2024/10/31 by Tomáš Roubal, Roubal, Tomáš, Jan Valdman +1
Mathematics · #49J52 #49J53 #FOS: Mathematics #Fixed Point Theorems Analysis #Optimization and Control (math.OC)
paper · pdf · doi:10.48550/arxiv.2410.23871
openalex publication_date 2024/10/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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: ℝ \longrightarrow ℝn that satisfies 0 ∈ F(t, x(t)) for each t ∈ [0, T] , where F is a set-valued mapping from ℝ × ℝn to ℝn . 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.