On the Geometry and Linear Convergence of Primal-Dual Dynamics

Abstract

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.

arXiv commentarXiv admin note: text overlap with arXiv:1905.04521

References in corpus (4)

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• Semi-Global Exponential Stability of Augmented Primal-Dual Gradient Dynamics for Constrained Convex Optimization
• On the Exponential Stability of Projected Primal-Dual Dynamics on a Riemannian Manifold