  In this paper we consider the problem of distributed nonlinear optimisation
of a separable convex cost function over a graph subject to cone constraints.
We show how to generalise, using convex analysis, monotone operator theory and
fixed-point theory, the primal-dual method of multipliers (PDMM), originally
designed for equality constraint optimisation and recently extended to include
linear inequality constraints, to accommodate for cone constraints. The
resulting algorithm can be used to implement a variety of optimisation
problems, including the important class of semidefinite programs with partially
separable structure, in a fully distributed fashion. We derive update equations
by applying the Peaceman-Rachford splitting algorithm to the monotonic
inclusion related to the lifted dual problem. The cone constraints are
implemented by a reflection method in the lifted dual domain where auxiliary
variables are reflected with respect to the intersection of the polar cone and
a subspace relating the dual and lifted dual domain. Convergence results for
both synchronous and stochastic update schemes are provided and an application
of the proposed algorithm is demonstrated to implement an approximate algorithm
for maximum cut problems based on semidefinite programming in a fully
distributed fashion.
