Infinite Programming

1. Front Matter

   Pages N2-XIV

   PDF

2. Openness, Closedness and Duality in Banach Spaces with Applications to Continuous Linear Programming

   • J. Ch. Pomerol

   Pages 1-15

3. Conditions for the Closedness of the Characteristic Cone Associated with an Infinite Linear System

   • M. A. Goberna, M. A. López

   Pages 16-28

4. Symmetric Duality: A Prelude

   • D. F. Karney

   Pages 29-36

5. Algebraic fundamentals of linear programming

   • Peter Nash

   Pages 37-52

6. On Regular Semi-Infinite Optimization

   • H. Th. Jongen, G. Zwier

   Pages 53-64

7. Semi-Infinite Programming and Continuum Physics

   • K. O. Kortanek

   Pages 65-78

8. On the computation of membrane-eigenvalues by semi-infinite programming methods

   • R. Hettich

   Pages 79-89

9. Lagrangian Methods for Semi-Infinite Programming Problems

   • G. A. Watson

   Pages 90-107

10. A New Primal Algorithm for Semi-Infinite Linear Programming

    • E. J. Anderson

    Pages 108-122

11. Extreme Points and Purification Algorithms in General Linear Programming

    • A. S. Lewis

    Pages 123-135

12. Network Programming in Continuous Time with Node Storage

    • A. B. Philpott

    Pages 136-153

13. The Theorem of Gale for Infinite Networks and Applications

    • Michael M. Neumann

    Pages 154-171

14. Nonlinear Optimal Control Problems as Infinite-Dimensional Linear Programming Problems

    • J. E. Rubio

    Pages 172-184

15. Continuity and Asymptotic Behaviour of the Marginal Function in Optimal Control

    • A. L. Dontchev

    Pages 185-193

16. Alternative Theorems for General Complementarity Problems

    • Jonathan M. Borwein

    Pages 194-203

17. Nonsmooth Analysis and Optimization for a Class of Nonconvex Mappings

    • Thomas W. Reiland, J. H. Chou

    Pages 204-218

18. Minimum Norm Problems in Normed Vector Lattices

    • H. Komiya

    Pages 219-225

19. “Stochastic Nonsmooth Analysis And Optimization In Banach Spaces”

    • Nikolaos S. Papageoraiou

    Pages 226-242

20. Back Matter

    Pages 243-246

    PDF

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.

• Programming
• duality
• economics
• inequality
• optimization

• Management Studies Group Engineering Department, Cambridge University, Mill Lane, Cambridge, UK

  Edward J. Anderson, Andrew B. Philpott

Policies and ethics