AI 中文总结
该研究针对无约束优化提出双梯度方法，通过双步原则确定步长，算法选使两梯度过程欧氏距离最小的步长，理论分析其收敛受方向夹角影响，还引入混合框架确保鲁棒性，数值结果表明双阶段为后续迭代创造良好初始条件。
AI 中文摘要
我们提出了一种基于梯度的无约束优化新策略，涉及两个并行迭代序列，通过“双步”原则协同确定步长。该算法并非单独最小化目标函数，而是选择使每次迭代中同时出现的两个基于梯度的过程之间的欧几里得距离最小的步长。理论分析表明，相互距离的收敛受搜索方向之间角度的控制。特别是当方向接近平行时，整个过程的有效性会降低。为确保对共线性的鲁棒性，我们引入了混合框架Twin - ABB$_{\min}$，当几何协作无效时切换到自适应Barzilai - Borwein方法。大量且很有前景的数值结果证明，双阶段为后续的BB型迭代创造了良好的初始条件。
英文摘要
We propose a new strategy for gradient-based unconstrained optimization, involving two parallel sequences of iterates that cooperate to determine their stepsizes via a \textit{Twin-Step} principle. Rather than minimizing the objective function individually, the algorithm selects steplengths that minimize the Euclidean distance between the two gradient based processes occurring simultaneously at each iteration. The theoretical analysis shows that the convergence of the mutual distance is governed by the angle between the search directions. In particular the effectiveness of the overall process degrades as the directions approach parallelism. To ensure robustness against collinearity, we introduce a hybrid framework, Twin-ABB$_{\min}$, which switches to the Adaptive Barzilai--Borwein method when the geometric cooperation becomes ineffective. Extensive and very promising numerical results evidence that the Twin phase creates favorable initial conditions for subsequent BB-type iterations.
