Abstract: The characterization of global minimizers of a quadratic function with a spherical constaint is well-understood from classical works of Gay and Moré-Sorensen. In this paper we give a complete characterization of local- nonglobal minimizers of this problem. Essentially, we prove that there exist at most one local-nonglobal minimizer, and that its Lagrange multiplier is the larger solution of a single nonlinear equation. This generalizes to the n- dimensional case a previous result of Celis-Dennis-Martínez-Tapia. We give an algorithm for computing the local-nonglobal minimizer, and we suggest some applications Ver menos