  This paper is concerned with multi-agent optimization problem. A distributed
randomized gradient-free mirror descent (DRGFMD) method is developed by
introducing a randomized gradient-free oracle in the mirror descent scheme
where the non-Euclidean Bregman divergence is used. The classical gradient
descent method is generalized without using subgradient information of
objective functions. The proposed algorithm is the first distributed
non-Euclidean zeroth-order method which achieves an $O(1/\sqrt{T})$ convergence
rate, recovering the best known optimal rate of distributed compact constrained
convex optimization. Also, the DRGFMD algorithm achieves an $O(\ln T/T)$
convergence rate for the strongly convex constrained optimization case. The
rate matches the best known non-compact constraint result. Moreover, a
decentralized reciprocal weighted average approximating sequence is
investigated and first used in distributed algorithm. A class of convergence
rates are also achieved for the algorithm with weighted averaging (DRGFMD-WA).
The technique on constructing the decentralized weighted average sequence
provides new insight in searching for minimizers in distributed algorithms.
