Projective Splitting Methods for Pairs of Monotone Operators

By embedding the notion of splitting within a general separator projection algorithmic framework, we develop a new class of splitting algorithms for the sum of two general maximal monotone operators in Hilbert space. Our algorithms are essentially standard projection methods, using splitting decomposition to construct separators. These projective algorithms converge under more general conditions than prior splitting methods, allowing the proximal parameter to vary from iteration to iteration, and even from operator to operator, while retaining convergence for essentially arbitrary pairs of operators. The new projective splitting class also contains noteworthy preexisting methods either as conventional special cases or excluded boundary cases.