Convergence Rates for the Relaxed Peaceman-Rachford Splitting Method on a Monotone Inclusion Problem

We consider the convergence behavior using the relaxed Peaceman–Rachford splitting method to solve the monotone inclusion problem $$0 \in (A + B)(u)$$ 0 ∈ ( A + B ) ( u ) , where $$A, B: \Re ^n \rightrightarrows \Re ^n$$ A , B : ℜ n ⇉ ℜ n are maximal $$\beta $$ β -strongly monotone operators, $$n \ge 1$$ n ≥ 1 and $$\beta > 0$$ β > 0 . Under a technical assumption, convergence of iterates using the method on the problem is proved when either A or B is single-valued, and the fixed relaxation parameter $$\theta $$ θ lies in the interval $$(2 + \beta , 2 + \beta + \min \{ \beta , 1/\beta \})$$ ( 2 + β , 2 + β + min { β , 1 / β } ) . With this convergence result, we address an open problem that is not settled in Monteiro et al. (Computat Optim Appl 70:763–790, 2018) on the convergence of these iterates for $$\theta \in (2 + \beta , 2 + \beta + \min \{ \beta , 1/\beta \})$$ θ ∈ ( 2 + β , 2 + β + min { β , 1 / β } ) . Pointwise convergence rate results and R-linear convergence rate results when $$\theta $$ θ lies in the interval $$[2 + \beta , 2 + \beta + \min \{\beta , 1/\beta \})$$ [ 2 + β , 2 + β + min { β , 1 / β } ) are also provided in the paper. Our analysis to achieve these results is atypical and hence novel. Numerical experiments on the weighted Lasso minimization problem are conducted to test the validity of the assumption.