  This book is devoted to finite-dimensional problems of non-convex non-smooth
optimization and numerical methods for their solution. The problem of
nonconvexity is studied in the book on two main models of nonconvex
dependencies: these are the so-called generalized differentiable functions and
locally Lipschitz functions. Non-smooth functions naturally arise in various
applications. In addition, they often appear in the theory of extremal problems
itself due to the operations of taking the maximum and minimum, decomposition
techniques, exact non-smooth penalties, and duality. The considered models of
nonconvexity are quite general and cover the majority of practically important
optimization problems; they clearly show all the difficulties of non-convex
optimization. The method of studying the generalized differentiable functions
is that for these functions a generalization of the concept of gradient is
introduced, a calculus is constructed, and various properties of nonconvex
problems are studied in terms of generalized gradients. As for numerical
methods, it is possible to extend the theory and algorithms of subgradient
descent of convex optimization to problems with generalized differentiable
functions. Methods for solving Lipschitz problems are characterized by the fact
that the original functions are approximated by smoothed ones and iterative
minimization procedures are applied to them. With this approach, it is possible
to approximate the gradients of smoothed functions by stochastic finite
differences and thus to construct methods without calculating gradients. A
similar approach can be justified in generalized differentiable and Lipschitz
stochastic programming. In these cases, various generalizations of the
classical stochastic approximation and stochastic quasi-gradient method are
obtained for solving constrained nonconvex nonsmooth stochastic programming
problems.
