)$ sufficiently close to $\bar{x}$ we have $\mathcal{G}^{\ell,\kappa}_{F,\bar{x}}(x) \neq \emptyset$. Then there exists some $\delta > 0$ such that fo

Figures (1)

  • Figure 1: Example \ref{['ex_simple_circuit']} (i) - numerical solution (left) obtained by Algorithm \ref{['algPathFollowingUniform']} and observed rates of convergence (right).

Theorems & Definitions (56)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 3.1
  • Proposition 3.1
  • Lemma 3.1
  • Lemma 3.2
  • ...and 46 more