  In this paper we consider large-scale composite optimization problems having
the objective function formed as a sum of two terms (possibly nonconvex), one
has (block) coordinate-wise Lipschitz continuous gradient and the other is
differentiable but nonseparable. Under these general settings we derive and
analyze two new coordinate descent methods. The first algorithm, referred to as
coordinate proximal gradient method, considers the composite form of the
objective function, while the other algorithm disregards the composite form of
the objective and uses the partial gradient of the full objective, yielding a
coordinate gradient descent scheme with novel adaptive stepsize rules. We prove
that these new stepsize rules make the coordinate gradient scheme a descent
method, provided that additional assumptions hold for the second term in the
objective function. We present a complete worst-case complexity analysis for
these two new methods in both, convex and nonconvex settings, provided that the
(block) coordinates are chosen random or cyclic. Preliminary numerical results
also confirm the efficiency of our two algorithms on practical problems.
