Risk-averse optimal control of random elliptic variational inequalities

TL;DR

This paper develops a risk-averse optimal control framework for elliptic variational inequalities with stochastic inputs, deriving optimality conditions and proposing a stochastic approximation algorithm with variance reduction.

Contribution

It introduces new stationarity conditions for stochastic variational inequalities and presents a novel path-following algorithm tailored for risk-averse control problems.

Findings

Derived KKT-type optimality conditions for penalised problems

Studied convergence of stationary points with respect to penalisation

Demonstrated the algorithm on a modified benchmark problem

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.