Distributed Optimization with Coupling Constraints Based on Dual Proximal Gradient Method in Multi-Agent Networks

In this paper, we aim to solve a distributed optimization problem with affine coupling constraints in a multi-agent network, where the cost function of the agents is composed of smooth and possibly non-smooth parts. To solve this problem, we resort to the dual problem by deriving the Fenchel conjugate, resulting in a consensus-based constrained optimization problem. Then, we propose a distributed dual proximal gradient algorithm, where the agents make decisions based on the information of immediate neighbors. Provided that the non-smooth parts in the primal cost functions are with some simple structures, we only need to update dual variables by some simple operations, by which the overall computational complexity can be reduced. An ergodic convergence rate of the proposed algorithm is derived and the feasibility is numerically verified by solving a social welfare optimization problem in the electricity market.