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In this paper, we investigate optimal control problems governed by semilinear elliptic variational inequalities involving constraints on the state, and more precisely the obstacle problem. Since we adopt a numerical point of view, we first relax the feasible domain of the problem, then using both mathematical programming methods and penalization methods we get optimality conditions with smooth Lagrange multipliers. Some numerical experiments using the Interior Point Optimizer (IPOPT), Nonlinear Interior point Trust Region Optimization (KNITRO) and Sequential Quadratic Optimization Technique (SNOPT) are presented to verify the efficiency of our approach.

A new relaxation method for optimal control of semilinear elliptic variational inequalities obstacle problems