Global minima for optimal control of the obstacle problem
For researchers in PDE-constrained optimization, this offers a practical way to verify global optimality in a class of nonconvex problems, though the approach is incremental.
The paper provides a condition to determine whether a solution of the first-order optimality conditions for a discretized optimal control problem with an obstacle PDE is a global minimum, and shows that this condition transfers to the limit problem under uniform penalization and discretization parameters. Numerical examples with unique global solutions are presented.
An optimal control problem subject to an elliptic obstacle problem is studied. We obtain a numerical approximation of this problem by discretising the PDE obtained via a Moreau--Yosida type penalisation. For the resulting discrete control problem we provide a condition that allows to decide whether a solution of the necessary first order conditions is a global minimum. In addition we show that the corresponding result can be transferred to the limit problem provided that the above condition holds uniformly in the penalisation and discretisation parameters. Numerical examples with unique global solutions are presented.