对于每个维度\(n\geq4\)，Barzilai - Borwein方法在二次函数的一个开集上不满足超线性收敛
Barzilai-Borwein Fails Superlinear Convergence on an Open Set of Quadratics for Every Dimension $n\geq 4$
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中文总结 AI 辅助
研究\(n\geq4\)时Barzilai - Borwein方法在二次函数上的收敛性，构造一族严格凸二次问题及初始点，证明长步BB1方法虽收敛但非超线性收敛，通过计算机辅助证明四维射影BB动力学的七周期情况来实现。
中文摘要 AI 辅助
Barzilai - Borwein（BB）方法在连续优化中表现出强大的实际性能，但其收敛动态仍知之甚少。一个核心未解决问题是，对于几乎每个严格凸二次问题和初始化，BB是否超线性收敛。本文给出否定答案。对于每个\(n\geq4\)的有限维度，构造了一族非空开的（具有正勒贝格测度）严格凸二次问题和初始点，长步Barzilai - Borwein方法（BB1）收敛但非根超线性收敛。明确常数\(\rho_{\min}=10^{-6},\rho_{\max}=0.61\)，梯度的每个谱分量由相应几何序列上下界限定。梯度范数、误差的能量范数满足双边几何估计，目标差距满足平方速率的相应估计，排除了超线性收敛。构造基于四维射影BB动力学的非共振吸引七周期的计算机辅助证明。
英文摘要
Barzilai--Borwein (BB) method has shown strong practical performance in continuous optimization, yet its convergence dynamics remains poorly understood. In particular, a central unresolved question is whether BB converges superlinearly for almost every strictly convex quadratic problem and initialization. We provide a negative answer to this question. Specifically, for every finite dimension $n\geq4$, we construct a nonempty open, hence positive-Lebesgue-measure, family of strictly convex quadratic problems and initial points for which the long Barzilai--Borwein method (BB1) converges but cannot converge root-superlinearly. More precisely, with the explicit constants $ρ_{\min}=10^{-6},ρ_{\max}=0.61$, every spectral component of the gradient is bounded above and below by the corresponding geometric sequence. Consequently, the gradient norm and the energy norm of the error satisfy two-sided geometric estimates with the same rates, while the objective gap satisfies the corresponding estimates with squared rates. In particular, all three quantities are bounded below by geometric sequences, ruling out superlinear convergence. The construction is highly nontrivial, based on a computer-assisted proof of a nonresonant, attracting seven-cycle of the projectivized BB dynamics in dimension four.
发表机构
- University of Minnesota（明尼苏达大学）
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