On the Exponential Stability of Projected Primal-Dual Dynamics on a Riemannian Manifold. Bansode, P. A., Chinde, V., Wagh, S. R., Pasumarthy, R., & Singh, N. M. arXiv:1905.04521 [math], May, 2019. arXiv: 1905.04521
On the Exponential Stability of Projected Primal-Dual Dynamics on a Riemannian Manifold [link]Paper  abstract   bibtex   
Equivalence of convex optimization and variational inequality is well established in the literature such that the latter is formally recognized as a fixed point problem of the former. Such equivalence is also known to exist between a saddle-point problem and the variational inequality. The variational inequality is a static problem which can be further studied within the dynamical settings using a framework called the projected dynamical system whose stationary points coincide with the static solutions of the associated variational inequality. Variational inequalities have rich properties concerning the monotonicity of its vector-valued map and the uniqueness of its solution, which can be extended to the convex optimization and saddle-point problems. Moreover, these properties also extend to the representative projected dynamical system. The objective of this paper is to harness rich monotonicity properties of the representative projected dynamical system to develop the solution concepts of the convex optimization problem and the associated saddle-point problem. To this end, this paper studies a linear inequality constrained convex optimization problem and models its equivalent saddle-point problem as a variational inequality. Further, the variational inequality is studied as a projected dynamical system [1] which is shown to converge to the saddle-point solution. By considering the monotonicity of the gradient of Lagrangian function as a key factor, this paper establishes exponential convergence and stability results concerning the saddle-points. Our results show that the gradient of the Lagrangian function is just monotone on the Euclidean space, leading to only Lyapunov stability of stationary points of the projected dynamical system. To remedy the situation, the underlying projected dynamical system is formulated on a Riemannian manifold whose Riemannian metric is chosen such that the gradient of the Lagrangian function becomes strongly monotone. Using a suitable Lyapunov function, the stationary points of the projected dynamical system are proved to be globally exponentially stable and convergent to the unique saddle-point.
@article{bansode_exponential_2019,
	title = {On the {Exponential} {Stability} of {Projected} {Primal}-{Dual} {Dynamics} on a {Riemannian} {Manifold}},
	url = {http://arxiv.org/abs/1905.04521},
	abstract = {Equivalence of convex optimization and variational inequality is well established in the literature such that the latter is formally recognized as a fixed point problem of the former. Such equivalence is also known to exist between a saddle-point problem and the variational inequality. The variational inequality is a static problem which can be further studied within the dynamical settings using a framework called the projected dynamical system whose stationary points coincide with the static solutions of the associated variational inequality. Variational inequalities have rich properties concerning the monotonicity of its vector-valued map and the uniqueness of its solution, which can be extended to the convex optimization and saddle-point problems. Moreover, these properties also extend to the representative projected dynamical system. The objective of this paper is to harness rich monotonicity properties of the representative projected dynamical system to develop the solution concepts of the convex optimization problem and the associated saddle-point problem. To this end, this paper studies a linear inequality constrained convex optimization problem and models its equivalent saddle-point problem as a variational inequality. Further, the variational inequality is studied as a projected dynamical system [1] which is shown to converge to the saddle-point solution. By considering the monotonicity of the gradient of Lagrangian function as a key factor, this paper establishes exponential convergence and stability results concerning the saddle-points. Our results show that the gradient of the Lagrangian function is just monotone on the Euclidean space, leading to only Lyapunov stability of stationary points of the projected dynamical system. To remedy the situation, the underlying projected dynamical system is formulated on a Riemannian manifold whose Riemannian metric is chosen such that the gradient of the Lagrangian function becomes strongly monotone. Using a suitable Lyapunov function, the stationary points of the projected dynamical system are proved to be globally exponentially stable and convergent to the unique saddle-point.},
	language = {en},
	urldate = {2022-01-19},
	journal = {arXiv:1905.04521 [math]},
	author = {Bansode, P. A. and Chinde, V. and Wagh, S. R. and Pasumarthy, R. and Singh, N. M.},
	month = may,
	year = {2019},
	note = {arXiv: 1905.04521},
	keywords = {/unread, Mathematics - Differential Geometry, Mathematics - Optimization and Control, ⛔ No DOI found},
}

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