A Selective Linearization Method For Multiblock Convex Optimization

We consider the problem of minimizing a sum of several convex nonsmooth functions. We introduce a new algorithm called the selective linearization method, which iteratively linearizes all but one of the functions and employs simple proximal steps. The algorithm is a form of multiple operator splitting in which the order of processing partial functions is not fixed, but rather determined in the course of calculations. Global convergence is proved and estimates of the convergence rate are derived. Specifically, the number of iterations needed to achieve solution accuracy $\varepsilon$ is of order $\mathcal{O}(\ln(1/\varepsilon)/\varepsilon)$.