In this paper, we aim at unifying, simplifying and improving the convergence rate analysis of Lagrangian-based methods for convex optimization problems. We first introduce the notion of nice primal algorithmic map, which plays a central role in the unification and in the simplification of the analysis of most Lagrangian-based methods. Equipped with a nice primal algorithmic map, we then introduce a versatile generic scheme, which allows for the design and analysis of Faster LAGrangian (FLAG) methods with new provably sublinear rate of convergence expressed in terms of function values and feasibility violation of the original (non-ergodic) generated sequence. To demonstrate the power and versatility of our approach and results, we show that most well-known iconic Lagrangian-based schemes admit a nice primal algorithmic map, and hence share the new faster rate of convergence results within their corresponding FLAG.

翻译：本文旨在统一、简化并改进凸优化问题中基于拉格朗日方法的收敛速率分析。我们首先引入“优原始算法映射”概念，该概念在统一和简化大多数基于拉格朗日方法的分析中起核心作用。在具备优原始算法映射的基础上，我们提出了一种通用架构，该架构能够设计并分析具有新亚线性收敛速率的更快拉格朗日（FLAG）方法，其收敛速率以原始（非遍历）生成序列的函数值和可行性违反程度表示。为证明本方法及结果的强大性和普适性，我们展示了大多数经典的基于拉格朗日的标志性方案均具有优原始算法映射，因此在其对应的FLAG中可共享这些新的更快收敛速率结果。