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Linear programming in infinite-dimensional spaces. Theory and applications. (English) Zbl 0632.90038
Wiley-Interscience Series in Discrete Mathematics and Optimization. A Wiley-Interscience Publication. Chichester etc.: John Wiley & Sons. XI, 172 p.; £22.95 (1987).
Linear programming (LP) is at the heart of mathematical programming. For the finite-dimensional and the semi-infinite case both theory and applications are well-established. At the time being, for general infinite-dimensional LP the theory is far from being complete. In the first part of the book, an attempt is made to survey results on the general case when both the number of the variables as well as the number of the constraints are infinite. After a set of motivating examples, linear programs are discussed in the algebraic setting of linear vector spaces including basic concepts of the simplex method. Adding topological structure results in a richer duality theory. Difficulties appearing in the general infinite case are discussed under various conditions. In the second part of the book the general results are applied to more concrete linear programs: semi-infinite programming, uniform approximation, continuous transportation, dynamic networks, and a special linear optimal control problem (continuous LP). A short survey of some other infinite linear programs and bibliography of about 180 titles is added. Each chapter ends with some notes on related literature and with some exercises. The book keeps a fair balance between general theory and more concrete problems which stimulated research in that area. With some background in finite-dimensional optimization and functional analysis the text is suitable for graduate level courses.
Reviewer: U.Zimmermann

MSC:
90C05 Linear programming
90C34 Semi-infinite programming
90C08 Special problems of linear programming (transportation, multi-index, data envelopment analysis, etc.)
93C05 Linear systems in control theory
90-02 Research exposition (monographs, survey articles) pertaining to operations research and mathematical programming
90C48 Programming in abstract spaces
65K10 Numerical optimization and variational techniques