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Relatively relaxed proximal point algorithms for generalized maximal monotone mappings and Douglas-Rachford splitting methods

The theory of maximal set-valued monotone mappings provide a powerful framework to the study of convex programming and variational inequalities. Based on the notion of relatively maximal relaxed monotonicity, the approximation solvability of a general class of inclusion problems is discussed, while generalizing most of investigations on weak convergence using the proximal point algorithm in a real Hilbert space setting. A well-known method of multipliers of constrained convex programming is a special case of the proximal point algorithm. The obtained results can be used to generalize the Yosida approximation, which, in turn, can be applied to generalize first-order evolution equations to the case of evolution inclusions. Furthermore, we observe that the Douglas-Rachford splitting method for finding the zero of the sum of two monotone operators is a specialization of the proximal point algorithm as well. This allows a further generalization and unification of a wide range of convex programming algorithms.