An introduction to nonsmooth analysis (Q2868666)

The main goal of nonsmooth analysis is to extend differentiable tools to the nonsmooth setting. This book is devoted to presenting the theory of the subdifferential of lower semicontinuous functions which is a generalization of the subdifferential of convex functions. Under this scope the book is divided into seven chapters together with useful exercises at the end of each chapter. The author first presents basic concepts and results which have an introductory nature. Since one of the most interesting aspects of the theory developed in the book is both geometrical and analytical, the notions of upper and lower semicontinuity are characterized in terms of epigraphs and hypographs. A short review of differentiable functions is given together with the introduction of the Moreau envelope and the Ekeland variational principle which are the tools for the theory. The author studies some properties of convex functions, the distance function to a convex set, the Moreau envelope of a convex function and the Minkowski separation theorem. The book addresses the subdifferential calculus for convex functions together with the equivalence between the different definitions of the subdifferential. In the next step, the author introduces the notion of subdifferential of a lower semicontinuous function which is a generalization of the subdifferential of convex functions. An important theorem on the equivalence of two natural definitions of the subdifferential via lower limits and differentiable supports is presented. The subdifferential of a function can be characterized geometrically via the regular normal cone of the epigraph. The subdifferential introduced here may be empty, but the set of points where it is nonempty is dense. The author is concerned with the subdifferential calculus. Note that almost all the results presented here are consequences of the powerful fuzzy sum rule. The mean value theorem and the multidirectional mean value theorem are presented. The author also deals with the generalized gradient of a locally Lipschitz function which is defined as the set of supports of the generalized directional derivatives. Since Lipschitz functions are differentiable a. e. in finite dimensions, it can be characterized as the convex hull of the set of limits of gradients. For vector functions the author presents the notion of generalized Jacobian which allows to establish the powerful general Lipschitz inverse function theorem. The author introduces the graphical derivative and the coderivative which have an advantage with respect to the generalized Jacobian. An interesting characterization of generalized Jacobian in terms of the graphical coderivative is presented. Note that nonsmooth analysis has many interesting applications in a wide range of fields, and the book presents some applications to the flow invariant sets of differential equations and the viscosity solutions of a first order Hamilton-Jacobi equation.NEWLINENEWLINEThis is an interesting concise book on nonsmooth analysis. It is suitable for applied mathematics and operations research. It is a good reference for researchers in optimization and applied mathematics.