GENERALIZED CONVEXITY,

GENERALIZED MONOTONICITY

AND APPLICATIONS

Nonconvex Optimization and Its Applications

Volume 77

Managing Editor:

Panos Pardalos

University of Florida, U.S.A.

Advisory Board:

J. R. Birge

University of Michigan, U.S.A.

Ding-Zhu Du

University of Minnesota, U.S.A.

C. A. Floudas

Princeton University, U.S.A.

J. Mockus

Lithuanian Academy of Sciences, Lithuania

H. D. Sherali

Virginia Polytechnic Institute and State University, U.S.A.

G. Stavroulakis

Technical University Braunschweig, Germany

H.Tuy

National Centre for Natural Science and Technology, Vietnam

GENERALIZED CONVEXITY,

GENERALIZED MONOTONICITY

AND APPLICATIONS

Proceedings of the International

Symposium on Generalized Convexity

and Generalized Monotonicity

Edited by

ANDREW EBERHARD

RMIT University, Australia

NICOLAS HADJISAVVAS

University of the Aegean, Greece

DINH THE LUC

University of Avignon, France

Springer

eBook ISBN: 0-387-23639-2

Print ISBN:

-387-23638-4

No part of this eBook may be reproduced or transmitted in any form or by any means, electronic,

mechanical, recording, or otherwise, without written consent from the Publisher

Created in the United States of America

Boston

Visit Springer's eBookstore at: http://ebooks.kluweronline.com

and the Springer Global Website Online at: http://www.springeronline.com

Contents

Preface

Part I

INVITED PAPERS

Algebraic Dynamics of Certain Gamma Function Values

J.M. Borwein and K. Karamanos

(Generalized) Convexity and Discrete Optimization

Rainer E. Burkard

Lipschitzian Stability of Parametric Constraint Systems in Infinite

Dimensions

Boris S. Mordukhovich

Monotonicity in the Framework of Generalized Convexity

Hoang Tuy

Part II CONTRIBUTED PAPERS

On the Contraction and Nonexpansiveness Properties of the Margi-

nal Mappings in Generalized Variational Inequalities Involving

co-Coercive Operators

Pham Ngoc Anh, Le Dung Muu, Van Hien Nguyen and Jean-Jacques Strodiot

A Projection-Type Algorithm for Pseudomonotone Nonlipschitzian

Multivalued Variational Inequalities

T. Q. Bao and P. Q. Khanh

Duality in Multiobjective Optimization Problems with Set Constraints

Riccardo Cambini and Laura Carosi

113

131

GENERALIZED CONVEXITY AND MONOTONICITY

Duality in Fractional Programming Problems with Set Constraints

Riccardo Cambini, Laura Carosi and Siegfried Schaible

On the Pseudoconvexity of the Sum of two Linear Fractional Functions

Alberto Cambini, Laura Martein and Siegfried Schaible

Bonnesen-type Inequalities and Applications

Raouf Chouikha

Characterizing Invex and Related Properties

D. Craven

Minty Variational Inequality and Optimization: Scalar and Vector

Case

Giovanni P. Crespi, Angelo Guerraggio and Matteo Rocca

Second Order Optimality Conditions for Nonsmooth Multiobjective

Optimization Problems

Giovanni

P. Crespi, Davide La Torre and Matteo Rocca

Second Order Subdifferentials Constructed using Integral Convolu-

tions Smoothing

Andrew

Eberhard, Michael Nyblom and Rajalingam Sivakumaran

Applying Global Optimization to a Problem in Short-Term Hy-

drothermal Scheduling

Albert Ferrer

for Nonsmooth Programming on a Hilbert Space

Misha G. Govil and Aparna Mehra

Identification of Hidden Convex Minimization Problems

Duan Li, Zhiyou Wu, Heung Wing Joseph Lee, Xinmin Yang and Liansheng

Zhang

On Vector Quasi-Saddle Points of Set-Valued Maps

Lai-Jiu Lin and Yu-Lin Tsai

New Generalized Invexity for Duality in Multiobjective Program-

ming Problems Involving N-Set Functions

147

161

173

183

193

213

229

263

287

299

311

321

Contents

S.K. Mishra, S.Y. Wang, K.K. Lai and J. Shi

Equilibrium Prices and Quasiconvex Duality

Phan Thien Thach

vii

341

Preface

In recent years there is a growing interest in generalized convex func-

tions and generalized monotone mappings among the researchers of ap-

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 cur-

rent theoretical studies of generalized convexity and monotonicity, and

numerous applications in optimization, variational inequalities, equilib-

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 orga-

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.

Taking this opportunity, we want to thank all speakers whose contri-

butions make up this volume, all referees whose cooperation helped in en-

suring the scientific quality of the papers, and all people from the Hanoi

Institute of Mathematics whose assistance was indispensable in running

the symposium. Our special thanks go to the Vietnam Academy of

Sciences and Technology, the Vietnam National Basic Research Project

“Selected problems of optimization and scientific computing” and the

Abdus Salam International Center for Theoretical Physics at Trieste,

Italy, for their generous support which made the meeting possible. Fi-

nally, we express our appreciation to Kluwer Academic Publishers for

including this volume into their series. We hope that the volume will

GENERALIZED CONVEXITY AND MONOTONICITY

be useful for students, researchers and those who are interested in this

emerging field of applied mathematics.

ANDREW EBERHARD

NICOLAS HADJISAVVAS

DINH THE LUC