The subject of the present book is sub differential calculus. The main
source of this branch of functional analysis is the theory of extremal
problems. For a start, we explicate the origin and statement of the
principal problems of sub differential calculus. To this end, consider
an abstract minimization problem formulated as follows: x E X, f(x) --]
inf. Here X is a vector space and f: X --+ iR is a numeric function
taking possibly infinite values. In these circumstances, we are usually
interested in the quantity inf f( x), the value of the problem, and in a
solution or an optimum plan of the problem (i. e., such an x that f(x) =
inf f(X», if the latter exists. It is a rare occurrence to solve an
arbitrary problem explicitly, i. e. to exhibit the value of the problem
and one of its solutions. In this respect it becomes necessary to
simplify the initial problem by reducing it to somewhat more manageable
modifications formulated with the details of the structure of the
objective function taken in due account. The conventional hypothesis
presumed in attempts at theoretically approaching the reduction sought
is as follows. Introducing an auxiliary function 1, one considers the
next problem: x EX, f(x) -l(x) --+ inf. Furthermore, the new problem is
assumed to be as complicated as the initial prob- lem provided that 1 is
a linear functional over X, i. e.