  This paper 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 non-convex. Unlike most
distributed optimization algorithms by taking the advantages of gradient, the
considered problem is allowed to be non-smooth, 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.
