Summary: A splitting method for two monotone operators \(A\) and \(B\) is an algorithm that attempts to converge to a zero of the sum \(A + B\) by solving a sequence of subproblems, each of which involves only the operator \(A\), or only the operator \(B\). Prior algorithms of this type can all in essence be categorized into three main classes, the Douglas/Peaceman–Rachford class, the forward-backward class, and the little-used double-backward class. Through a certain “extended” solution set in a product space, we construct a fundamentally new class of splitting methods for pairs of general maximal monotone operators in Hilbert space.
Our algorithms are essentially standard projection methods, using splitting decomposition to construct separators. We prove convergence through Fejér monotonicity techniques, but showing Fejér convergence of a different sequence to a different set than in earlier splitting methods. Our 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.