  Novel coordinate descent (CD) methods are proposed for minimizing nonconvex
functions consisting of three terms: (i) a continuously differentiable term,
(ii) a simple convex term, and (iii) a concave and continuous term. First, by
extending randomized CD to nonsmooth nonconvex settings, we develop a
coordinate subgradient method that randomly updates block-coordinate variables
by using block composite subgradient mapping. This method converges
asymptotically to critical points with proven sublinear convergence rate for
certain optimality measures. Second, we develop a randomly permuted CD method
with two alternating steps: linearizing the concave part and cycling through
variables. We prove asymptotic convergence to critical points and sublinear
complexity rate for objectives with both smooth and concave parts. Third, we
extend accelerated coordinate descent (ACD) to nonsmooth and nonconvex
optimization to develop a novel randomized proximal DC algorithm whereby we
solve the subproblem inexactly by ACD. Convergence is guaranteed with at most a
few number of ACD iterations for each DC subproblem, and convergence complexity
is established for identification of some approximate critical points. Fourth,
we further develop the third method to minimize certain ill-conditioned
nonconvex functions: weakly convex functions with high Lipschitz constant to
negative curvature ratios. We show that, under specific criteria, the ACD-based
randomized method has superior complexity compared to conventional gradient
methods. Finally, an empirical study on sparsity-inducing learning models
demonstrates that CD methods are superior to gradient-based methods for certain
large-scale problems.
