发表机构
Beijing International Center for Mathematical Research, Peking University; School of Mathematical Sciences, Peking University（北京大学北京国际数学研究中心; 北京大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究精确线搜索BFGS算法的最小Q阶，构造满足特定条件的目标函数，证明其最小Q阶为1，且算法虽Q超线性收敛但无大于1的固定幂次控制相邻误差。
AI 中文摘要
Powell提出，经典超线性收敛所基于的光滑性假设是否会对精确线搜索变尺度方法的相邻迭代点之间施加固定幂次定律。我们针对BFGS算法否定了这一问题：在光滑强凸设定下，最小可能的相邻迭代Q阶为1，且该边界可由单次非终止运行达到。在每个至少为2的有限维空间中，对于任意规定的半径和海森矩阵容差，我们构造了一个无穷可微、全局强凸的目标函数，该函数在对应球外等于标准二次函数，且其海森矩阵在算子范数下与单位矩阵的偏差在规定容差内。该目标函数的唯一极小点在原点，且该处海森矩阵为单位矩阵。采用单位矩阵初始化并在该球内启动的精确线搜索BFGS算法，虽收敛为Q超线性，但不存在大于1的固定幂次能控制所有足够晚的相邻误差。
英文摘要
Powell asked whether the smoothness assumptions underlying classical superlinear convergence force a fixed power law between adjacent iterates of exact-line-search variable-metric methods. We answer this question negatively for BFGS: within the smooth strongly convex setting, the smallest possible adjacent-iterate Q-order is one, and this boundary is attained by a single nonterminating run. In every finite dimension at least two, and for any prescribed radius and Hessian tolerance, we construct an infinitely differentiable, globally strongly convex objective that equals the standard quadratic outside the corresponding ball and whose Hessian remains within the prescribed tolerance of the identity in operator norm. The objective has its unique minimizer at the origin and identity Hessian there. Exact-line-search BFGS, initialized with the identity matrix and started inside that ball, converges Q-superlinearly, yet no fixed power greater than one controls all sufficiently late adjacent errors.
Comments19 pages, 2 tables
