  This paper studies a class of distributed optimization algorithms by a set of
agents, where each agent has only access to its own local convex objective
function, and jointly minimizes the sum of the functions. The communications
among agents are described by a sequence of time-varying directed graphs which
are assumed to be uniformly strongly connected. A column stochastic mixing
matrices is employed in the algorithm, which also exactly steers all the agents
to asymptotically converge to a global and consensual optimal solution even
under the assumption that the step-sizes are uncoordinated. Two fairly standard
conditions for achieving the geometrical convergence rate are established under
the assumption that the objective functions are strong convexity and have
Lipschitz continuous gradient. The theoretical analysis shows that the
distributed algorithm is capable of driving the whole network to geometrically
converge to an optimal solution of the convex optimization problem as long as
the uncoordinated step-sizes do not exceed some upper bounds. We also give an
explicit analysis for the convergence rate of our algorithm through a different
approach. Finally, simulation results illustrate the feasibility of the
proposed algorithm and the theoretical analysis throughout this paper.
