A Finite Algorithm for Solving Infinite Dimensional Optimization Problems

Irwin Schochetman and Robert Smith

Annals of Operations Research, 2001, vol. 101, issue 1, 119-142

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. Copyright Kluwer Academic Publishers 2001

Keywords: continuous time optimization; optimal control; infinite horizon optimization; production control (search for similar items in EconPapers)
Date: 2001
References: Add references at CitEc
Citations:

Downloads: (external link)
http://hdl.handle.net/10.1023/A:1010964322204 (text/html)
Access to full text is restricted to subscribers.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:annopr:v:101:y:2001:i:1:p:119-142:10.1023/a:1010964322204

Ordering information: This journal article can be ordered from
http://www.springer.com/journal/10479

DOI: 10.1023/A:1010964322204

Access Statistics for this article

Annals of Operations Research is currently edited by Endre Boros

More articles in Annals of Operations Research from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().