  We suggest an adaptive version of a partial linearization method for
composite optimization problems. The goal function is the sum of a smooth
function and a non necessary smooth convex separable function, whereas the
feasible set is the corresponding Cartesian product. The method consists in
selective component-wise steps together with a special control of a tolerance
sequence. This technique is destined to reduce the computational expenses per
iteration and maintain the basic convergence properties. We also establish its
convergence rates and describe some examples of applications. Preliminary
results of computations illustrate usefulness of the new method.
