拓展了两种优化算法的收敛性，突破了传统非单调性限制。
Spingarn's Method and Progressive Decoupling Beyond Elicitable Monotonicity
- 提出改进版渐进解耦算法，对子空间和正交补分别设松弛参数
- 证明在非单调性受限条件下仍可收敛，突破原有理论边界
- 适用于需要处理复杂约束的优化问题，适合优化与机器学习研究者
Spingarn的偏逆法和渐进解耦算法用于求解涉及算子与线性子空间法向量锥之和的包含问题，即关联问题。尽管已有成功应用，其收敛性分析仅限于所谓的可引出单调情形，即非单调性仅允许出现在关联子空间的正交补上。本文提出渐进解耦+，是标准渐进解耦的推广版本，为关联子空间及其正交补分别引入独立松弛参数。我们证明了在松弛参数与各自子空间非单调性相关联的条件下算法收敛，并表明Spingarn方法与标准渐进解耦亦可扩展至非可引出单调设置。分析基于渐进解耦+与预条件近端点算法的等价性，发展了一般非单调设定下的局部收敛分析。
原文摘要 · Abstract (English)
Spingarn's method of partial inverses and the progressive decoupling algorithm address inclusion problems involving the sum of an operator and the normal cone of a linear subspace, known as linkage problems. Despite their success, existing convergence results are limited to the so-called elicitable monotone setting, where nonmonotonicity is allowed only on the orthogonal complement of the linkage subspace. In this paper, we introduce progressive decoupling+, a generalized version of standard progressive decoupling that incorporates separate relaxation parameters for the linkage subspace and its orthogonal complement. We prove convergence under conditions that link the relaxation parameters to the nonmonotonicity of their respective subspaces and show that the special cases of Spingarn's method and standard progressive decoupling also extend beyond the elicitable monotone setting. Our analysis hinges upon an equivalence between progressive decoupling+ and the preconditioned proximal point algorithm, for which we develop a general local convergence analysis in a certain nonmonotone setting.
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