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In the paper, we develop a composite version of Mirror Prox algorithm forsolving convex-concave saddle point problems and monotone variationalinequalities of special structure, allowing to cover saddle point/variationalanalogies of what is usually called "composite minimization" (minimizing a sumof an easy-to-handle nonsmooth and a general-type smooth convex functions "asif" there were no nonsmooth component at all). We demonstrate that thecomposite Mirror Prox inherits the favourable (and unimprovable already in thelarge-scale bilinear saddle point case) $O(1/\epsilon)$ efficiency estimate ofits prototype. We demonstrate that the proposed approach can be naturallyapplied to Lasso-type problems with several penalizing terms (e.g. actingtogether $\ell_1$ and nuclear norm regularization) and to problems of thestructure considered in the alternating directions methods, implying in bothcases methods with the $O(\epsilon^{-1})$ complexity bounds.

Mirror Prox Algorithm for Multi-Term Composite Minimization and Semi-Separable Problems