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. In contrast to most existing algorithms for optimisation with inequality constraints, the proposed algorithm does not need any slack variables. 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 extended 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.

翻译：本文研究了在图结构下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可能同时携带线性等式和/或不等式约束。我们展示了如何修改最初为线性等式约束设计的原始-对偶乘子法（PDMM），使其也能处理不等式约束。与大多数现有不等式约束优化算法不同，所提算法无需引入任何松弛变量。通过凸分析、单调算子理论和不动点理论，我们证明了如何将Peaceman-Rachford分裂应用于扩展对偶问题相关的单调包含关系，从而推导出修改后PDMM算法的更新方程。为纳入不等式约束，我们对相关对偶变量施加了非负约束。这一附加约束导致网络数据交换中引入反射算子以替代等式约束PDMM中使用的置换算子。本文给出了同步和随机更新方案下PDMM的收敛性证明，后者包括异步更新方案及存在传输损耗的更新方案。