Distributed Nonlinear Conic Optimization with Partially Separable Structure

Abstract. In this paper, we consider the problem of distributed nonlinear optimization of a separable convex cost function over a graph subject to cone constraints. We show how to generalize using convex analysis, monotone operator theory, and fixed-point theory, the primal-dual method of multipliers (PDMM), originally designed for equality constraint optimization and recently extended to include linear inequality constraints, so that it can also accommodate cone constraints. The resulting algorithm can be applied to a variety of optimization problems, including the important class of semidefinite programs with partially separable structure, in a fully distributed fashion without relying on interior-point methods. 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 are provided for both synchronous and stochastic update schemes, and the proposed algorithm is demonstrated through an application to fully distributed sensor localization based on semidefinite programming.