  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) $O(1/\epsilon)$ efficiency estimate of
its prototype. We demonstrate that the proposed approach can be naturally
applied to Lasso-type problems with several penalizing terms (e.g. acting
together $\ell_1$ and nuclear norm regularization) and to problems of the
structure considered in the alternating directions methods, implying in both
cases methods with the $O(\epsilon^{-1})$ complexity bounds.
