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

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.