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Semi-Infinite Programming and Applications

  • Chapter
Mathematical Programming The State of the Art

Abstract

An important list of topics in the physical and social sciences involves continuum concepts and modelling with infinite sets of inequalities in a finite number of variables. Topics include: engineering design, variational inequalities and saddle value problems, nonlinear parabolic and bang-bang control, experimental regression design and the theory of moments, continuous linear programming, geometric programming, sequential decision theory, and fuzzy set theory. As an optimization involving only finitely many variables, semi-infinite programming can be studied with various reductions to finiteness, such as finite subsystems of the infinite inequality system or finite probability measures.

This survey develops the theme of finiteness in three main directions: (1) a duality theory emphasizing a perfect duality and classification analogous to finite linear programming, (2) a numerical treatment emphasizing discretizations, cutting plane methods, and nonlinear systems of duality equations, and (3) separably-infinite programming emphasizing its uniextremal duality as an equivalent to saddle value, biextremal duality. The focus throughout is on the fruitful interaction between continuum concepts and a variety of finite constructs.

This paper is dedicated to Professor Abraham Charnes on his his 65th birthday.

Preparation of this paper was supported in part by National Science Foundation Grant ECS-8209951.

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Authors and Affiliations

  1. Department of Numerical Analysis and Computing Science, Royal Institute of Technology, S-10044, Stockholm 70, Sweden

    S.-Å. Gustafson

  2. Department of Mathematics, Carnegie-Mellon University, Schenley Park, Pittsburgh, PA, 15213, USA

    K. O. Kortanek

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  1. S.-Å. Gustafson
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  1. Institut für Ökonometrie und Operations Research, Universität Bonn, Nassestraße 2, D-5300, Bonn 1, Germany

    Achim Bachem  & Bernhard Korte  & 

  2. Lehrstuhl für Angewandte Mathematik II, Universität Augsburg, Memminger Straße 6, D-8900, Augsburg, Germany

    Martin Grötschel

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Gustafson, SÅ., Kortanek, K.O. (1983). Semi-Infinite Programming and Applications. In: Bachem, A., Korte, B., Grötschel, M. (eds) Mathematical Programming The State of the Art. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-68874-4_7

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  • DOI: https://doi.org/10.1007/978-3-642-68874-4_7

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