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A finite algorithm for solving infinite dimensional optimization problems
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DOI：10.1023/A:1010964322204.png)
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Abstract
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We consider the general optimization problem (P) of selecting a continuous function x over a sigma -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(,T) 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 is an element of I, of approximating problems of the form min(x is an element ofX)i c(i)(x) for each i is an element of I, where (1) the net of sets [X-i](I) converges to X in the sense of Kuratowski and (2) the net (c(i)), of functions converges to c uniformly on X. We show that for large i, any opitmal 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 delta 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 delta of x* on Q. We illustrate the theory and algorithm with a problem in continuous time production control over an infinite horizon.
Keywords:
continuous time optimization
optimal control
infinite horizon optimization
production control
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