Distributed Nonsmooth Optimization with Coupled Inequality Constraints   via Modified Lagrangian Function
  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. 
Shu Liang and Xianlin Zeng and Yiguang Hong
math.OC