Second-Order Conditions for the Existence of Augmented Lagrange Multipliers for Sparse Optimization

Chao Kan () and Wen Song ()
Additional contact information
Chao Kan: Harbin Normal University
Wen Song: Harbin Normal University

Journal of Optimization Theory and Applications, 2024, vol. 201, issue 1, No 5, 103-129

Abstract: Abstract In this paper, we consider the augmented Lagrangian duality for optimization problems with sparsity and abstract set constraints and present second-order conditions for the existence of augmented Lagrange multipliers by calculating the second-order epi-derivative of the augmented Lagrangian. The ingredient of the augmented Lagrangian here includes the indicator function of a sparse set and a composition of the Moreau envelope of the indicator function of a second-order regular set and a twice continuously differentiable mapping. The main process depends heavily on the calculation of the second-order epi-derivative of the indicator function of sparse set which is shown to be second-order regular and also parabolically regular. The second-order sufficient conditions for the sparse nonlinear programming, the sparse inverse covariance selection problem, and the sparse second-order cone programming are obtained as special cases of our general results. We prove that the existence of augmented Lagrange multipliers ensures the exactness of penalty functions and the stability of augmented solutions under small perturbations of the corresponding augmented Lagrange multipliers.

Keywords: Augmented Lagrange multiplier; Augmented Lagrangian duality; Exact penalty; Second-order epi-derivative; Sparsity; 49K35; 49J52; 90C30 (search for similar items in EconPapers)
Date: 2024
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s10957-024-02382-w Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:joptap:v:201:y:2024:i:1:d:10.1007_s10957-024-02382-w

Ordering information: This journal article can be ordered from
http://www.springer. ... cs/journal/10957/PS2

DOI: 10.1007/s10957-024-02382-w

Access Statistics for this article

Journal of Optimization Theory and Applications is currently edited by Franco Giannessi and David G. Hull

More articles in Journal of Optimization Theory and Applications from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().