A second-order adaptive Douglas–Rachford dynamic method for maximal $$\alpha $$-monotone operators

Abstract

The Douglas–Rachford splitting method is a classical and powerful method that is widely used in engineering fields for finding a zero of the sum of two operators. In this paper, we begin by proposing an abstract second-order dynamic system involving a generalized cocoercive operator to find a zero of the operator in a real Hilbert space. Then we develop a second-order adaptive Douglas–Rachford dynamic system for finding a zero of the sum of two operators, one of which is strongly monotone while the other one is weakly monotone. With proper tuning of the parameters such that the adaptive Douglas–Rachford operator is quasi-nonexpansive, we demonstrate that the trajectory of the proposed adaptive system converges weakly to a fixed point of the adaptive operator. When the strong monotonicity strictly outweighs the weak one, we further derive the strong convergence of shadow trajectories to the solution of the original problem. Finally, two simulation examples are reported to corroborate the effectiveness of the proposed adaptive system.