论文标题
BALPA：一种用于非平衡优化的平衡原始二算法，并应用于分布式优化
BALPA: A Balanced Primal-Dual Algorithm for Nonsmooth Optimization with Application to Distributed Optimization
论文作者
论文摘要
在本文中，我们提出了一种新型的原始双偶近距离拆分算法（PD-PSA），名为BALPA，因为具有相等性约束的复合优化问题，其中损耗函数由平滑项组成，由平滑项组成，由线性映射组成。在BALPA中，双重更新被设计为具有时间变化的二次功能的近端点，该功能平衡了原始更新和双重更新的实现，并保留了经典PD-PSA的接近诱导的功能。此外，通过这种平衡，BALPA消除了对于复合优化问题的经典PD-PSA的效率低下，其中线性映射的欧几里得规范或相等性约束映射很大。因此，BALPA不仅继承了简单结构的优势和经典PD-PSA的容易实现，而且还可以确保当这些规范很大时快速收敛。此外，我们提出了一个随机版本的BALPA（S-BALPA），并将开发的BALPA应用于分布式优化，以设计新的分布式优化算法。此外，分别对BALPA和S-BALPA进行了全面的收敛分析。最后，数值实验证明了所提出的算法的效率。
In this paper, we propose a novel primal-dual proximal splitting algorithm (PD-PSA), named BALPA, for the composite optimization problem with equality constraints, where the loss function consists of a smooth term and a nonsmooth term composed with a linear mapping. In BALPA, the dual update is designed as a proximal point for a time-varying quadratic function, which balances the implementation of primal and dual update and retains the proximity-induced feature of classic PD-PSAs. In addition, by this balance, BALPA eliminates the inefficiency of classic PD-PSAs for composite optimization problems in which the Euclidean norm of the linear mapping or the equality constraint mapping is large. Therefore, BALPA not only inherits the advantages of simple structure and easy implementation of classic PD-PSAs but also ensures a fast convergence when these norms are large. Moreover, we propose a stochastic version of BALPA (S-BALPA) and apply the developed BALPA to distributed optimization to devise a new distributed optimization algorithm. Furthermore, a comprehensive convergence analysis for BALPA and S-BALPA is conducted, respectively. Finally, numerical experiments demonstrate the efficiency of the proposed algorithms.
