A new splitting method for monotone inclusions of three operators

In this article, we consider monotone inclusions in real Hilbert spaces and suggest a new splitting method. The associated monotone inclusions consist of the sum of one bounded linear monotone operator and one inverse strongly monotone operator and one maximal monotone operator. The new method, at each iteration, first implements one forward-backward step as usual and next implements a descent step, and it can be viewed as a variant of a proximal-descent algorithm in a sense. Its most important feature is that, at each iteration, it needs evaluating the inverse strongly monotone part once only in the forward-backward step and, in contrast, the original proximal-descent algorithm needs evaluating this part twice both in the forward-backward step and in the descent step. Moreover, unlike a recent work, we no longer require the adjoint operation of this bounded linear monotone operator in the descent step. Under standard assumptions, we analyze weak and strong convergence properties of this new method. Rudimentary experiments indicate the superiority of our suggested method over several recently-proposed ones for our test problems.