# Infinite Programming

## Proceedings of an International Symposium on Infinite Dimensional Linear Programming Churchill College, Cambridge, United Kingdom, September 7–10, 1984

• Edward J. Anderson
• Andrew B. Philpott
Conference proceedings

Part of the Lecture Notes in Economics and Mathematical Systems book series (LNE, volume 259)

1. Front Matter
Pages N2-XIV
2. M. A. Goberna, M. A. López
Pages 16-28
3. D. F. Karney
Pages 29-36
4. Peter Nash
Pages 37-52
5. H. Th. Jongen, G. Zwier
Pages 53-64
6. K. O. Kortanek
Pages 65-78
7. R. Hettich
Pages 79-89
8. G. A. Watson
Pages 90-107
9. E. J. Anderson
Pages 108-122
10. A. S. Lewis
Pages 123-135
11. A. B. Philpott
Pages 136-153
12. Michael M. Neumann
Pages 154-171
13. J. E. Rubio
Pages 172-184
14. A. L. Dontchev
Pages 185-193
15. Jonathan M. Borwein
Pages 194-203
16. Thomas W. Reiland, J. H. Chou
Pages 204-218
17. H. Komiya
Pages 219-225
18. Nikolaos S. Papageoraiou
Pages 226-242
19. Back Matter
Pages 243-246

### Introduction

Infinite programming may be defined as the study of mathematical programming problems in which the number of variables and the number of constraints are both possibly infinite. Many optimization problems in engineering, operations research, and economics have natural formul- ions as infinite programs. For example, the problem of Chebyshev approximation can be posed as a linear program with an infinite number of constraints. Formally, given continuous functions f,gl,g2, ••• ,gn on the interval [a,b], we can find the linear combination of the functions gl,g2, ... ,gn which is the best uniform approximation to f by choosing real numbers a,xl,x2, •.. ,x to n minimize a t€ [a,b]. This is an example of a semi-infinite program; the number of variables is finite and the number of constraints is infinite. An example of an infinite program in which the number of constraints and the number of variables are both infinite, is the well-known continuous linear program which can be formulated as follows. T minimize ~ c(t)Tx(t)dt t b(t) , subject to Bx(t) + fo Kx(s)ds x(t) .. 0, t € [0, T] • If x is regarded as a member of some infinite-dimensional vector space of functions, then this problem is a linear program posed over that space. Observe that if the constraint equations are differentiated, then this problem takes the form of a linear optimal control problem with state IV variable inequality constraints.

### Keywords

Programming duality economics inequality optimization

### Editors and affiliations

• Edward J. Anderson
• 1
• Andrew B. Philpott
• 1
1. 1.Management Studies Group Engineering DepartmentCambridge UniversityMill Lane, CambridgeUK

### Bibliographic information

• DOI https://doi.org/10.1007/978-3-642-46564-2
• Copyright Information Springer-Verlag Berlin Heidelberg 1985
• Publisher Name Springer, Berlin, Heidelberg
• eBook Packages
• Print ISBN 978-3-540-15996-4
• Online ISBN 978-3-642-46564-2
• Series Print ISSN 0075-8442
• Buy this book on publisher's site
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