## The U-Lagrangian Of The Maximum Eigenvalue Function (1998)

Citations: | 16 - 3 self |

### BibTeX

@MISC{Oustry98theu-lagrangian,

author = {François Oustry},

title = {The U-Lagrangian Of The Maximum Eigenvalue Function},

year = {1998}

}

### Years of Citing Articles

### OpenURL

### Abstract

. In this paper we apply the U-Lagrangian theory to the maximum eigenvalue function 1 and to its precomposition with ane matrix-valued mappings. We rst give geometrical interpretations of the U-objects that we introduce. We also show that the U-Lagrangian of 1 has a Hessian which can be explicitly computed; the second-order development of the U-Lagrangian provides a second-order development of 1 along a characteristic smooth manifold: the set of symmetric matrices whose maximal eigenvalues have a xed multiplicity. The same results can be obtained when we precompose 1 with an ane matrix-valued mapping A, provided that this mapping satises a regularity condition (transversality condition). We show that the Hessian of the U-Lagrangian of 1 A coincides with the reduced Hessian encountered in Sequential Quadratic Programming. Finally, we use the U-Lagrangian to derive second-order algorithms for minimizing 1 A. Key words. Eigenvalue optimization, convex optimization, generalize...