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In this paper, a gradient-free distributed algorithm is introduced to solve aset constrained optimization problem under a directed communication network.Specifically, at each time-step, the agents locally compute a so-calledpseudo-gradient to guide the updates of the decision variables, which can beapplied in the fields where the gradient information is unknown, not availableor non-existent. A surplus-based method is adopted to remove the doublystochastic requirement on the weighting matrix, which enables theimplementation of the algorithm in graphs having no associated doublystochastic weighting matrix. For the convergence results, the proposedalgorithm is able to obtain the exact convergence to the optimal value with anypositive, non-summable and non-increasing step-sizes. Furthermore, when thestep-size is also square-summable, the proposed algorithm is guaranteed toachieve the exact convergence to an optimal solution. In addition to thestandard convergence analysis, the convergence rate of the proposed algorithmis also investigated. Finally, the effectiveness of the proposed algorithm isverified through numerical simulations.

Gradient-Free Distributed Optimization with Exact Convergence