On the use of Jordan algebras for improving global convergence of an augmented Lagrangian method in nonlinear semidefinite programming
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.
scientific article; zbMATH DE number 7379832
scientific article