Linear convergence of the generalized PPA and several splitting methods for the composite inclusion problem

For the inclusion problem involving two maximal monotone operators, under the metric subregularity of the composite operator, we derive the linear convergence of the generalized proximal point algorithm and several splitting algorithms, which include the over-relaxed forward-backward splitting algorithm, the generalized Douglas-Rachford splitting algorithm and Davis' three-operator splitting algorithm. To the best of our knowledge, this linear convergence condition is weaker than the existing ones that almost all require the strong monotonicity of the composite operator. Withal, we give some sufficient conditions to ensure the metric subregularity of the composite operator. At last, the preliminary numerical performances on some toy examples support the theoretical results.