A Finite Algorithm for Solving Infinite Dimensional Optimization Problems

Abstract

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 Y t 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 {P i }, i∈I, of approximating problems of the form min x∈X i c i(x) for each i∈I, where (1) the net of sets {X i } I converges to X in the sense of Kuratowski and (2) the net {c i } I of functions converges to c uniformly on X. We show that for large i, any optimal solution x * i to the approximating problem (P i ) arbitrarily well approximates some optimal solution x * to (P). It follows that if (P) is well-posed, i.e., lim sup X i * is a singleton {x *}, then any net {x i *} I of (P i )-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.