  This technical note considers a distributed convex optimization problem with
nonsmooth cost functions and coupled nonlinear inequality constraints. To solve
the problem, we first propose a modified Lagrangian function containing local
multipliers and a nonsmooth penalty function. Then we construct a distributed
continuous-time algorithm by virtue of a projected primal-dual subgradient
dynamics. Based on the nonsmooth analysis and Lyapunov function, we obtain the
existence of the solution to the nonsmooth algorithm and its convergence.
