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Semi-Infinite Programming and Applications

In: Mathematical Programming The State of the Art

Author

Listed:
  • S.-Å. Gustafson

    (Royal Institute of Technology, Department of Numerical Analysis and Computing Science)

  • K. O. Kortanek

    (Carnegie-Mellon University, Department of Mathematics)

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.

Suggested Citation

  • S.-Å. Gustafson & K. O. Kortanek, 1983. "Semi-Infinite Programming and Applications," Springer Books, in: Achim Bachem & Bernhard Korte & Martin Grötschel (ed.), Mathematical Programming The State of the Art, pages 132-157, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-68874-4_7
    DOI: 10.1007/978-3-642-68874-4_7
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