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

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