  We consider a group of computation units trying to cooperatively solve a
distributed optimization problem with shared linear equality and inequality
constraints. Assuming that the computation units are communicating over a
network whose topology is described by a time-invariant directed graph, by
combining saddle-point dynamics with Lie bracket approximation techniques we
derive a methodology that allows to design distributed continuous-time
optimization algorithms that solve this problem under minimal assumptions on
the graph topology as well as on the structure of the constraints. We discuss
several extensions as well as special cases in which the proposed procedure
becomes particularly simple.
