Strong Variational Sufficiency for Nonlinear Semidefinite Programming and its Implications
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2022
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| _version_ | 1866910335466733568 |
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| author | Wang, Shiwei Ding, Chao Zhang, Yangjing Zhao, Xinyuan |
| author_facet | Wang, Shiwei Ding, Chao Zhang, Yangjing Zhao, Xinyuan |
| contents | 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 non-polyhedral 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 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 strong SOSC in the absence of constraint qualifications. Moreover, under the strong SOSC, we can apply the semi-smooth Newton method to solve the ALM subproblems of NLSDP as the positive definiteness of the generalized Hessian of augmented Lagrangian function is satisfied. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_04448 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Strong Variational Sufficiency for Nonlinear Semidefinite Programming and its Implications Wang, Shiwei Ding, Chao Zhang, Yangjing Zhao, Xinyuan Optimization and Control 49J52, 90C22, 90C46 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 non-polyhedral 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 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 strong SOSC in the absence of constraint qualifications. Moreover, under the strong SOSC, we can apply the semi-smooth Newton method to solve the ALM subproblems of NLSDP as the positive definiteness of the generalized Hessian of augmented Lagrangian function is satisfied. |
| title | Strong Variational Sufficiency for Nonlinear Semidefinite Programming and its Implications |
| topic | Optimization and Control 49J52, 90C22, 90C46 |
| url | https://arxiv.org/abs/2210.04448 |