AI 中文总结
该研究提出带仿射约束复合半单调包含的通用分裂框架，通过算子-向量对分离隐式包含与仿射约束，可构造适配算法，扩展现有方法并获更优收敛保证。
AI 中文摘要
我们提出了一种带仿射约束的复合包含问题的通用框架，涵盖单调和半单调两种情形。核心思想是通过算子-向量对重新阐释带约束的包含问题，将隐式包含与仿射约束分离开：前者通过可能带预条件的预解式计算处理，后者通过辅助变量中的显式前向步处理。这得到了适用于多算子包含、线性耦合包含及带仿射约束的块可分包含的单一抽象迭代格式。算子-向量对选择的灵活性使得可系统构造适配问题的分裂算法，包括针对若干重要问题类别的新方案。利用约束子空间诱导的正交分解，我们开展了统一且简化的收敛分析，并在半单调性假设下建立了弱收敛和强收敛保证。当应用于多算子包含时，该框架允许一般的有界线性算子系数，而非仅标量系数，因此可容纳预条件预解式。所得方案恢复了若干现有方法，同时将其扩展至此前未覆盖的情形，且在若干重要情形下，所需假设更弱，并具有可证更大的容许参数范围。
英文摘要
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.
