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Volume 36, Issue 3
A Complete Characterization of the Robust Isolated Calmness of Nuclear Norm Regularized Convex Optimization Problems

Ying Cui & Defeng Sun

J. Comp. Math., 36 (2018), pp. 441-458.

Published online: 2018-06

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  • Abstract

In this paper, we provide a complete characterization of the robust isolated calmness of the Karush-Kuhn-Tucker (KKT) solution mapping for convex constrained optimization problems regularized by the nuclear norm function. This study is motivated by the recent work in [8], where the authors show that under the Robinson constraint qualification at a local optimal solution, the KKT solution mapping for a wide class of conic programming problems is robustly isolated calm if and only if both the second order sufficient condition (SOSC) and the strict Robinson constraint qualification (SRCQ) are satisfied. Based on the variational properties of the nuclear norm function and its conjugate, we establish the equivalence between the primal/dual SOSC and the dual/primal SRCQ. The derived results lead to several equivalent characterizations of the robust isolated calmness of the KKT solution mapping and add insights to the existing literature on the stability of nuclear norm regularized convex optimization problems.

  • Keywords

Robust isolated calmness, Nuclear norm, Second order sufficient condition, Strict Robinson constraint qualification.

  • AMS Subject Headings

90C25, 90C31, 65K10

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

yingcui@usc.edu (Ying Cui)

defeng.sun@polyu.edu.hk (Defeng Sun)

  • BibTex
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@Article{JCM-36-441, author = {Ying and Cui and yingcui@usc.edu and 6806 and The Daniel J. Epstein Department of Industrial and Systems Engineering, University of Southern California, Los Angeles, California, USA and Ying Cui and Defeng and Sun and defeng.sun@polyu.edu.hk and 5403 and Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China and Defeng Sun}, title = {A Complete Characterization of the Robust Isolated Calmness of Nuclear Norm Regularized Convex Optimization Problems}, journal = {Journal of Computational Mathematics}, year = {2018}, volume = {36}, number = {3}, pages = {441--458}, abstract = {

In this paper, we provide a complete characterization of the robust isolated calmness of the Karush-Kuhn-Tucker (KKT) solution mapping for convex constrained optimization problems regularized by the nuclear norm function. This study is motivated by the recent work in [8], where the authors show that under the Robinson constraint qualification at a local optimal solution, the KKT solution mapping for a wide class of conic programming problems is robustly isolated calm if and only if both the second order sufficient condition (SOSC) and the strict Robinson constraint qualification (SRCQ) are satisfied. Based on the variational properties of the nuclear norm function and its conjugate, we establish the equivalence between the primal/dual SOSC and the dual/primal SRCQ. The derived results lead to several equivalent characterizations of the robust isolated calmness of the KKT solution mapping and add insights to the existing literature on the stability of nuclear norm regularized convex optimization problems.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1709-m2017-0034}, url = {http://global-sci.org/intro/article_detail/jcm/12270.html} }
TY - JOUR T1 - A Complete Characterization of the Robust Isolated Calmness of Nuclear Norm Regularized Convex Optimization Problems AU - Cui , Ying AU - Sun , Defeng JO - Journal of Computational Mathematics VL - 3 SP - 441 EP - 458 PY - 2018 DA - 2018/06 SN - 36 DO - http://doi.org/10.4208/jcm.1709-m2017-0034 UR - https://global-sci.org/intro/article_detail/jcm/12270.html KW - Robust isolated calmness, Nuclear norm, Second order sufficient condition, Strict Robinson constraint qualification. AB -

In this paper, we provide a complete characterization of the robust isolated calmness of the Karush-Kuhn-Tucker (KKT) solution mapping for convex constrained optimization problems regularized by the nuclear norm function. This study is motivated by the recent work in [8], where the authors show that under the Robinson constraint qualification at a local optimal solution, the KKT solution mapping for a wide class of conic programming problems is robustly isolated calm if and only if both the second order sufficient condition (SOSC) and the strict Robinson constraint qualification (SRCQ) are satisfied. Based on the variational properties of the nuclear norm function and its conjugate, we establish the equivalence between the primal/dual SOSC and the dual/primal SRCQ. The derived results lead to several equivalent characterizations of the robust isolated calmness of the KKT solution mapping and add insights to the existing literature on the stability of nuclear norm regularized convex optimization problems.

Ying Cui & Defeng Sun. (2020). A Complete Characterization of the Robust Isolated Calmness of Nuclear Norm Regularized Convex Optimization Problems. Journal of Computational Mathematics. 36 (3). 441-458. doi:10.4208/jcm.1709-m2017-0034
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