发表机构
School of Engineering and Applied Sciences, Harvard University（哈佛大学生物工程与应用科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出余像原理，用于构造线性锥规划一阶方法（如ADMM、PDHG）状态空间中的慢区域，解释并刻画了长平台上的停滞现象。
AI 中文摘要
线性锥规划的一阶方法常常在长平台上停滞。现有分析刻画了在运行过程中或哪些问题实例上会出现慢收敛；我们则转而询问状态空间中哪些位置存在慢收敛。我们为参数化平均不动点迭代引入了慢区域族。在慢区域上，一步仅将状态移动其到不动点集距离的一小部分，因此从那里出发的轨道在任意多次迭代中几乎保持其初始距离。随后，我们发展了余像原理来构造这些慢区域。一个中心族和一个邻近的花瓣族共享相同的极限参数，因此它们的残差场变得接近，而它们的不动点集或前向漂移则保持远离。花瓣几何随后为中心族证明了慢区域的存在。我们验证了ADMM、sGS-ADMM和PDHG的常设假设，并给出了一个LP、SOCP和SDP的图集，展示慢区域可以多么多样化。
英文摘要
First-order methods for linear conic programming often stall on long plateaus. Existing analyses characterize when during a run or on which problem instances slow convergence occurs; we instead ask where in the state space slow convergence is present. We introduce the slow region family for parameterized averaged fixed-point iterations. On a slow region, one step moves the state by only a small fraction of its distance to the fixed-point set, so an orbit starting there keeps almost its initial distance for arbitrarily many iterations. We then develop the afterimage principle to construct them. A center family and a nearby petal family share the same limit parameter, so their residual fields become close, while their fixed-point sets or forward drifts stay far apart. The petal geometry then certifies a slow region for the center. We verify the standing assumptions for ADMM, sGS-ADMM, and PDHG, and give an LP, SOCP, and SDP gallery showing how varied slow regions can be.
