DescriptionWe consider the problem of minimizing a sum of several convex non-smooth functions. In this thesis, 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. It proposes one of the first operator-splitting type methods which are globally convergent for an arbitrary number of operators without artificial duplication of variables. This algorithm is a multi-block extension of the alternating linearization (ALIN) method for solving structured non-smooth convex optimization problems. Global convergence is proved and estimates of the convergence rate are derived. Specifically, under a strong convexity condition, the number of iterations needed to achieve solution accuracy ε is of order O(ln(1/ε)/ε). The convergence rate analysis technique invented by us can also be used to derive the rate of convergence of the classical bundle method and ALIN method, for which no convergence rate estimate has been available so far. We report results of extensive comparison experiments in structured regularization problems such as large-scale fused lasso regularization problems and overlapping group lasso problems. The numerical results demonstrate the efficacy and accuracy of the method.