发表机构
King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对正常闭凸函数提出球近端点方法的精细收敛理论，给出指数收敛率、半径序列收敛条件及自适应半径规则，无需光滑性或强凸性假设。
AI 中文摘要
我们研究了针对正常、闭凸函数的精确欧几里得球近端点方法，其中每次迭代在当前点为中心的球上最小化目标函数。在平方距离的减少中保留目标间隙，可得到目标值、平稳性以及到最小化器的对称Bregman距离的更精细界限。对于常数半径 $t>0$ 和到解集的初始距离 $D_0>0$，经过 $K$ 次迭代后的目标间隙至多为其初始值乘以 $\exp(-2K^2t^2/D_0^2)$。我们刻画了任意正半径序列的收敛性。如果它们的和发散，则只要存在最小化器，该方法在有限次迭代内达到最小化器；否则，目标值收敛到下确界，且迭代点逃逸出每个有界集合。如果半径可求和，则迭代点收敛到可能非最优的点。自收缩性给出了有界迭代的有限轨迹长度和常数半径终止界限 $O_d(1+D_0/t)$，其隐含常数仅依赖于维度。一个多面体族表明，与维度无关的二次界限在渐近意义上是尖锐的。我们还确定了最小成功的几何衰减因子，并分析了基于次梯度或目标间隙的自适应半径规则、上图重构和松弛更新。这些结果共同加强了该方法的基础和收敛性保证，而无需假设光滑性或强凸性。
英文摘要
We study the exact Euclidean ball-proximal point method for proper, closed, convex functions, where each iteration minimizes the objective over a ball centered at the current point. Retaining the objective gap in the decrease of squared distance yields sharper bounds on objective values, stationarity, and the symmetric Bregman distance to a minimizer. For constant radius $t>0$ and initial distance $D_0>0$ to the solution set, the objective gap after $K$ iterations is at most its initial value multiplied by $\exp(-2K^2t^2/D_0^2)$. We characterize convergence for arbitrary positive radius sequences. If their sum diverges, the method reaches a minimizer in finitely many iterations whenever one exists; otherwise, the objective values converge to the infimum and the iterates escape every bounded set. If the radii are summable, the iterates converge to a possibly nonoptimal point. Self-contraction gives finite trajectory length for bounded iterates and a constant-radius termination bound $O_d(1+D_0/t)$, whose implicit constant depends only on the dimension. A polyhedral family shows that the dimension-independent quadratic bound remains asymptotically sharp. We also identify a minimum successful geometric decay factor and analyze adaptive radius rules based on subgradients or objective gaps, epigraph reformulation, and relaxed updates. Together, these results strengthen the foundations and convergence guarantees of the method without assuming smoothness or strong convexity.
