On inertia and schur complement in optimization

Abstract

Partitioned symmetric matrices, in particular the Hessian of the Lagrangian, play a fundamental role in nonlinear optimization. For this type of matrices S.-P. Han and O. Fujiwara recently presented an inertia theorem under a certain regularity assumption. We prove that this theorem is true without any regularity assumption. Then we consider matrix extensions preserving the sign of the determinant. Such extensions are shown to be related with the positive definiteness of some Schur complement. Under a regularity assumption this shows, from the viewpoint of linear algebra, the equivalence of strong stability in the sense of M. Kojima and strong regularity in the sense of S. M. Robinson. Finally, we discuss the inertia of a typical one-parameter family of symmetric matrices, occurring in various places in optimization (augmented Lagrangians, focal-point theory, etc.).