  We consider the problem of minimizing a sum of several convex non-smooth
functions. We introduce a new algorithm called the selective linearization
method, which iteratively linearizes all but one of the functions and employs
simple proximal steps. The algorithm is a form of multiple operator splitting
in which the order of processing partial functions is not fixed, but rather
determined in the course of calculations. Global convergence is proved and
estimates of the convergence rate are derived. Specifically, the number of
iterations needed to achieve solution accuracy $\varepsilon$ is of order
$\mathcal{O}\big(\ln(1/\varepsilon)/\varepsilon\big)$. We also illustrate the
operation of the algorithm on structured regularization problems.
