Nonsingularity and symmetry for linear normal maps

Normal maps are single-valued, generally nonsmooth functions expressing conditions for the solution of variational problems such as those of optimization or equilibrium. Normal maps arising from linear transformations are particularly important, both in their own right and as predictors of the behavior of related nonlinear normal maps. They are called (locally or globally)nonsingular if the functions appearing in them are (local or global) homeomorphisms satisfying a Lipschitz condition. We show here that when the linear transformation giving rise to such a normal map has a certain symmetry property, the necessary and sufficient condition for nonsingularity takes a particularly simple and convenient form, being simply a positive definiteness condition on a certain subspace.This paper in dedicated to Phil Wolfe on the occasion of his 65th birthday.The research reported here was sponsored by the National Science Foundation under Grant CCR-9109345, by the Air Force Systems Command, USAF, under Grant AFOSR-91-0089, by the U.S. Army Research Office under Contract No. DAAL03-89-K-0149, and by the U.S. Army Strategic Defense Command under Contract DASG60-91-C-0144. The US Government has certain rights in this material, and is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon.