Numerical Sensitivity Analysis for the Quantity of Interest in PDE-Constrained Optimization

PDE-constrained optimization problems involving inequality constraints for the design variables are considered. The optimization problems and hence their solutions are subject to perturbations in the data. An output functional (quantity of interest) is given which depends on both the optimal state and design variables. Conditions are derived such that the quantity of interest at the optimal solution is once and twice differentiable with respect to the perturbation parameters. A procedure is devised for the efficient evaluation of these derivatives. Numerical examples are given.