Duality principles in nonconvex systems. Theory, methods and applications
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: Much of the book contains material that is new, both in its manner of presentation and in its research development. With especially 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 framework in different fields and to provide ideas, concepts and powerful methods for solving nonconvex nonsmooth problems arising naturally in engineering and science.
classical mono-duality of convex static equilibria
scientific article; zbMATH DE number 1404560