The U-Lagrangian of the maximum eigenvalue function

Authors
Citation
F. Oustry, The U-Lagrangian of the maximum eigenvalue function, SIAM J OPTI, 9(2), 1999, pp. 526-549
Citations number
45
Categorie Soggetti
Mathematics
Journal title
SIAM JOURNAL ON OPTIMIZATION
ISSN journal
10526234 → ACNP
Volume
9
Issue
2
Year of publication
1999
Pages
526 - 549
Database
ISI
SICI code
1052-6234(19990420)9:2<526:TUOTME>2.0.ZU;2-A
Abstract
In this paper we apply the U-Lagrangian theory to the maximum eigenvalue fu nction lambda(1) and to its precomposition with affine matrix-valued mappin gs. We first give geometrical interpretations of the U-objects that we intr oduce. We also show that the U-Lagrangian of lambda(1) has a Hessian which can be explicitly computed; the second-order development of the U-Lagrangia n provides a second-order development of lambda(1) along a characteristic s mooth manifold: the set of symmetric matrices whose maximal eigenvalues hav e a fixed multiplicity. The same results can be obtained when we precompose lambda(1) with an affine matrix-valued mapping A, provided that this mappi ng satisfies a regularity condition (transversality condition). We show tha t the Hessian of the U-Lagrangian of lambda(1) circle A coincides with the reduced Hessian encountered in sequential quadratic programming. Finally, w e use the U-Lagrangian to derive second-order algorithms for minimizing lam bda(1) circle A.