Distributed Mirror Descent Algorithm With Bregman Damping for Nonsmooth Constrained Optimization

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 <tex-math notation="LaTeX">O(1/t)</tex-math> rate. For practical implementation, we further give a discrete-time algorithm based on the proposed dynamics with an <tex-math notation="LaTeX">O(1/√k)</tex-math> convergence rate.