A complete characterization of the robust isolated calmness of nuclear norm regularized convex optimization problems*

Ying Cui, Defeng Sun

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

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.

Original languageEnglish (US)
Pages (from-to)441-458
Number of pages18
JournalJournal of Computational Mathematics
Volume36
Issue number3
DOIs
StatePublished - 2019
Externally publishedYes

Bibliographical note

Funding Information:
Acknowledgement The second author is on leave from National University of Singapore and his research is supported in part by the Academic Research Fund under Grant R-146-000-207-112. We would like to thank the two anonymous referees and the guest editor for their many helpful comments on improving the quality of this paper.

Publisher Copyright:
© 2019 Global Science Press. All rights reserved.

Keywords

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

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