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 convex–concave 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) 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 𝓁1 and nuclear norm regularization) and to problems of semi‐separable structures considered in the alternating directions methods, implying in both cases methods with the complexity bounds.