Strong variational sufficiency is a newly proposed property, which turns out to be of great use in the convergence analysis of multiplier methods. However, what this property implies for nonpolyhedral problems remains a puzzle. In this paper, we prove the equivalence between the strong variational sufficiency and the strong second-order sufficient condition (SOSC) for nonlinear semidefinite programming (NLSDP) without requiring the uniqueness of the multiplier or any other constraint qualifications. Based on this characterization, the local convergence property of the augmented Lagrangian method (ALM) for NLSDP can be established under the strong SOSC in the absence of constraint qualifications. Moreover, under the strong SOSC, we can apply the semismooth Newton method to solve the ALM subproblems of NLSDP because the positive definiteness of the generalized Hessian of augmented Lagrangian function is satisfied.

Publication:

SIAM Journal on Optimization ( Volume: 33, Issue: 4, 2023)

https://doi.org/10.1137/22M1530161

Author:

Shiwei Wang

School of Mathematical Sciences, University of Chinese Academy of Science, Beijing, People’s Republic of China and Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, People’s Republic of China

Chao Ding

School of Mathematical Sciences, University of Chinese Academy of Science, Beijing, People’s Republic of China and Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, People’s Republic of China

Email: dingchao@amss.ac.cn

Yangjing Zhang

Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, People’s Republic of China

Xinyuan Zhao

Department of Mathematics, Beijing University of Technology, Beijing, People’s Republic of China