Mirror Prox algorithm for multi-term composite minimization and semi-separable problems

In the paper, we develop a composite version of Mirror Prox algorithm for solving convexconcave saddle point problems and monotone variational inequalities of special structure, allowing to cover saddle point/variational analogies of what is usually called “composite minimization” (minimizing a sum of an easy-to-handle nonsmooth and a general-type smooth convex functions “as if” there were no nonsmooth component at all). We demonstrate that the composite Mirror Prox inherits the favourable (and unimprovable already in the large-scale bilinear saddle point case) O(1/ ) efficiency estimate of its prototype. We demonstrate that the proposed approach can be successfully applied to Lasso-type problems with several penalizing terms (e.g. acting together `1 and nuclear norm regularization) and to problems of semi-separable structures considered in the alternating directions methods, implying in both cases methods with the O(1/ ) complexity bounds.