On the use of Jordan algebras for improving global convergence of an augmented Lagrangian method in nonlinear semidefinite programming
type: Item
instance of: scholarly article
title: On the use of Jordan algebras for improving global convergence of an augmented Lagrangian method in nonlinear semidefinite programming
zbMATH Open document ID: 1482.90137
author: Ellen H. Fukuda
author: Gabriel Haeser
author: D. O. Santos
author: L. D. Secchin
author: R. Andreani
published in: Computational Optimization and Applications
publication date: 2021-08-09T00:00:00Z
review text: Let \(\mathcal{E}\) be a finite dimensional real inner product space, \(f:\mathbb{R}^{n}\rightarrow \mathbb{R}\) and \(g:\mathbb{R}^{n}\rightarrow \mathcal{E}\) be continuously differentiable functions, and \(\mathcal{K\subseteq E}\) be a symmetric cone. This paper deals with the problem of minimizing \(f\left( x\right) \) subject to \(g\left( x\right) \in \mathcal{K}\). The authors consider a suitable extension of the CAKKT necessary optimality condition introduced by \textit{R. Andreani} et al. [SIAM J. Optim 20, No. 6, 3533--3554 (2010; Zbl 1217.90148)] for nonlinear programming problems, and prove that, in the case when \(\mathcal{K}\) is the cone of of semidefinite \(m\times m\) real symmetric matrices, it is stronger than the corresponding extension of the AKKT condition introduced by \textit{R. Andreani} et al. [Optimization 60, No. 4--6, 627--641 (2011; Zbl 1225.90123)]. The same holds true for arbitrary symmetric cones under an additional assumption. They also show that the augmented Lagrangian method and the primal-dual interior point method proposed by \textit{R. Andreani} et al. [Math. Program. 180, No. 1--2 (A), 203--235 (2020; Zbl 1434.90121)] and \textit{H. Yamashita} et al. [Math. Program. 135, No. 1--2 (A), 89--121 (2012; Zbl 1273.90150)], respectively, for nonlinear semidefinite programming problems generate sequences satisfying the CAKKT condition, thus improving the global convergence results obtained in those references.
Mathematics Subject Classification ID: 90C22
Mathematics Subject Classification ID: 90C30
zbMATH DE Number: 7379832
zbMATH Keywords: augmented Lagrangian method
zbMATH Keywords: nonlinear semidefinite programming
zbMATH Keywords: symmetric cones
zbMATH Keywords: optimality conditions
zbMATH Keywords: primal-dual interior point method
reviewed by: Juan-Enrique Martínez-Legaz
MaRDI profile type: Publication
full work available at URL: s10589-021-00281-8
OpenAlex ID: W3163615686
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DOI: 10.1007/S10589-021-00281-8
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description: scientific article; zbMATH DE number 7379832