On the use of Jordan Algebras for improving global convergence of an Augmented Lagrangian method in nonlinear semidefinite programming

R. Andreani (), E. H. Fukuda (), G. Haeser (), D. O. Santos () and L. D. Secchin ()
Additional contact information
R. Andreani: University of Campinas
E. H. Fukuda: Kyoto University
G. Haeser: University of São Paulo
D. O. Santos: Federal University of São Paulo
L. D. Secchin: Federal University of Espírito Santo

Computational Optimization and Applications, 2021, vol. 79, issue 3, No 4, 633-648

Abstract: Abstract Jordan Algebras are an important tool for dealing with semidefinite programming and optimization over symmetric cones in general. In this paper, a judicious use of Jordan Algebras in the context of sequential optimality conditions is done in order to generalize the global convergence theory of an Augmented Lagrangian method for nonlinear semidefinite programming. An approximate complementarity measure in this context is typically defined in terms of the eigenvalues of the constraint matrix and the eigenvalues of an approximate Lagrange multiplier. By exploiting the Jordan Algebra structure of the problem, we show that a simpler complementarity measure, defined in terms of the Jordan product, is stronger than the one defined in terms of eigenvalues. Thus, besides avoiding a tricky analysis of eigenvalues, a stronger necessary optimality condition is presented. We then prove the global convergence of an Augmented Lagrangian algorithm to this improved necessary optimality condition. The results are also extended to an interior point method. The optimality conditions we present are sequential ones, and no constraint qualification is employed; in particular, a global convergence result is available even when Lagrange multipliers are unbounded.

Keywords: Nonlinear semidefinite programming; Symmetric cones; Optimality conditions; Constraint qualifications; Augmented Lagrangian method; 90C30; 90C33; 90C46; 65K05 (search for similar items in EconPapers)
Date: 2021
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (2)

Downloads: (external link)
http://link.springer.com/10.1007/s10589-021-00281-8 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:coopap:v:79:y:2021:i:3:d:10.1007_s10589-021-00281-8

Ordering information: This journal article can be ordered from
http://www.springer.com/math/journal/10589

DOI: 10.1007/s10589-021-00281-8

Access Statistics for this article

Computational Optimization and Applications is currently edited by William W. Hager

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