Distributed Mirror Descent Algorithm With Bregman Damping for Nonsmooth Constrained Optimization
Guanpu Chen, Gehui Xu, Weijian Li, Yiguang Hong · IEEE Transactions on Automatic Control · 2023
To efficiently solve the nonsmooth distributed optimization with both local constraints and coupled constraints, we propose a distributed continuous-time algorithm based on the mirror descent (MD) method. In this article, we introduce the Bregman damping into distributed MD-based dynamics, which not only successfully applies the MD idea to the distributed primal-dual framework, but also ensures the boundedness of all variables and the convergence of the entire dynamics. Our approach generalizes the classic distributed projection-based dynamics, and establishes a connection between MD methods and distributed Euclidean-projected approaches. Also, we prove the convergence of the proposed distributed dynamics with an$ \mathcal {O}(1/t)$rate. For practical implementation, we further give a discrete-time algorithm based on the proposed dynamics with an$ \mathcal {O}(1/\sqrt{k})$convergence rate.