Distributed Multiproximal Algorithm for Nonsmooth Convex Optimization With Coupled Inequality Constraints

This article studies a class of distributed nonsmooth convex optimization problems subject to local set constraints and coupled nonlinear inequality constraints. In particular, each local objective function consists of one differentiable convex function and multiple nonsmooth convex functions. By applying multiple proximal splittings and derivative feedback techniques, a new distributed continuous-time multiproximal algorithm is developed, whose dynamics satisfies Lipschitz continuity even if the considered problem is nonsmooth. Compared with previous results that rely on either the differentiability or strong convexity of local objective functions, the proposed algorithm can be applied to more general functions, which are only convex and not necessarily smooth. Moreover, in contrast to some results that require some specific initial conditions, the developed algorithm is free of initialization. The convergence analysis of the proposed algorithm is conducted by applying Lyapunov stability theory. It is shown that the states of all the agents achieve consensus at an optimal solution. Finally, a numerical example is presented to demonstrate the effectiveness of the proposed algorithm.