On the Geometry and Linear Convergence of Primal-Dual Dynamics

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 dynami...

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Bibliographic Details
Published inarXiv.org
Main Authors Bansode, P, Chinde, V, Wagh, S R, Pasumarthy, R, Singh, N M
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 05.10.2020
Subjects
Convergence
Fields (mathematics)
Lagrangian function
Linear functions
Optimization
Riemann manifold
Saddle points
Stability analysis
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Summary: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.
ISSN:2331-8422