On Semismooth∗ Path-Following Method for Parametric Inclusions

This paper investigates a path-following method inspired by the semismooth ∗ approach for solving algebraic inclusions, with a primary emphasis on the role of strong subregularity and its crucial role in ensuring the robustness and stability of the path-following method, 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: [0, T] Rⁿ x : [ 0 , T ] ⟶ R n that satisfies 0 ∈ F(t, x(t)) 0 ∈ F ( t , x ( t ) ) for each t ∈ [0, T] t ∈ [ 0 , T ] , where F F is a set-valued mapping from R × Rⁿ R × R n to Rⁿ R n . We consider an approach based on mappings exhibiting uniform semismooth ∗ properties along continuous trajectories, thereby maintaining a uniform grid error throughout the interval. Error estimates demonstrate an $o(h)$ o ( h ) grid error with respect to the discretization parameter under natural regularity assumptions. Numerical experiments applied to electric circuit and elastoplastic models confirm the efficiency and robustness of the method.