  The class of nonsmooth codifferentiable functions was introduced by professor
V.F.~Demyanov in the late 1980s. He also proposed a method for minimizing these
functions called the method of codifferential descent (MCD). However, until now
almost no theoretical results on the performance of this method on particular
classes of nonsmooth optimization problems were known. In the first part of the
paper, we study the performance of the method of codifferential descent on a
class of nonsmooth convex functions satisfying some regularity assumptions,
which in the smooth case are reduced to the Lipschitz continuity of the
gradient. We prove that in this case the MCD has the iteration complexity bound
$\mathcal{O}(1 / \varepsilon)$. In the second part of the paper we obtain new
global optimality conditions for piecewise affine functions in terms of
codifferentials. With the use of these conditions we propose a modification of
the MCD for minimizing piecewise affine functions (called the method of global
codifferential descent) that does not use line search, and discards those
"pieces" of the objective functions that are no longer useful for the
optimization process. Then we prove that the MCD as well as its modification
proposed in the article find a point of global minimum of a nonconvex piecewise
affine function in a finite number of steps.
