We propose a decomposition framework for theparallel optimization of the sum of a differentiable (possiblynonconvex) function and a (block) separable nonsmooth, convexone. The latter term is usually employed to enforce structure inthe solution, typically sparsity. Our framework is very flexible andincludes both fully parallel Jacobi schemes and Gauss–Seidel (i.e.,sequential) ones, as well as virtually all possibilities “in between”with only a subset of variables updated at each iteration. Ourtheoretical convergence results improve on existing ones, andnumerical results on LASSO, logistic regression, and some nonconvexquadratic problems show that the new method consistentlyoutperforms existing algorithms.
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