  In this paper we analyze several new methods for solving nonconvex
optimization problems with the objective function formed as a sum of two terms:
one is nonconvex and smooth, and another is convex but simple and its structure
is known. Further, we consider both cases: unconstrained and linearly
constrained nonconvex problems. For optimization problems of the above
structure, we propose random coordinate descent algorithms and analyze their
convergence properties. For the general case, when the objective function is
nonconvex and composite we prove asymptotic convergence for the sequences
generated by our algorithms to stationary points and sublinear rate of
convergence in expectation for some optimality measure. Additionally, if the
objective function satisfies an error bound condition we derive a local linear
rate of convergence for the expected values of the objective function. We also
present extensive numerical experiments for evaluating the performance of our
algorithms in comparison with state-of-the-art methods.
