论文标题
团结Nesterov和重型球方法，用于一组最小化器的均匀全球渐近稳定性
Uniting Nesterov and Heavy Ball Methods for Uniform Global Asymptotic Stability of the Set of Minimizers
论文作者
论文摘要
我们提出了一种混合控制算法，该算法保证了连续可区分的凸目标功能的独特最小化器的快速收敛和均匀的全局渐近稳定性。使用混合系统工具开发的算法采用了一种团结的控制策略，其中Nesterov的加速梯度下降被“全球”使用，并且相对于最小化器使用了“本地”重球方法。在不了解其位置的情况下，提出的混合控制策略在这些加速方法之间切换，以确保没有振荡的最小化合物收敛，并具有（混合）收敛速率，从而保留了个体优化算法的收敛速率。我们分析了所得闭环系统的关键特性，包括解决方案的存在，均匀的全球渐近稳定性和收敛速率。此外，分析了Nesterov方法的稳定性，并提出了现有文献中收敛率的扩展。数值结果验证了发现并证明了统一算法的鲁棒性。
We propose a hybrid control algorithm that guarantees fast convergence and uniform global asymptotic stability of the unique minimizer of a continuously differentiable, convex objective function. The algorithm, developed using hybrid system tools, employs a uniting control strategy, in which Nesterov's accelerated gradient descent is used "globally" and the heavy ball method is used "locally," relative to the minimizer. Without knowledge of its location, the proposed hybrid control strategy switches between these accelerated methods to ensure convergence to the minimizer without oscillations, with a (hybrid) convergence rate that preserves the convergence rates of the individual optimization algorithms. We analyze key properties of the resulting closed-loop system including existence of solutions, uniform global asymptotic stability, and convergence rate. Additionally, stability properties of Nesterov's method are analyzed, and extensions on convergence rate results in the existing literature are presented. Numerical results validate the findings and demonstrate the robustness of the uniting algorithm.
