A new splitting method for systems of monotone inclusions in Hilbert spaces

Abstract

In this article, we consider the problem of finding a zero of systems of monotone inclusions in real Hilbert spaces. Furthermore, each monotone inclusion consists of three operators and the third is linearly composed. We suggest a splitting method for solving them: At each iteration, for each monotone inclusion, it mainly needs computations of three resolvents for individual operator. This method can be viewed as a powerful extension of the classical Douglas–Rachford splitting. Under the weakest possible assumptions, by introducing and using the characteristic operator, we analyze its weak convergence. The most striking feature is that it merely requires each scaling factor for individual operator be positive. Numerical results indicate practical usefulness of this method, together with its special cases, in solving our test problems of separable structure.