The forward–backward splitting method and its convergence rate for the minimization of the sum of two functions in Banach spaces

Abstract

In this paper, the forward–backward splitting method for the minimization of the sum of two functions have been extended to Banach spaces. The regularization term of the forward–backward splitting method is constructed by the norm with p-power. We prove the functional values of the sequences generated by this method converge with an asymptotic rate $$n^{1-p}$$ to the optimal value of the problem and establish the linear convergence under an error bound assumption.