Risk-averse optimal control of random elliptic variational inequalities

¹ Weierstrass Institute, Mohrenstrasse 39, 10117 Berlin, Germany
² Fachbereich Mathematik, MIN Fakultät, Universität Hamburg, Bundesstrrasse 55, 20146, Hamburg, Germany
³ Department of Scientific Computing and Numerical Analysis, Simula Research Laboratory, Kristian Augusts gate 23, 0164 Oslo, Norway

^* Corresponding author: caroline.geiersbach@uni-hamburg.de

Received: 10 October 2022
Accepted: 9 May 2025

Abstract

We consider a risk-averse optimal control problem governed by an elliptic variational inequality (VI) subject to random inputs. By deriving KKT-type optimality conditions for a penalised and smoothed problem and studying convergence of the stationary points with respect to the penalisation parameter, we obtain two forms of stationarity conditions. The lack of regularity with respect to the uncertain parameters and complexities induced by the presence of the risk measure give rise to new challenges unique to the stochastic setting. We also propose a path-following stochastic approximation algorithm using variance reduction techniques and demonstrate the algorithm on a modified benchmark problem.