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Convex and Set-Valued Analysis

Aram V. Arutyunov and Valeri Obukhovskii
Walter de Gruyter
Publication Date: 
Number of Pages: 
De Gruyter Graduate
[Reviewed by
Allen Stenger
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This introductory text consists of two largely-independent halves: one on convex sets and convex functions, and one on set-valued functions (that is, each function value is a subset of a set rather than an element of a set; also known as multifunction, set-valued map, or multivalued function).

Both subjects are useful in optimization and in mathematical economics, although the presentation in the present book is very abstract and is weak on examples and applications. One of the attractions of convex analysis is that many of the ideas of differential calculus can be applied more generally to convex functions and semicontinuous functions. Set-valued functions are useful in situations where the values are only known to within some range and not exactly; again it is possible to extend the ideas of differential calculus and draw some useful (although inexact) conclusions.

The book gives linear programming as an example of convex analysis, and proves an existence theorem and a duality theorem about linear programs. It also proves a finite-dimensional version of the Krein–Milman theorem from functional analysis.

One important application area for set-valued analysis is differential inclusions, which are like differential equations except that the derivative at each point belongs to a set rather than being a specific value. The book has a chapter on these, and also proves the existence of a solution for the first-order differential inclusion (analog of Picard’s theorem). The proof is also analogous and depends on a fixed-point theorem.

Another good introduction to convex analysis, that covers generally the same areas but has many more examples, is Borwein & Lewis’s Convex Analysis and Nonlinear Optimization. This book also has some set-valued analysis but is not as in-depth as the present book.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His personal web page is His mathematical interests are number theory and classical analysis.

See the table of contents in the publisher's webpage.