论文标题
混合重力系统：不确定性优化的重置方法
Hybrid Heavy-Ball Systems: Reset Methods for Optimization with Uncertainty
论文作者
论文摘要
凸优化的动量方法通常依赖于基于问题参数知识的算法参数的精确选择，以实现快速收敛，并防止振荡严重限制这些算法对网络物理系统的应用。为了解决这些问题，我们提出了两个动态系统，称为混合重力系统和杂交启发的重力系统，它们采用反馈机制将动量状态推向零时，每当它指向不需要的方向。我们描述了所提出的系统及其离散时间对应物之间的关系，这些关系基于线性矩阵不等式得出条件，以确保在连续时间和离散时间内确保指数率。我们提供数值LMI结果，以说明在模拟问题参数不确定性的环境中，我们的重置机制对收敛速率的影响。最后，我们从数值上证明了求解强烈凸和非刺激问题时，我们在数值上证明了所提出系统的振荡。
Momentum methods for convex optimization often rely on precise choices of algorithmic parameters, based on knowledge of problem parameters, in order to achieve fast convergence, as well as to prevent oscillations that could severely restrict applications of these algorithms to cyber-physical systems. To address these issues, we propose two dynamical systems, named the Hybrid Heavy-Ball System and Hybrid-inspired Heavy-Ball System, which employ a feedback mechanism for driving the momentum state toward zero whenever it points in undesired directions. We describe the relationship between the proposed systems and their discrete-time counterparts, deriving conditions based on linear matrix inequalities for ensuring exponential rates in both continuous time and discrete time. We provide numerical LMI results to illustrate the effects of our reset mechanisms on convergence rates in a setting that simulates uncertainty of problem parameters. Finally, we numerically demonstrate the efficiency and avoidance of oscillations of the proposed systems when solving both strongly convex and non-strongly convex problems.
