Regularization for state constrained optimal control problems by half spaces based decoupling

In this paper, we study an abstract constrained optimization problem which subsumes a common model for the optimal control of linear partial differential equations. Our emphasis is the case when the ordering cone for the optimization problem has an empty interior. To circumvent this difficulty, we propose a new regularization approach which is based on the decoupling of the ordering cone by using certain half-spaces. The approach results in a family of regular optimization problems having a simpler structure than the original optimization problem. Existence theorems, convergence analysis, and optimality conditions are given for the regularized problems. We present two numerical examples and the results are quite encouraging, showing the potential of the proposed approach.