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Duality Principles in Nonconvex Systems : Theory, Methods and Applications :  Theory, Methods and Applications - David Yang Gao

Duality Principles in Nonconvex Systems : Theory, Methods and Applications

Theory, Methods and Applications

Hardcover Published: January 2000
ISBN: 9780792361459
Number Of Pages: 454

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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.

Preface
Acknowledgments
Symmetry in Convex Systems
Mono-Duality in Static Systemsp. 3
The First Problem in The Calculus of Variationsp. 4
Fundamental Lemma and Euler Equationp. 12
Linear Operators and Bilinear Formsp. 20
Legendre Transformation and Dualityp. 27
Alternative Variational Problems: Lagrange Equations and Multipliersp. 33
Saddle Lagrange Duality Theoryp. 38
Bi-Duality in Dynamical Systemsp. 59
Particle Dynamics: Newton and Einsteinp. 60
Convex Hamiltonian Systemsp. 63
Least Action Principle: Legendre and Jacobian Conditionp. 65
Initial-Value Problems and Dissipative Hamiltonian Systemsp. 69
Complementary Hamiltonian Principles and Euler-Lagrange Equationsp. 74
Super-Lagrangian Dualityp. 81
Symmetry Breaking: Triality Theory in Nonconvex Systems
Tri-Duality in Nonconvex Systemsp. 99
Constitutive Symmetry Breaking in Convex Systemsp. 100
Geometrical Symmetry Breaking: Framework in Nonconvex Systemsp. 110
Quadratic Canonical Transformation and the Gap Functionalp. 121
Complementary Energy Variational Principle and Analytic Solutionsp. 127
Tri-Extremum Principles and Triality Theoryp. 136
Canonical Dual Transformation for Nonconvex Dynamical Systemsp. 146
Multi-Duality and Classifications of General Systemsp. 167
The First Type of Sequential Canonical Dual Transformationp. 168
The Second Type of Sequential Canonical Dual Transformationp. 173
Canonical Systems: The Classificationsp. 179
Generalized Tri-Duality Principlesp. 191
Framework for Geometrically Linear Canonical Systemsp. 194
Framework for Boundary-Value Problemsp. 206
Nonlinear Systems and Commentaryp. 212
Duality in Canonical Systems
Duality in Geometrically Linear Systemsp. 219
Extended Variational Problems and Fenchel Duality Theoryp. 220
Perturbation and Rockafellar Duality Theoryp. 243
Extended Lagrange Duality Theoryp. 252
Hamilton and Clarke Duality Theoriesp. 261
Duality in Variational Inequality and Complementarity Problemsp. 268
Duality in Finite Deformation Systemsp. 283
Finite Deformation Theoryp. 284
Primal, Dual and Polar Variational Problemsp. 295
Canonical Strain Measures and Complementary Gap Functionalp. 303
Tri-Duality Theory in Finite Deformation Problemsp. 314
Minimal Hyper-Surface Problemsp. 324
Applications, Open Problems and Concluding Remarksp. 347
Constitutive Nonlinearity: Plastic Limit Analysisp. 348
Contact Problems of Extended Elastoplastic Beam Theoryp. 356
Geometrical Nonlinearity: von Karman Platep. 365
Large Deformation Beam Theoryp. 372
Optimal Shape Designs and Eigenvalue Problemsp. 381
Miscellaneous Open Problemsp. 390
Duality in Linear Analysisp. 401
Linear Operators and Adjointnessp. 416
Nonlinear Operatorsp. 422
Referencesp. 433
Indexp. 449
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9780792361459
ISBN-10: 0792361458
Series: Nonconvex Optimization and Its Applications
Audience: General
Format: Hardcover
Language: English
Number Of Pages: 454
Published: January 2000
Publisher: Springer
Country of Publication: NL
Dimensions (cm): 23.39 x 15.6  x 2.69
Weight (kg): 0.84

Earn 542 Qantas Points
on this Book