  We propose a single time-scale stochastic subgradient method for constrained
optimization of a composition of several nonsmooth and nonconvex functions. The
functions are assumed to be locally Lipschitz and differentiable in a
generalized sense. Only stochastic estimates of the values and generalized
derivatives of the functions are used. The method is parameter-free. We prove
convergence with probability one of the method, by associating with it a system
of differential inclusions and devising a nondifferentiable Lyapunov function
for this system. For problems with functions having Lipschitz continuous
derivatives, the method finds a point satisfying an optimality measure with
error of order $1/\sqrt{N}$, after executing $N$ iterations with constant
stepsize.
