Adaptive Douglas--Rachford Splitting Algorithm from a Yosida Approximation Standpoint

The adaptive Douglas--Rachford splitting algorithm iteratively applies the operator $T=\kappa_{n}Q_{A}Q_{B}+(1-\kappa_{n}){Id}$ to solve the inclusion problem $\text{zer}(A+B)$. By taking a Yosida approximation standpoint, we express in canonical form $Q_{A}Q_{B}=({Id}-(\gamma+\lambda)\ ^{\gamma}\!A)\circ({Id}-(\gamma+\lambda)\ ^{\lambda}\!B)$. We extend the domain of indices $\gamma, \lambda$ to the entire real line, so that the adaptive algorithm is able to encompass a forward-backward splitting algorithm into one unified framework. Convergence results for both primal and dual problems are proved for different combinations of (strongly and weakly) monotone and comonotone operators. Under the “monotone + comonotone” assumption, we obtain a better rate bound for linear convergence than the classical Douglas--Rachford algorithm.