On Semismooth∗ Path-Following Method for Parametric Inclusions

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 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] \longrightarrow \mathbb{R}^{n} $ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>x</mml:mi> <mml:mo>:</mml:mo> <mml:mo>[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> <mml:mo>]</mml:mo> <mml:mo>⟶</mml:mo> <mml:msup> <mml:mi>R</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:math> that satisfies $0 \in F(t, x(t)) $ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>0</mml:mn> <mml:mo>∈</mml:mo> <mml:mi>F</mml:mi> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> <mml:mo>)</mml:mo> </mml:math> for each $t \in [0, T] $ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>t</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> <mml:mo>]</mml:mo> </mml:math> , where $F $ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>F</mml:mi> </mml:math> is a set-valued mapping from $\mathbb{R} \times \mathbb{R}^{n} $ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>R</mml:mi> <mml:mo>×</mml:mo> <mml:msup> <mml:mi>R</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:math> to $\mathbb{R}^{n} $ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>R</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:math> . 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 $o(h)$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>o</mml:mi> <mml:mo>(</mml:mo> <mml:mi>h</mml:mi> <mml:mo>)</mml:mo> </mml:math> 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.