In this paper, we consider the problem of distributed optimisation of a separable convex cost function over a graph, where every edge and node in the graph could carry both linear equality and/or inequality constraints. We show how to modify the primal-dual method of multipliers (PDMM), originally designed for linear equality constraints, such that it can handle inequality constraints as well. The proposed algorithm does not need any slack variables, which is similar to the recent work [1] which extends the alternating direction method of multipliers (ADMM) for addressing decomposable optimisation with linear equality and inequality constraints. Using convex analysis, monotone operator theory and fixed-point theory, we show how to derive the update equations of the modified PDMM algorithm by applying Peaceman-Rachford splitting to the monotonic inclusion related to the lifted dual problem. To incorporate the inequality constraints, we impose a non-negativity constraint on the associated dual variables. This additional constraint results in the introduction of a reflection operator to model the data exchange in the network, instead of a permutation operator as derived for equality constraint PDMM. Convergence for both synchronous and stochastic update schemes of PDMM are provided. The latter includes asynchronous update schemes and update schemes with transmission losses. Experiments show that PDMM converges notably faster than extended ADMM of [1].

翻译：本文研究了图结构上可分离凸代价函数的分布式优化问题，其中图中的每条边和节点均可同时承载线性等式和/或不等式约束。我们展示了如何改进最初为线性等式约束设计的原始-对偶乘法器方法（PDMM），使其能够处理不等式约束。所提出的算法无需任何松弛变量，这与近期拓展交替方向乘子法（ADMM）以处理可分解线性等式与不等式约束的研究[1]具有相似性。通过运用凸分析、单调算子理论与不动点理论，我们阐明了如何通过将Peaceman-Rachford分裂应用于提升对偶问题相关单调包含关系，推导修正PDMM算法的更新方程。为纳入不等式约束，我们对关联对偶变量施加非负性约束。这一附加约束导致网络中数据交换需引入反射算子（而非等式约束PDMM中的置换算子）进行建模。本文提供了PDMM同步与随机更新方案的收敛性证明，后者涵盖异步更新方案及存在传输损耗的更新方案。实验结果表明，PDMM的收敛速度显著优于文献[1]的扩展ADMM方法。