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Generalized concavity. (English) Zbl 0679.90029
Mathematical Concepts and Methods in Science and Engineering, 36. New York etc.: Plenum Press. x, 332 p. (1988).
The book represents a detailed study of various approaches to the generalization of concave functions, which occupy a significant position in economics, engineering, management science and so on, because of the following two important properties they have: 1) a local maximizer is also a global minimizer, 2) the usual first order necessary conditions over an open set (i.e. the vanishing of the gradient) are also sufficient.
The book is divided into nine chapters. The first chapter has an introductory character. The second chapter reviews the well-kown properties of the concave functions and concave mathematical programming problems.
Chapter 3 deals with various types of generalized concave functions. The properties of the following cases of the generalized concave functions are investigated: quasiconcave, semistrictly quasiconcave and strictly quasiconcave functions, further pseudoconcave, strictly pseudoconcave functions. Necessary and sufficient optimality conditions for mathematical programming problems with these types of generalized concave functions are investigated.
Chapter 4 considers four models of economic behaviour, in which each of the types of generalized concavity studied in the preceding chapters arise in their natural application. Some economic duality theorems are proved and their applications are discussed.
Chapter 5 and 6 provide some hints how to recognize whether a given function of n variables has a generalized concavity property. The characterization of the generalized concavity properties of quadratic functions are studied.
Chapter 7 investigates the generalized concavity concepts in the framework of the fractional programming theory and gives some applications.
Chapter 8 is devoted to the investigation of properties of transconcave functions and (h,\(\phi)\)-concave functions. A function f defined over a convex set C, \(C\subseteq E^ n\), is transconcave if there exists a monotonically increasing function G of one variable such that the function \(h(x)=G(f(x))\) is a concave function over C. It is shown that the class of h(\(\phi)\)-concave functions contains the earlier class of transconcave functions and that if f is a differentiable (h,\(\phi)\)- concave function, then \(\nabla f(x^*)=0\) implies that \(x^*\) is a global maximizer of f.
Chapter 9 provides an introduction to another two classes of generalized concave functions, namely the class of F-concave functions and the class of arcwise connected functions. Various families of arcwise connected functions generalize naturally the classes of generalized concave functions described above by replacing in the characterization of the behaviour of the function the straight line segment by continuous arcs.
The book provides a good review of the present state of the area of generalized concavity.
Reviewer: K.Zimmermann

90-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to operations research and mathematical programming
90C30 Nonlinear programming
26B25 Convexity of real functions of several variables, generalizations
49-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to calculus of variations and optimal control
49M37 Numerical methods based on nonlinear programming
49K10 Optimality conditions for free problems in two or more independent variables
49N15 Duality theory (optimization)
90C32 Fractional programming