Generalized Concavity In Optimization And Economics

Originally published: New York: Plenum Press, 1988.

Concavity plays a central role in mathematical economics, engineering, management science, and optimization theory. The reason is that concavity of functions is used as a hypothesis in most of the important theorems concerning extremum problems. In other words, concavity is usually a sufficient condition for satisfying the underlying assumptions of these theorems, but concavity is definitely not a necessary condition. In fact, there are large families of functions that are nonconcave and yet have properties similar to those of concave functions. Such functions are called generalized concave functions, and this book is about the various generalizations of concavity, mainly in the context of economics and optimization. Although hundreds of articles dealing with generalized concavity have appeared in scientific journals, numerous textbooks have specific chapters on this subject, and scientific meetings devoted to generalized concavity have been held and their proceedings published, this book is the first attempt to present generalized concavity in a unified framework. We have collected results dealing with this subject mainly from the economics and optimization literature, and we hope that the material presented here will be useful in applications and will stimulate further research. The writing of this book constituted a unique experience for the authors in international scientific cooperation-cooperation that extended over many years and at times spanned three continents. It was an extremely fruitful and enjoyable experience, which we will never forget.

Generalizations of convex functions have been used in a variety of fields such as economics. business administration. engineering. statistics and applied sciences.· In 1949 de Finetti introduced one of the fundamental of generalized convex functions characterized by convex level sets which are now known as quasiconvex functions. Since then numerous types of generalized convex functions have been defined in accordance with the need of particular applications.· In each case such functions preserve soine of the valuable properties of a convex function. In addition to generalized convex functions this volume deals with fractional programs. These are constrained optimization problems which in the objective function involve one or several ratios. Such functions are often generalized convex. Fractional programs arise in management science. economics and numerical mathematics for example. In order to promote the circulation and development of research in this field. an international workshop on "Generalized Concavity. Fractional Programming and Economic Applications" was held at the University of Pisa. Italy. May 30 - June 1. 1988. Following conferences on similar topics in Vancouver. Canada in 1980 and in Canton. USA in 1986. it was the first such conference organized in Europe. It brought together 70 scientists from 11 countries. Organizers were Professor A. Cambini. University of Pisa. Professor E. Castagnoli. Bocconi University. Milano. Professor L. Martein. University of Pisa. Professor P. Mazzoleni. University of Verona and Professor S. Schaible. University of California. Riverside.

The book contains invited papers by well-known experts on a wide range of topics (economics, variational analysis, probability etc.) closely related to convexity and generalized convexity, and refereed contributions of specialists from the world on current research on generalized convexity and applications, in particular, to optimization, economics and operations research.

The authors have written a rigorous yet elementary and self-contained book to present, in a unified framework, generalized convex functions. The book also includes numerous exercises and two appendices which list the findings consulted.

Generalizations of the classical concept of a convex function have been proposed in various fields such as economics, management science, engineering, statistics and applied sciences during the second half of this century. In addition to new results in more established areas of generalized convexity, this book presents several important developments in recently emerging areas. Also, a number of interesting applications are reported.