Duality principles in nonconvex systems. Theory, methods and applications
type: Item
instance of: scholarly article
title: Duality principles in nonconvex systems. Theory, methods and applications
zbMATH Open document ID: 0940.49001
author: David Y. Gao
published in: Nonconvex Optimization and Its Applications
publication date: 2000-02-21T00:00:00Z
review text: Duality allows us to associate a dual problem with a variational problem and to study the relationship between the two problems. This is useful in mathematical economics where the dual problem can be stated in terms of the price, in mechanics where the primal and the dual problems are two well-known forms of the conservation principles, characterizing the displacements and the constraints, respectively, in numerics where the dual problem may help us to solve the primal problem. In addition, the dual problem enables us to define the generalized solution of a variational problem which has no classical solution. The present very important and useful book is of mathematics motivated by duality in natural phenomena, with particular emphasis on the mechanics rather than a book of mathematical analysis, proofs and applications. The book is therefore written in a dual way: by using the very simple examples, involving an elastostatic string and the dynamics of particle and then gives a rigorous analysis of the classical mono-duality of one-dimensional convex static equilibria, the nice bi-duality in dynamical systems, through the interesting two-duality in nonconvex problems to the complicated multi-duality in canonical systems. A potentially powerful sequential dual canonical transformation method is developed heuristically and illustrated by use of many interesting simple 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, nonsmooth post-bifurcation, large deformation mechanics, structural limit analysis, differential geometry and nonconvex dynamical systems. The book divides naturally into three closely interconnected parts with a total of seven chapters. Each chapter provides some motivation, both at the beginning and throughout, and concludes with substantial applications and commentaries. Main result: Muc…
Mathematics Subject Classification ID: 49-02
Mathematics Subject Classification ID: 90-02
Mathematics Subject Classification ID: 49N15
Mathematics Subject Classification ID: 90C46
zbMATH DE Number: 1404560
zbMATH Keywords: classical mono-duality of convex static equilibria
zbMATH Keywords: dual problem
zbMATH Keywords: variational problem
zbMATH Keywords: bi-duality in dynamical systems
zbMATH Keywords: constrained global optimization
zbMATH Keywords: multi-well phase transitions
zbMATH Keywords: nonsmooth post-bifurcation
zbMATH Keywords: large deformation mechanics
zbMATH Keywords: structural limit analysis
zbMATH Keywords: differential geometry
zbMATH Keywords: nonconvex dynamical systems
reviewed by: Jan Lovíšek
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1404560