论文标题
关于原始偶二动力学的几何和线性收敛
On the Geometry and Linear Convergence of Primal-Dual Dynamics
论文作者
论文摘要
本文提出了一种基于变异质量的原始二极管动力学，当应用于求解线性不等式约束优化问题时，它具有全球稳定的鞍点解决方案。提出了一个riemannian几何框架，其中我们首先将提出的动力学构建在带有riemannian度量的纤维束设置中，该设置捕获了riemannian指标，该指标捕获了梯度的几何形状（Lagrangian函数的几何形状）。通过在Riemannian歧管上使用自然梯度适应来获得强烈的单调梯度矢量场。 Lyapunov稳定性分析证明，这种适应性导致了全球稳定的鞍点解决方案。此外，通过数字模拟，我们显示了Riemannian公制中的关键参数的缩放会导致加速收敛到鞍点解决方案。
The paper proposes a variational-inequality based primal-dual dynamic that has a globally exponentially stable saddle-point solution when applied to solve linear inequality constrained optimization problems. A Riemannian geometric framework is proposed wherein we begin by framing the proposed dynamics in a fiber-bundle setting endowed with a Riemannian metric that captures the geometry of the gradient (of the Lagrangian function). A strongly monotone gradient vector field is obtained by using the natural gradient adaptation on the Riemannian manifold. The Lyapunov stability analysis proves that this adaption leads to a globally exponentially stable saddle-point solution. Further, with numeric simulations we show that the scaling a key parameter in the Riemannian metric results in an accelerated convergence to the saddle-point solution.
