Distributed convex optimization with a row-stochastic matrix over directed graphs

Abstract

This work considers a convex optimization problem on basis of the multi-agent system where the global objective function is the sum of all agents' individual cost functions, and each of that is only be aware by itself. The agents' communication graph is depicted to be strongly connected. A striking feature in this paper is that a row stochastic matrix is exploited in the algorithm, which exactly drives each agent to gradually converge to some common optimum value in spite of the fixed step-size. Moreover, to obtain the conclusion, we give the assumption about the smoothness and convexity of the objective function. Thereupon, we can see that the method we give can steer the network to an optimal solution with geometrical convergence, so long as the fixed step-size does not exceed a certain superior limit. Finally, we still more give a simulation experiment to represent the efficacy of our algorithm and to verify our theoretical findings.