Unified convergence analysis of a second-order method of multipliers for nonlinear conic programming  

作　　者：Liang Chen Junyuan Zhu Xinyuan Zhao 

机构地区：[1]School of Mathematics,Hunan University,Changsha 410082,China [2]Hunan Provincial Key Laboratory of Intelligent Information Processing and Applied Mathematics,Changsha 410082,China [3]School of Mathematics,Beijing University of Technology,Beijing 100124,China

出　　处：《Science China Mathematics》2022年第11期2397-2422,共26页中国科学：数学（英文版）

基　　金：supported by National Natural Science Foundation of China (Grant No. 11801158);the Hunan Provincial Natural Science Foundation of China (Grant No. 2019JJ50040);the Fundamental Research Funds for the Central Universities in China;supported by National Natural Science Foundation of China (Grant No. 11871002);the General Program of Science and Technology of Beijing Municipal Education Commission (Grant No. KM201810005004)

摘　　要：In this paper,we accomplish the unified convergence analysis of a second-order method of multipliers(i.e.,a second-order augmented Lagrangian method)for solving the conventional nonlinear conic optimization problems.Specifically,the algorithm that we investigate incorporates a specially designed nonsmooth(generalized)Newton step to furnish a second-order update rule for the multipliers.We first show in a unified fashion that under a few abstract assumptions,the proposed method is locally convergent and possesses a(nonasymptotic)superlinear convergence rate,even though the penalty parameter is fixed and/or the strict complementarity fails.Subsequently,we demonstrate that for the three typical scenarios,i.e.,the classic nonlinear programming,the nonlinear second-order cone programming and the nonlinear semidefinite programming,these abstract assumptions are nothing but exactly the implications of the iconic sufficient conditions that are assumed for establishing the Q-linear convergence rates of the method of multipliers without assuming the strict complementarity.

关 键 词：second-order method of multipliers augmented Lagrangian method convergence rate generalized Newton method second-order cone programming semidefinite programming 

分 类 号：O221[理学—运筹学与控制论]