提出一种高效分布式优化算法，解决网络中局部目标函数之和的优化问题。
A distributed semismooth Newton based augmented Lagrangian method for distributed optimization
- 基于增广拉格朗日法与半光滑牛顿法，分步求解子问题。
- 通过分布式加速近端梯度法计算牛顿方向，避免传递完整海森矩阵。
- 理论保证收敛性，实验显示优于现有先进算法。
本文提出一种新型分布式半光滑牛顿增广拉格朗日方法，用于求解网络中由局部代价函数之和定义的优化问题，通信仅限于相邻代理之间。具体地，将原问题等价重写为约束形式，并采用增广拉格朗日法求解。每个子问题通过分布式半光滑牛顿法近似求解。充分利用广义海森矩阵结构，设计了一种分布式加速近端梯度法以高效计算牛顿方向，避免传输完整的海森矩阵。理论分析证明了所提算法的收敛性。数值实验表明，该算法在效率和性能上均优于现有先进分布式算法。
原文摘要 · Abstract (English)
This paper proposes a novel distributed semismooth Newton based augmented Lagrangian method for solving a class of optimization problems over networks, where the global objective is defined as the sum of locally held cost functions, and communication is restricted to neighboring agents. Specifically, we employ the augmented Lagrangian method to solve an equivalently reformulated constrained version of the original problem. Each resulting subproblem is solved inexactly via a distributed semismooth Newton method. By fully leveraging the structure of the generalized Hessian, a distributed accelerated proximal gradient method is proposed to compute the Newton direction efficiently, eliminating the need to communicate with full Hessian matrices. Theoretical results are also obtained to guarantee the convergence of the proposed algorithm. Numerical experiments demonstrate the efficiency and superiority of our algorithm compared to state-of-the-art distributed algorithms.
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