  This paper proposes a multi-scale method to design a continuous-time
distributed algorithm for constrained convex optimization problems by using
multi-agents with Markov switched network dynamics and noisy inter-agent
communications. Unlike most previous work which mainly puts emphasis on dealing
with fixed network topology, this paper tackles the challenging problem of
investigating the joint effects of stochastic networks and the inter-agent
communication noises on the distributed optimization dynamics, which has not
been systemically studied in the past literature. Also, in sharp contrast to
previous work in constrained optimization, we depart from the use of projected
gradient flow which is non-smooth and hard to analyze; instead, we design a
smooth optimization dynamics which leads to easier convergence analysis and
more efficient numerical simulations. Moreover, the multi-scale method
presented in this paper generalizes previously known distributed convex
optimization algorithms from the fixed network topology to the switching case
and the stochastic averaging obtained in this paper is a generalization of the
existing deterministic averaging.
