Duality Principles in Nonconvex Systems

Theory, Methods and Applications

  • David Yang Gao

Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 39)

Table of contents

  1. Front Matter
    Pages i-xviii
  2. Symmetry in Convex Systems

    1. Front Matter
      Pages 1-1
    2. David Yang Gao
      Pages 3-57
    3. David Yang Gao
      Pages 59-95
  3. Symmetry Breaking: Triality Theory in Nonconvex Systems

    1. Front Matter
      Pages 97-97
    2. David Yang Gao
      Pages 99-166
  4. Duality in Canonical Systems

    1. Front Matter
      Pages 217-217
    2. David Yang Gao
      Pages 219-282
    3. David Yang Gao
      Pages 283-346
  5. Back Matter
    Pages 401-454

About this book


Motivated by practical problems in engineering and physics, drawing on a wide range of applied mathematical disciplines, this book is the first to provide, within a unified framework, a self-contained comprehensive mathematical theory of duality for general non-convex, non-smooth systems, with emphasis on methods and applications in engineering mechanics. Topics covered include the classical (minimax) mono-duality of convex static equilibria, the beautiful bi-duality in dynamical systems, the interesting tri-duality in non-convex problems and the complicated multi-duality in general canonical systems. A potentially powerful sequential canonical dual transformation method for solving fully nonlinear problems is developed heuristically and illustrated by use of many interesting examples as well as extensive applications in a wide variety of nonlinear systems, including differential equations, variational problems and inequalities, constrained global optimization, multi-well phase transitions, non-smooth post-bifurcation, large deformation mechanics, structural limit analysis, differential geometry and non-convex dynamical systems.
With exceptionally coherent and lucid exposition, the work fills a big gap between the mathematical and engineering sciences. It shows how to use formal language and duality methods to model natural phenomena, to construct intrinsic frameworks in different fields and to provide ideas, concepts and powerful methods for solving non-convex, non-smooth problems arising naturally in engineering and science. Much of the book contains material that is new, both in its manner of presentation and in its research development. A self-contained appendix provides some necessary background from elementary functional analysis.
Audience: The book will be a valuable resource for students and researchers in applied mathematics, physics, mechanics and engineering. The whole volume or selected chapters can also be recommended as a text for both senior undergraduate and graduate courses in applied mathematics, mechanics, general engineering science and other areas in which the notions of optimization and variational methods are employed.


Mathematica applied mathematics deformation dynamical systems engineering mechanics functional analysis mechanics optimization

Authors and affiliations

  • David Yang Gao
    • 1
  1. 1.Department of MathematicsVirginia Polytechnic Institute and State UniversityBlacksburgUSA

Bibliographic information

  • DOI
  • Copyright Information Springer-Verlag US 2000
  • Publisher Name Springer, Boston, MA
  • eBook Packages Springer Book Archive
  • Print ISBN 978-1-4419-4825-0
  • Online ISBN 978-1-4757-3176-7
  • Series Print ISSN 1571-568X
  • Buy this book on publisher's site