In practical applications of optimization we often get into the situation where
the objective function to be minimized/maximized is not necessarily differentiable.
The source of the nonsmoothness may be the objective function itself, its possible
interior function or both. For example, economics tax models typically consists of
several different elements which at their intersections have discontinuous gradients.
In optimal control problems governed by partial differential systems the
smoothness may be caused by the technological constraints. We may add that there
also exist so called stiff problems which are smooth analytically but nonsmooth
numerically, where the gradient varies too rapidly and the classical methods will
fail. Thus we cannot directly use the methods demanding differential information
while the usual methods that do not require gradients are often very inefficient.
Instead of gradients we must use the so called generalized gradients (or subgra-
dients) which allow us to generalize the effective smooth derivative methods for
nonsmooth problems.
The aim of this book is to introduce various methods for nonsmooth
optimization and to apply these methods to solve discretized nonsmooth optimal control
problems of systems governed by boundary value problems. The material of this
work is divided into three parts, which are organized as follows. The first part
consists of nonsmooth analysis and optimization theory. The material of this part
is planned to be at the introductory level. We concentrate only on the results
relevant for optimization and have simplified the presentation by considering only
functions defined on finite dimensional Euclidean spaces. To give as self-contained
a description as possible, we shall give relatively detailed proofs of all the theorems
and lemmas.