A Finite Algorithm for Solving Infinite Dimensional Optimization Problems

We consider the general optimization problem (P ) of selecting a continuous function x over a σ -compact Hausdorff space T to a metric space A, from a feasible region X of such functions, so as to minimize a functional c on X. We require that X consist of a closed equicontinuous family of functions lying in the product (over T ) of compact subsets Yt of A. (An important special case is the optimal control problem of finding a continuous time control function x that minimizes its associated discounted cost c(x) over the infinite horizon.) Relative to the uniform-on-compacta topology on the function space C(T, A) of continuous functions from T to A, the feasible region X is compact. Thus optimal solutions x∗ to (P ) exist under the assumption that c is continuous. We wish to approximate such an x∗ by optimal solutions to a net {Pi}, i ∈ I , of approximating problems of the form minx∈Xi ci (x) for each i ∈ I , where (1) the net of sets {Xi}I converges to X in the sense of Kuratowski and (2) the net {ci}I of functions converges to c uniformly on X. We show that for large i, any optimal solution x∗ i to the approximating problem (Pi) arbitrarily well approximates some optimal solution x∗ to (P ). It follows that if (P ) is well-posed, i.e., lim supX∗ i is a singleton {x∗}, then any net {x∗ i }I of (Pi )-optimal solutions converges in C(T, A) to x∗. For this case, we construct a finite algorithm with the following property: given any prespecified error δ and any compact subset Q of T , our algorithm computes an i in I and an associated x∗ i in X∗ i which is within δ of x∗ on Q. We illustrate the theory and algorithm with a problem in continuous time production control over an infinite horizon.