Decentralized dual methods play significant roles in large-scale optimization, which effectively resolve many constrained optimization problems in machine learning and power systems. In this article, we focus on studying a class of totally non-smooth constrained composite optimization problems over multi-agent systems, where the mutual goal of agents in the system is to optimize a sum of two separable non-smooth functions consisting of a strongly-convex function and another convex (not necessarily strongly-convex) function. Agents in the system conduct parallel local computation and communication in the overall process without leaking their private information. In order to resolve the totally non-smooth constrained composite optimization problem in a fully decentralized manner, we devise a synchronous decentralized dual proximal (SynDe-DuPro) gradient algorithm and its asynchronous version (AsynDe-DuPro) based on the randomized block-coordinate method. Both SynDe-DuPro and AsynDe-DuPro algorithms are theoretically proved to achieve the globally optimal solution to the totally non-smooth constrained composite optimization problem relied on the quasi-Fejér monotone theorem. As a main result, AsynDe-DuPro algorithm attains the globally optimal solution without requiring all agents to be activated at each iteration and thus is more robust than most existing synchronous algorithms. The practicability of the proposed algorithms and correctness of the theoretical findings are demonstrated by the experiments on a constrained Decentralized Sparse Logistic Regression (DSLR) problem in machine learning and a Decentralized Energy Resources Coordination (DERC) problem in power systems.

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非光滑约束复合优化问题的分散对偶近邻梯度算法

分散对偶方法在大规模优化中起着重要作用，有效地解决了机器学习和电力系统中许多受限的优化问题。在本文中，我们着重研究一类基于多智能体系统的完全非光滑约束复合优化问题，其中系统中智能体的共同目标是优化两个可分离的非光滑函数之和，该函数包含一个-凸函数和另一个凸（不一定是强凸）函数。系统中的代理在整个过程中进行并行的本地计算和通信，而不会泄漏其私有信息。为了以完全分散的方式解决完全非平滑约束的复合优化问题，我们设计了一种基于随机块坐标法的同步分散双近邻（SynDe-DuPro）梯度算法及其异步版本（AsynDe-DuPro）。理论上证明，SynDe-DuPro算法和AsynDe-DuPro算法都可以基于拟Fejér单调定理，针对完全非光滑约束的复合优化问题实现全局最优解。作为主要结果，AsynDe-DuPro算法无需在每次迭代中激活所有代理即可获得全局最优解决方案，因此比大多数现有的同步算法更强大。