Generalized Convexity and Optimization

Generalized Convexity and Optimization

4.1 Introduction In this chapter, the role of generalized convexity in Optimization is stressed. After presenting the Fritz John and Karush–Kuhn–Tucker necessary optimality conditions, which are proven by means of separation theorems, ...

Author: Alberto Cambini

Publisher: Springer Science & Business Media

ISBN: 9783540708766

Category: Mathematics

Page: 248

View: 260

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.
Categories: Mathematics

Generalized Convexity and Vector Optimization

Generalized Convexity and Vector Optimization

These conditions are more general than those of existing ones in the literature. ... relationships between vector variational inequalities and vector optimization problems under some convexity or generalized convexity assumptions.

Author: Shashi K. Mishra

Publisher: Springer Science & Business Media

ISBN: 9783540856719

Category: Mathematics

Page: 294

View: 793

The present lecture note is dedicated to the study of the optimality conditions and the duality results for nonlinear vector optimization problems, in ?nite and in?nite dimensions. The problems include are nonlinear vector optimization problems, s- metric dual problems, continuous-time vector optimization problems, relationships between vector optimization and variational inequality problems. Nonlinear vector optimization problems arise in several contexts such as in the building and interpretation of economic models; the study of various technolo- cal processes; the development of optimal choices in ?nance; management science; production processes; transportation problems and statistical decisions, etc. In preparing this lecture note a special effort has been made to obtain a se- contained treatment of the subjects; so we hope that this may be a suitable source for a beginner in this fast growing area of research, a semester graduate course in nonlinear programing, and a good reference book. This book may be useful to theoretical economists, engineers, and applied researchers involved in this area of active research. The lecture note is divided into eight chapters: Chapter 1 brie?y deals with the notion of nonlinear programing problems with basic notations and preliminaries. Chapter 2 deals with various concepts of convex sets, convex functions, invex set, invex functions, quasiinvex functions, pseudoinvex functions, type I and generalized type I functions, V-invex functions, and univex functions.
Categories: Mathematics

Generalized Convexity Generalized Monotonicity and Applications

Generalized Convexity  Generalized Monotonicity and Applications

Proceedings of the 7th International Symposium on Generalized Convexity and Generalized Monotonicity Andrew Eberhard, Nicolas Hadjisavvas, D.T. Luc. However, convexity or reverse convexity is not always the natural property to be ...

Author: Andrew Eberhard

Publisher: Springer Science & Business Media

ISBN: 9780387236391

Category: Business & Economics

Page: 350

View: 725

In recent years there is a growing interest in generalized convex fu- tions and generalized monotone mappings among the researchers of - plied mathematics and other sciences. This is due to the fact that mathematical models with these functions are more suitable to describe problems of the real world than models using conventional convex and monotone functions. Generalized convexity and monotonicity are now considered as an independent branch of applied mathematics with a wide range of applications in mechanics, economics, engineering, finance and many others. The present volume contains 20 full length papers which reflect c- rent theoretical studies of generalized convexity and monotonicity, and numerous applications in optimization, variational inequalities, equil- rium problems etc. All these papers were refereed and carefully selected from invited talks and contributed talks that were presented at the 7th International Symposium on Generalized Convexity/Monotonicity held in Hanoi, Vietnam, August 27-31, 2002. This series of Symposia is or- nized by the Working Group on Generalized Convexity (WGGC) every 3 years and aims to promote and disseminate research on the field. The WGGC (http://www.genconv.org) consists of more than 300 researchers coming from 36 countries.
Categories: Business & Economics

Generalized Convexity Generalized Monotonicity Recent Results

Generalized Convexity  Generalized Monotonicity  Recent Results

Just as convex functions are characterized by a monotone gradient (subdifferential in the nonsmooth case), different kinds of generalized convex functions [1] give rise to gradientmaps (subdifferentials) with certain generalized ...

Author: Jean-Pierre Crouzeix

Publisher: Springer Science & Business Media

ISBN: 9781461333418

Category: Mathematics

Page: 471

View: 591

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.
Categories: Mathematics

Generalized Convexity

Generalized Convexity

Proceedings of the IVth International Workshop on Generalized Convexity Held at Janus Pannonius University Pécs, Hungary, August 31–September 2, 1992 Sandor Komlosi, Tamas Rapcsak, Siegfried Schaible. Now consider the linear map G(z) ...

Author: Sandor Komlosi

Publisher: Springer Science & Business Media

ISBN: 9783642468025

Category: Business & Economics

Page: 404

View: 705

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.
Categories: Business & Economics

Handbook of Generalized Convexity and Generalized Monotonicity

Handbook of Generalized Convexity and Generalized Monotonicity

A well-known feature of convexity is that it is closely related to monotonicity: a differentiable function is convex if and only ... Similar connections were discovered between generalized convex functions and certain classes of maps, ...

Author: Nicolas Hadjisavvas

Publisher: Springer Science & Business Media

ISBN: 9780387233932

Category: Mathematics

Page: 672

View: 818

Studies in generalized convexity and generalized monotonicity have significantly increased during the last two decades. Researchers with very diverse backgrounds such as mathematical programming, optimization theory, convex analysis, nonlinear analysis, nonsmooth analysis, linear algebra, probability theory, variational inequalities, game theory, economic theory, engineering, management science, equilibrium analysis, for example are attracted to this fast growing field of study. Such enormous research activity is partially due to the discovery of a rich, elegant and deep theory which provides a basis for interesting existing and potential applications in different disciplines. The handbook offers an advanced and broad overview of the current state of the field. It contains fourteen chapters written by the leading experts on the respective subject; eight on generalized convexity and the remaining six on generalized monotonicity.
Categories: Mathematics

Generalized Convexity and Related Topics

Generalized Convexity and Related Topics

Intensive study of generalized convex objects began about three decades ago when the theory of convex analysis nearly reached its perfect stage of development with the pioneering contributions of Fenchel, Moreau, Rockafellar and others.

Author: Igor V. Konnov

Publisher: Springer Science & Business Media

ISBN: 9783540370079

Category: Business & Economics

Page: 472

View: 596

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.
Categories: Business & Economics

Generalized Convexity and Generalized Monotonicity

Generalized Convexity and Generalized Monotonicity

Proceedings of the 6th International Symposium on Generalized Convexity/Monotonicity, Samos, September 1999 Nicolas Hadjisavvas, Juan E. Martinez-Legaz, Jean-Paul Penot. Recently, these classes of s-increasing convex functions have been ...

Author: Nicolas Hadjisavvas

Publisher: Springer Science & Business Media

ISBN: 9783642566455

Category: Mathematics

Page: 410

View: 833

Various generalizations of convex functions have been introduced in areas such as mathematical programming, economics, management science, engineering, stochastics and applied sciences, for example. Such functions preserve one or more properties of convex functions and give rise to models which are more adaptable to real-world situations than convex models. Similarly, generalizations of monotone maps have been studied recently. A growing literature of this interdisciplinary field has appeared, and a large number of international meetings are entirely devoted or include clusters on generalized convexity and generalized monotonicity. The present book contains a selection of refereed papers presented at the 6th International Symposium on Generalized Convexity/Monotonicity, and aims to review the latest developments in the field.
Categories: Mathematics

Generalized Convexity Nonsmooth Variational Inequalities and Nonsmooth Optimization

Generalized Convexity  Nonsmooth Variational Inequalities  and Nonsmooth Optimization

4.1 Introduction A well-known aspect of a differentiable convex function is that it is closely related to monotonicity. It is known that a differentiable function is convex if and only if its gradient is a monotone map. Generalized ...

Author: Qamrul Hasan Ansari

Publisher: CRC Press

ISBN: 9781439868201

Category: Business & Economics

Page: 298

View: 290

Until now, no book addressed convexity, monotonicity, and variational inequalities together. Generalized Convexity, Nonsmooth Variational Inequalities, and Nonsmooth Optimization covers all three topics, including new variational inequality problems defined by a bifunction. The first part of the book focuses on generalized convexity and generalized monotonicity. The authors investigate convexity and generalized convexity for both the differentiable and nondifferentiable case. For the nondifferentiable case, they introduce the concepts in terms of a bifunction and the Clarke subdifferential. The second part offers insight into variational inequalities and optimization problems in smooth as well as nonsmooth settings. The book discusses existence and uniqueness criteria for a variational inequality, the gap function associated with it, and numerical methods to solve it. It also examines characterizations of a solution set of an optimization problem and explores variational inequalities defined by a bifunction and set-valued version given in terms of the Clarke subdifferential. Integrating results on convexity, monotonicity, and variational inequalities into one unified source, this book deepens your understanding of various classes of problems, such as systems of nonlinear equations, optimization problems, complementarity problems, and fixed-point problems. The book shows how variational inequality theory not only serves as a tool for formulating a variety of equilibrium problems, but also provides algorithms for computational purposes.
Categories: Business & Economics

Generalized Convexity and Fractional Programming with Economic Applications

Generalized Convexity and Fractional Programming with Economic Applications

Proceedings of the International Workshop on “Generalized Concavity, Fractional Programming and Economic Applications” Held at the University of Pisa, Italy, May 30 – June 1, 1988 Alberto Cambini, Erio Castagnoli, Laura Martein, ...

Author: Alberto Cambini

Publisher: Springer Science & Business Media

ISBN: 9783642467097

Category: Mathematics

Page: 361

View: 588

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.
Categories: Mathematics