  This paper studies distributed algorithms for the extended monotropic
optimization problem, which is a general convex optimization problem with a
certain separable structure. The considered objective function is the sum of
local convex functions assigned to agents in a multi-agent network, with
private set constraints and affine equality constraints. Each agent only knows
its local objective function, local constraint set, and neighbor information.
We propose two novel continuous-time distributed subgradient-based algorithms
with projected output feedback and derivative feedback, respectively, to solve
the extended monotropic optimization problem. Moreover, we show that the
algorithms converge to the optimal solutions under some mild conditions, by
virtue of variational inequalities, Lagrangian methods, decomposition methods,
and nonsmooth Lyapunov analysis. Finally, we give two examples to illustrate
the applications of the proposed algorithms.
