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 all 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 a new provably sublinear rate of convergence expressed in terms of functions 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 all 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：在本文中,我们的目标是统一、简化和改进对基于拉格朗日法的方法的趋同率分析,以便解决锥形优化问题。我们首先引入了精美原始算法图的概念,该图在统一和简化分析所有基于拉格朗日法的方法方面发挥着核心作用。我们用一个精美原始算法图,然后引入一个多功能通用方案,以便设计和分析快速拉格朗日法(FLAG)方法,其新的可辨的次线性趋同率表现在功能值和违反原始(非电子)生成序列的可行性方面。为了展示我们的方法和结果的力量和多功能,我们展示出所有著名的标志性拉格朗日法办法都接受一个良好的原始算法图,从而在相应的FLAG中分享新的更快的趋同率。