论文标题
基于增强Lagrangian的非凸的限制优化的一阶原始偶二方法
A First-Order Primal-Dual Method for Nonconvex Constrained Optimization Based On the Augmented Lagrangian
论文作者
论文摘要
非线性约束的非凸和非平滑优化模型在机器学习，统计和数据分析中起着越来越重要的作用。在本文中，基于增强的拉格朗日函数，我们引入了一种灵活的一阶原始偶对偶，被称为Nonconvex辅助问题，用于求解一类非线性约束的非凸dvex和非平滑优化问题的增强lagrangian（Napp-al）的原则。我们证明，Napp-al以O（1/\ sqrt {K}）的速率收敛到固定解，其中k是迭代的数量。此外，在额外的误差绑定条件下（在论文中称为VP-EB），我们进一步表明收敛速率实际上是线性的。最后，我们表明著名的库迪卡·洛哈西维奇（Kurdyka-Lojasiewicz）的财产和度量次限制意味着上述VP-EB条件。
Nonlinearly constrained nonconvex and nonsmooth optimization models play an increasingly important role in machine learning, statistics and data analytics. In this paper, based on the augmented Lagrangian function we introduce a flexible first-order primal-dual method, to be called nonconvex auxiliary problem principle of augmented Lagrangian (NAPP-AL), for solving a class of nonlinearly constrained nonconvex and nonsmooth optimization problems. We demonstrate that NAPP-AL converges to a stationary solution at the rate of o(1/\sqrt{k}), where k is the number of iterations. Moreover, under an additional error bound condition (to be called VP-EB in the paper), we further show that the convergence rate is in fact linear. Finally, we show that the famous Kurdyka- Lojasiewicz property and the metric subregularity imply the afore-mentioned VP-EB condition.
