Efficiency of Coordinate Descent Methods for Structured Nonconvex Optimization

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