This book on canonical duality theory provides a comprehensive review of its philosophical origin, physics foundation, and mathematical statements in both finite- and infinite-dimensional spaces. A ground-breaking methodological theory, canonical duality theory can be used for modeling complex systems within a unified framework and for solving a large class of challenging problems in multidisciplinary fields in engineering, mathematics, and the sciences. This volume places a particular emphasis on canonical duality theory's role in bridging the gap between non-convex analysis/mechanics and global optimization.
With 18 total chapters written by experts in their fields, this volume provides a nonconventional theory for unified understanding of the fundamental difficulties in large deformation mechanics, bifurcation/chaos in nonlinear science, and the NP-hard problems in global optimization. Additionally, readers will find a unified methodology and powerful algorithms for solving challenging problems in complex systems with real-world applications in non-convex analysis, non-monotone variational inequalities, integer programming, topology optimization, post-buckling of large deformed structures, etc. Researchers and graduate students will find explanation and potential applications in multidisciplinary fields.
David Gao is a the Alex Rubinov chair professor of mathematics in the Faculty of Sciences and Technology at the Federation University Australia. His research interests span a broad range of topics, including nonlinear analysis and mechanics, mathematical modeling, simulation and control of complex systems, global optimization, and information technology. He has published over 10 books and 180 refereed papers and is a founding editor of the series Advances in Mechanics and Mathematics.
Dr. Vittorio Latorre received his PhD in the Department of Computer, Control and Management Engineering at the Sapienza University of Rome. His research interests include global optimization, chaotic dynamics, information technology, and large-scale computation.
Dr. Ning Ruan is a senior research fellow in the Faculty of Sciences and Technology at the Federation University Australia. Her research interests include global optimization, operations research, sensor network, system engineering, applied mathematics, and large scale computation.
Preface.- Canonical Duality-Triality Theory: Bridge Between Nonconvex Analysis/Mechanics and Global Optimization in Complex System.- Analytic Solutions to Large Deformation Problems Governed by Generalized Neo-Hookean Model.- Analytic Solutions to 3-D Finite Deformation Problems Governed by St Venant-Kirchhoff Material.- Remarks on Analytic Solutions and Ellipticity in Anti-Plane Shear Problems of Nonlinear Elasticity.- Canonical Duality Method for Solving Kantorovich Mass Transfer Problem.- Triality Theory for General Unconstrained Global Optimization Problems.- Canonical Duality Theory for Solving Non-Monotone Variational Inequality Problems.- Canonical Dual Approach for Contact Mechanics Problems with Friction.- Canonical Duality Theory for Solving Nonconvex/Discrete Constrained Global Optimization Problems.- On D.C. Optimization Problems.- Canonical Primal-Dual Method for Solving Non-convex Minimization Problems.- Unified Interior Point Methodology for Canonical Duality in Global Optimization.- Canonical Duality Theory for Topology Optimization.- Improved Canonical Dual Finite Element Method and Algorithm for Post Buckling Analysis of Nonlinear Gao Beam.- Global Solutions to Spherically Constrained Quadratic Minimization via Canonical Duality Theory.- Global Optimal Solution to QuadraticDiscrete Programming Problem with Inequality Constraints.- Global Optimal Solution to Quadratic Discrete Programming Problem with Inequality Constraints.- On Minimal Distance Between Two Surfaces.