A gradient-free distributed optimization method for convex sum of nonconvex cost functions

This article presents a special type of distributed optimization problems, where the summation of agents' local cost functions (i.e., global cost function) is convex, but each individual can be nonconvex. Unlike most distributed optimization algorithms by taking the advantages of gradient, the considered problem is allowed to be nonsmooth, and the gradient information is unknown to the agents. To solve the problem, a Gaussian-smoothing technique is introduced and a gradient-free method is proposed. We prove that each agent's iterate approximately converges to the optimal solution both with probability 1 and in mean, and provide an upper bound on the optimality gap, characterized by the difference between the functional value of the iterate and the optimal value. The performance of the proposed algorithm is demonstrated by a numerical example and an application in privacy enhancement.