Consistent Approximations and Approximate Functions and Gradients in Optimal Control

Because of the unavoidable use of numerical integration methods, such as Runge--Kutta or finite elements, the numerical solution of optimal control problems, with either ODE or PDE dynamics, is governed by a discretization parameter such as the integration mesh-size. Usually, when explicit integration techniques are used, function and derivative values can be computed exactly for the discretized problems. Recently, we have come across some examples where function and derivative values of the explicitly discretized problems had to be approximated by the outcome of N iterations of a solver. Consequently, the discretization of these problems is controlled by two parameters: the mesh-size and the number of iterations of the solver. Referring to [E. Polak, Optimization: Algorithms and Consistent Approximations, Springer-Verlag, 1997], we find a theory for solving optimization problems that require discretization. It deals with two situations. In the first, which is referred to as that of consistent approximations, it is assumed that an infinite dimensional optimization problem can be suitably approximated by a family of progressively higher dimensional optimization problems. In this case, strategies, in the form of algorithm models, are presented for "diagonalizing" the solution process. In the second situation, it is assumed that numerical solution of the dynamic equations does not result in a family of finite dimensional consistent approximations (e.g., when implicit integration methods are used). For this case, the theory provides models for the implementation of conceptual algorithms. Unfortunately, neither of these situations envisions the possibility of two discretization parameters. In this paper, we present new algorithm models that can be used with two discretization parameters. The first one controls the mesh-size of an explicit integration scheme, and the second one controls the precision with which functions and gradients associated with a fixed mesh-size are computed. The result can be seen as a framework of quasi-consistent approximations. We implemented these new algorithm models using an approximate steepest descent method for the solution of two problems: a two-point boundary value problem in which the discretized linear ODE dynamics are solved approximately using the Gauss--Seidel method and a distributed control problem in which the discretized dynamics are solved using a domain decomposition algorithm which can be implemented on parallelized computers. Our numerical results show that these new algorithms perform quite well and are fairly insensitive to the selection of user-set parameters. Also, they appear to be superior to some alternative, ad hoc schemes.