非正则线性增广拉格朗日乘子法的收敛速率研究  

CONVERGENCE RATE OF THE INDEFINITE LINEARIZED AUGMENTED LAGRANGIAN METHOD

作　　者：马峰 倪明放 Ma Feng;Ni Mingfang(High-Tech Institute of Xi'an,Xi'an 710025;Zhuhai College of Science and Technology,Zhuhai 519041)

机构地区：[1]火箭军工程大学作战保障学院,西安710025 [2]珠海科技学院,珠海519041

出　　处：《高等学校计算数学学报》2022年第1期36-49,共14页Numerical Mathematics A Journal of Chinese Universities

基　　金：Supported by the NSFC Grant 12171481;NSF of Shaanxi Province grant 2020JQ-485.

摘　　要：Recently,an indefinite linearized augmented Lagrangian method(IL-ALM)was proposed for the convex programming problems with linear constraints.The IL-ALM differs from the linearized augmented Lagrangian method in that the augmented Lagrangian is linearized by adding an indefinite quadratic proximal term.But,it preserves the algorithmic feature of the linearized ALM and usually has the advantage to improve the performance.The IL-ALM is proved to be convergent from contraction perspective,but its convergence rate is still missing.This is mainly because that the indefinite setting destroys the structures when we directly employ the contraction frameworks.In this paper,we derive the convergence rate for this algorithm by using a different analysis.We prove that a worst-case O(1/t)convergence rate is still hold for this algorithm,where t is the number of iterations.Additionally we show that the customized proximal point algorithm can employ larger step sizes by proving its equivalence to the linearized ALM.

关 键 词：Convex programming augmented Lagrangian method proximal point algorithm convergence analysis indefinite regularization 

分 类 号：O221.2[理学—运筹学与控制论] O224[理学—数学]