A Splitting Framework for Composite Semimonotone Inclusions

Abstract: We introduce a general framework for composite inclusion problems with affine constraints, covering both monotone and semimonotone regimes. The central idea is a new interpretation of the constrained inclusion through an operator-vector pair that separates the implicit inclusion from the affine constraint: the former is handled through possibly preconditioned resolvent evaluations, while the latter is handled through an explicit forward step in an auxiliary variable. This yields a single abstract iteration applicable to multioperator inclusions, linearly coupled inclusions, and block-separable inclusions with affine constraints. The freedom in choosing the operator-vector pair enables the systematic construction of problem-adapted splitting algorithms, including new schemes for several important problem classes. Exploiting the orthogonal decomposition induced by the constraint subspace, we develop a unified and streamlined convergence analysis and establish weak and strong convergence guarantees under semimonotonicity assumptions. When specialized to multioperator inclusions, the framework permits general bounded linear operator coefficients, rather than only scalar coefficients, and therefore accommodates preconditioned resolvents. The resulting schemes recover several existing methods while extending them to previously uncovered regimes, and in several important cases, require weaker assumptions and admit provably larger admissible parameter ranges.