This book is aimed at the approximation of set-valued functions with compact sets in an Euclidean space as values. The interest in set-valued functions is rather new. Such functions arise in various modern areas such as control theory, dynamical systems and optimization. The authors' motivation also comes from the newer field of geometric modeling, in particular from the problem of reconstruction of 3D objects from 2D cross-sections. This is reflected in the focus of this book, which is the approximation of set-valued functions with general (not necessarily convex) sets as values, while previous results on this topic are mainly confined to the convex case. The approach taken in this book is to adapt classical approximation operators and to provide error estimates in terms of the regularity properties of the approximated set-valued functions. Specialized results are given for functions with 1D sets as values.

This book is aimed at the approximation of set-valued functions with compact sets in an Euclidean space as values. The interest in set-valued functions is rather new. Such functions arise in various modern areas such as control theory, dynamical systems and optimization. The authors' motivation also comes from the newer field of geometric modeling, in particular from the problem of reconstruction of 3D objects from 2D cross-sections. This is reflected in the focus of this book, which is the approximation of set-valued functions with general (not necessarily convex) sets as values, while previous results on this topic are mainly confined to the convex case. The approach taken in this book is to adapt classical approximation operators and to provide error estimates in terms of the regularity properties of the approximated set-valued functions. Specialized results are given for functions with 1D sets as values.

This book provides a comprehensive overview of the authors' pioneering contributions to nonlinear set-valued analysis by topological methods. The coverage includes fixed point theory, degree theory, the KKM principle, variational inequality theory, the Nash equilibrium point in mathematical economics, the Pareto optimum in optimization, and applications to best approximation theory, partial equations and boundary value problems.Self-contained and unified in presentation, the book considers the existence of equilibrium points of abstract economics in topological vector spaces from the viewpoint of Ky Fan minimax inequalities. It also provides the latest developments in KKM theory and degree theory for nonlinear set-valued mappings.

A collection of 28 refereed papers grouped according to four broad topics: duality and optimality conditions, optimization algorithms, optimal control, and variational inequality and equilibrium problems. Suitable for researchers, practitioners and postgrads.

From its origins in the minimization of integral functionals, the notion of variations has evolved greatly in connection with applications in optimization, equilibrium, and control. This book develops a unified framework and provides a detailed exposition of variational geometry and subdifferential calculus in their current forms beyond classical and convex analysis. Also covered are set-convergence, set-valued mappings, epi-convergence, duality, and normal integrands.

Optimization, simulation and control play an increasingly important role in science and industry. Because of their numerous applications in various disciplines, research in these areas is accelerating at a rapid pace. This volume brings together the latest developments in these areas of research as well as presents applications of these results to a wide range of real-world problems. The book is composed of invited contributions by experts from around the world who work to develop and apply new optimization, simulation and control techniques either at a theoretical level or in practice. Some key topics presented include: equilibrium problems, multi-objective optimization, variational inequalities, stochastic processes, numerical analysis, optimization in signal processing, and various other interdisciplinary applications. This volume can serve as a useful resource for researchers, practitioners, and advanced graduate students of mathematics and engineering working in research areas where results in optimization, simulation and control can be applied.

Comprehensive and state-of-the art study of the basic concepts and principles of variational analysis and generalized differentiation in both finite-dimensional and infinite-dimensional spaces Presents numerous applications to problems in the optimization, equilibria, stability and sensitivity, control theory, economics, mechanics, etc.