Direct optimization methods for solving a complex state-constrained optimal control problem in microeconomics

We analyze and solve a complex optimal control problem in microeconomics which has been investigated earlier in the literature. The complexity of the control problem originates from four control variables appearing linearly in the dynamics and several state inequality constraints. Thus, the control problem offers a considerable challenge to the numerical analyst. We implement a hybrid optimization approach which combines two direct optimization methods. The first step consists in solving the discretized control problem by nonlinear programming methods. The second step is a refinement step where, in addition to the discretized control and state variables, the junction times between bang–bang, singular and boundary subarcs are optimized. The computed solutions are shown to satisfy precisely the necessary optimality conditions of the Maximum Principle where the state constraints are directly adjoined to the Hamiltonian. Despite the complexity of the control structure, we are able to verify sufficient optimality conditions which are based on the concavity of the maximized Hamiltonian.