Generalized Concavity

Originally published: New York: Plenum Press, 1988.

Convexity of sets in linear spaces, and concavity and convexity of functions, lie at the root of beautiful theoretical results that are at the same time extremely useful in the analysis and solution of optimization problems, including problems of either single objective or multiple objectives. Not all of these results rely necessarily on convexity and concavity; some of the results can guarantee that each local optimum is also a global optimum, giving these methods broader application to a wider class of problems. Hence, the focus of the first part of the book is concerned with several types of generalized convex sets and generalized concave functions. In addition to their applicability to nonconvex optimization, these convex sets and generalized concave functions are used in the book's second part, where decision-making and optimization problems under uncertainty are investigated. Uncertainty in the problem data often cannot be avoided when dealing with practical problems. Errors occur in real-world data for a host of reasons. However, over the last thirty years, the fuzzy set approach has proved to be useful in these situations. It is this approach to optimization under uncertainty that is extensively used and studied in the second part of this book. Typically, the membership functions of fuzzy sets involved in such problems are neither concave nor convex. They are, however, often quasiconcave or concave in some generalized sense. This opens possibilities for application of results on generalized concavity to fuzzy optimization. Despite this obvious relation, applying the interface of these two areas has been limited to date. It is hoped that the combination of ideas and results from the field of generalized concavity on the one hand and fuzzy optimization on the other hand outlined and discussed in Generalized Concavity in Fuzzy Optimization and Decision Analysis will be of interest to both communities. Our aim is to broaden the classes of problems that the combination of these two areas can satisfactorily address and solve.

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.

This book contains the proceedings of the 4th International Conference on Data Analysis and Processing held in Cefalu' (Palermo, ITALY) on September 23-25 1987. The aim of this Conference, now at its fourth edition, was to give a general view of the actual research in the area of methods and systems for achieving artificial vision as well as to have an up-dated information of the current activity in Europe. A number of invited speakers presented overviews of statistical classification problems and methods, non conventional archi tectures, mathematical morphology, robotic vision, analysis of range images in vision systems, pattern matching algorithms and astronomical data processing. Finally a survey of the discussion on the contribution of AI to Image Analysis is given. The papers presented at the Conference have been subdivided in four sections: knowledge based approaches, basic pattern recognition tools, multi features system based solutions, image analysis-applications. We must thank the IBM-Italia and the Digital Equipment Corpo ration for sponsoring this Conference. We feel that the days spent at Cefalu' were an important step toward the mutual exchange of scientific information within the image processing community. v. Cantoni Pavia University V. Di Gesu' Palermo University S. Levialdi Rome University v CONTENTS INVITED LECTURES . • • • • . • • • 3 Morphological Optics.

A function is convex if its epigraph is convex. This geometrical structure has very strong implications in terms of continuity and differentiability. Separation theorems lead to optimality conditions and duality for convex problems. A function is quasiconvex if its lower level sets are convex. Here again, the geo metrical structure of the level sets implies some continuity and differentiability properties for quasiconvex functions. Optimality conditions and duality can be derived for optimization problems involving such functions as well. Over a period of about fifty years, quasiconvex and other generalized convex functions have been considered in a variety of fields including economies, man agement science, engineering, probability and applied sciences in accordance with the need of particular applications. During the last twenty-five years, an increase of research activities in this field has been witnessed. More recently generalized monotonicity of maps has been studied. It relates to generalized convexity off unctions as monotonicity relates to convexity. Generalized monotonicity plays a role in variational inequality problems, complementarity problems and more generally, in equilibrium prob lems.

This volume includes a selection of refereed papers presented at the GAMM/IFIP-Workshop on "Stochastic Optimization: Numerical Methods and Technical Applications", held at the Federal Armed Forces University Munich, May 29 - 31, 1990. The objective of this meeting was to bring together scientists from Stochastic Programming and from those Engineering areas, where Mathematical Programming models are common tools, as e. g. Optimal Structural Design, Power Dispatch, Acid Rain Management etc. The first, theoretical part includes the papers by S. D. Flam. H. Niederreiter, E. Poechinger and R. Schultz. The second part on methods and applications contains the articles by N. Baba, N. Grwe and W. Roemisch, J. Mayer, E. A. Mc Bean and A. Vasarhelyi.

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.