A Finite Algorithm for Solving Infinite Dimensional Optimization Problems

We consider the general optimization problem (P) of selecting a continuous function x over a -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}, iI, of approximating problems of the form min_xXici(x) for each iI, 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., limsupX_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 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 of x^* on Q. We illustrate the theory and algorithm with a problem in continuous time production control over an infinite horizon.