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Consider the problem of minimizing the sum of a smooth (possibly non-convex)and a convex (possibly nonsmooth) function involving a large number ofvariables. A popular approach to solve this problem is the block coordinatedescent (BCD) method whereby at each iteration only one variable block isupdated while the remaining variables are held fixed. With the recent advancesin the developments of the multi-core parallel processing technology, it isdesirable to parallelize the BCD method by allowing multiple blocks to beupdated simultaneously at each iteration of the algorithm. In this work, wepropose an inexact parallel BCD approach where at each iteration, a subset ofthe variables is updated in parallel by minimizing convex approximations of theoriginal objective function. We investigate the convergence of this parallelBCD method for both randomized and cyclic variable selection rules. We analyzethe asymptotic and non-asymptotic convergence behavior of the algorithm forboth convex and non-convex objective functions. The numerical experimentssuggest that for a special case of Lasso minimization problem, the cyclic blockselection rule can outperform the randomized rule.

Parallel Successive Convex Approximation for Nonsmooth Nonconvex Optimization