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Fast primal-dual algorithm via dynamical system for a linearly constrained convex optimization problem

AUTOMATICA 2022年 146卷

作者： He, Xin Hu, Rong Fang, Ya-Ping Sichuan Univ Dept Math Chengdu Sichuan Peoples R China Chengdu Univ Informat Technol Dept Appl Math Chengdu Sichuan Peoples R China

By time discretization of a second-order primal-dual dynamical system with damping alpha/t where an inertial construction in the sense of Nesterov is needed only for the primal variable, we propose a fast primal-dual algorithm for a linear equality constrained convex optimization problem. Under a suitable scaling condition, we show that the proposed algorithm enjoys a fast convergence rate for the objective residual and the feasibility violation, and the decay rate can reach O(1/k(alpha-1)) at the most. We also study convergence properties of the corresponding primal-dual dynamical system to better understand the acceleration scheme. Finally, we report numerical experiments to demonstrate the effectiveness of the proposed algorithm. (c) 2022 Elsevier Ltd. All rights reserved.

关键词： linearly constrained convex optimization problem Primal-dual algorithm Inertial primal-dual dynamical system Convergence rate Nesterov's acceleration

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Inertial primal-dual dynamics with damping and scaling for linearly constrained convex optimization problems

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APPLICABLE ANALYSIS 2023年第15期102卷 4114-4139页

作者： He, Xin Hu, Rong Fang, Ya-Ping Sichuan Univ Dept Math Chengdu Sichuan Peoples R China Chengdu Univ Informat Technol Dept Appl Math Chengdu Sichuan Peoples R China

We propose an inertial primal-dual dynamic with damping and scaling coefficients, which involves inertial terms both for primal and dual variables, for a linearly constrained convex optimization problem in a Hilbert setting. With different choices of damping and scaling coefficients, by a Lyapunov analysis approach, we investigate the asymptotic properties of the dynamic and prove its fast convergence results. Our results can be viewed as extensions of the existing ones on inertial dynamical systems for the unconstrained convex optimization problem to the primal-dual one for the linearly constrained convex optimization problem.

关键词： Inertial primal-dual dynamical system linearly constrained convex optimization problem damping and scaling Lyapunov analysis approach convergence rate

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