On the Geometry and Linear Convergence of Primal-Dual Dynamics

TL;DR

This paper introduces a Riemannian geometric framework for primal-dual dynamics in constrained optimization, demonstrating global exponential stability and accelerated convergence through natural gradient adaptation.

Contribution

It develops a novel Riemannian geometric approach to primal-dual dynamics, ensuring stability and efficiency in solving linear inequality constrained problems.

Findings

Proposes a variational-inequality based primal-dual dynamic with exponential stability.

Uses Riemannian geometry and natural gradient to enhance convergence.

Numerical results show parameter scaling accelerates convergence.

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.