Distributed Nonlinear Conic Optimisation with partially separable Structure

TL;DR

This paper develops a distributed primal-dual algorithm for nonlinear convex optimization with cone constraints, extending existing methods to handle complex cone constraints and applying it to semidefinite programming problems in a distributed setting.

Contribution

It generalizes the primal-dual method of multipliers to include cone constraints using convex analysis and fixed-point theory, enabling distributed solutions for complex optimization problems.

Findings

Algorithm converges for both synchronous and stochastic updates.

Applicable to semidefinite programs with partially separable structure.

Demonstrated on distributed maximum cut problem using semidefinite programming.

Abstract

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.