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.
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)
