用于$(L_0, L_1)$-光滑凸优化的不精确增广拉格朗日方法

中文摘要 AI 辅助

增广拉格朗日方法是求解约束凸优化问题最有效的方法之一。然而，对增广拉格朗日框架内一阶方法的经典复杂度分析通常依赖于目标函数具有利普希茨连续梯度的假设，该假设排除了一类梯度可能无界增长的重要广义光滑函数。本文研究一种不精确增广拉格朗日方法，用于求解具有$(L_0, L_1)$-光滑目标函数的线性约束凸优化问题。我们证明，增广拉格朗日子问题保留了$(L_0, L_1)$-光滑结构，其参数取决于惩罚系数。该性质使我们能够采用近期为广义光滑优化设计的加速一阶方法，而非经典的光滑优化方法。特别地，我们将不精确增广拉格朗日框架与基于截断梯度下降和加速优化的两阶段加速过程相结合。

英文摘要

Augmented Lagrangian methods are among the most effective approaches for solving constrained convex optimization problems. However, classical complexity analyses of first-order methods applied within the augmented Lagrangian framework usually rely on the assumption that the objective function has a Lipschitz continuous gradient. This assumption excludes an important class of generalized smooth functions whose gradients may grow unboundedly. In this paper, we study an inexact augmented Lagrangian method for solving linearly constrained convex optimization problems with $(L_0,L_1)$-smooth objective functions. We show that the augmented Lagrangian subproblems preserve the $(L_0,L_1)$-smooth structure, with parameters depending on the penalty coefficient. This property allows us to employ recent accelerated first-order schemes designed for generalized smooth optimization instead of classical smooth optimization methods. In particular, we combine the inexact augmented Lagrangian framework with a two-stage acceleration procedure based on clipped gradient descent and accelerated optimization.

发表机构

• Moscow Institute of Physics and Technology（莫斯科物理技术学院）
• Innopolis University（因诺波利斯大学）
• Adyghe State University（阿迪格国立大学）
• V. I. Vernadsky Crimean Federal University（V.I.维尔纳茨基克里米亚联邦大学）

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