A variable metric Douglas-Rachford splitting method for nonconvex composite optimization

The Douglas-Rachford splitting method (DRSM) has been highly successful in solving composite optimization and structured inclusions over the last decades and it has continued its success in recent years. However, DRSM usually suffers from the slow convergence for ill-posed problems. To ameliorate the efficiency of DRSM, we develop a variable metric DRSM (VMDR) with provable convergence by deploying the second-order oracle. Under the Kurdyka-{\L}ojasiewicz assumption, we analyze the global convergence of VMDR and establish its convergence rate from two different viewpoints. Alternatively, a latent pitfall of VMDR is the computational effort on solving the weighted proximity, which is often handled by internally nested subroutines. Thereby, we develop an inexact VMDR (iVMDR) algorithmic framework by devising two inexact criteria (one involves the summable error sequence, and the other exploits the verifiable functional residual). Numerical experiments on binary classification and image reconstruction demonstrate the compelling performance of the proposed method.

Recommended citation: Z.H. Jia, W.X. Zhang, X.J. Cai, D.R. Han. (2025). manuscript. 1-28.