Back to Search
Start Over
On the use of Jordan Algebras for improving global convergence of an Augmented Lagrangian method in nonlinear semidefinite programming
- Source :
- Repositório Institucional da USP (Biblioteca Digital da Produção Intelectual), Universidade de São Paulo (USP), instacron:USP
- Publication Year :
- 2021
- Publisher :
- Springer Science and Business Media LLC, 2021.
-
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.
- Subjects :
- Semidefinite programming
021103 operations research
Control and Optimization
Jordan algebra
Augmented Lagrangian method
Applied Mathematics
MÉTODOS NUMÉRICOS
MathematicsofComputing_NUMERICALANALYSIS
0211 other engineering and technologies
010103 numerical & computational mathematics
02 engineering and technology
01 natural sciences
Measure (mathematics)
Computational Mathematics
symbols.namesake
Lagrange multiplier
ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION
Convergence (routing)
symbols
Applied mathematics
0101 mathematics
Interior point method
Eigenvalues and eigenvectors
Mathematics
Subjects
Details
- ISSN :
- 15732894 and 09266003
- Volume :
- 79
- Database :
- OpenAIRE
- Journal :
- Computational Optimization and Applications
- Accession number :
- edsair.doi.dedup.....81802759c721963eb44e2c2e29d40f55