Convergence analysis of the generalized Douglas-Rachford splitting method under Hölder subregularity assumptions

In solving convex optimization and monotone inclusion problems, operator-splitting methods are often employed to transform optimization and inclusion problems into fixed-point equations as the equations obtained from operator-splitting methods are often easy to be solved by standard techniques. For the inclusion problem involving two maximally monotone operators, under the Hölder metric subregularity of the concerned operator, which is weaker than the strong monotonicity of the operator, we derive relationships between the convergence rate of the generalized Douglas-Rachford splitting algorithm and the order of the Hölder metric subregularity of the concerned operator. Moreover, for general multifunctions in Hilbert spaces, by proximal coderivative, we provided some dual sufficient conditions for Hölder metric subregularity.