  This paper addresses the problem of nonconvex nonsmooth decentralised
optimisation in multi-agent networks with undirected connected communication
graphs. Our contribution lies in introducing an algorithmic framework designed
for the distributed minimisation of the sum of a smooth (possibly nonconvex and
non-separable) function and a convex (possibly nonsmooth and non-separable)
regulariser. The proposed algorithm can be seen as a modified version of the
ADMM algorithm where, at each step, an "inner loop" needs to be iterated for a
number of iterations. The role of the inner loop is to aggregate and
disseminate information across the network. We observe that a naive
decentralised approach (one iteration of the inner loop) may not converge. We
establish the asymptotic convergence of the proposed algorithm to the set of
stationary points of the nonconvex problem where the number of iterations of
the inner loop increases logarithmically with the step count of the ADMM
algorithm. We present numerical results demonstrating the proposed method's
correctness and performance.
