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Generalized Convexity, Generalized Monotonicity: Recent Results

  • Jean-Pierre Crouzeix
  • Juan-Enrique Martinez-Legaz
  • Michel Volle

Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 27)

Table of contents

  1. Front Matter
    Pages i-xv
  2. Generalized Convexity

    1. Front Matter
      Pages 1-1
    2. Adrian S. Lewis, Jong-Shi Pang
      Pages 75-110
    3. Andrew Eberhard, Michael Nyblom, Danny Ralph
      Pages 111-157
    4. A. M. Rubinov, B. M. Glover
      Pages 159-183
    5. A. Jourani, M. Théra
      Pages 185-198
    6. Lucia Blaga, Liana Lupşa
      Pages 199-208
    7. Marco Castellani, Massimo Pappalardo
      Pages 219-233
  3. Generalized Monotonicity

  4. Optimality Conditions and Duality

    1. Front Matter
      Pages 303-303
    2. Fabián Flores-Bazán, Juan-Enrique Martínez-Legaz
      Pages 305-329
    3. B. Lemaire, M. Volle
      Pages 331-345
    4. Alberto Cambini, Sándor Komlósi, Laura Martein
      Pages 347-356
    5. C. R. Bector, S. Chandra, V. Kumar
      Pages 373-386
  5. Vector Optimization

    1. Front Matter
      Pages 388-388
    2. Giorgio Giorgi, Angelo Guerraggio
      Pages 389-405
    3. A. Beato-Moreno, P. Ruiz-Canales, P.-L. Luque-Calvo, R. Blanquero-Bravo
      Pages 425-438
    4. Riccardo Cambini
      Pages 439-451
    5. Alberto Cambini, Laura Martein
      Pages 453-467
  6. Back Matter
    Pages 469-470

About this book

Introduction

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.

Keywords

complementarity derivatives duality equilibrium inequality Mathematica Optimality Conditions optimization probability sets variational inequalities vector optimization

Editors and affiliations

  • Jean-Pierre Crouzeix
    • 1
  • Juan-Enrique Martinez-Legaz
    • 2
  • Michel Volle
    • 3
  1. 1.Université Blaise PascalClermont-FerrandFrance
  2. 2.Universitat Autònoma de BarcelonaBarcelonaSpain
  3. 3.Université d’AvignonAvignonFrance

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4613-3341-8
  • Copyright Information Springer-Verlag US 1998
  • Publisher Name Springer, Boston, MA
  • eBook Packages Springer Book Archive
  • Print ISBN 978-1-4613-3343-2
  • Online ISBN 978-1-4613-3341-8
  • Series Print ISSN 1571-568X
  • Buy this book on publisher's site