Why Lagrange Multipliers with Extreme Magnitudes Give Extrema of Definite Hermitian Forms on Quadric Surfaces

Abstract

This work explains the geometry of analytic theorems by Forsythe and Golub, Gander, and More to the effect that the constrained minimum (respectively, maximum) of a positive definite Hermitian form on a level set of any nonnecessarily definite Hermitian polynomial corresponds to the Lagrange multiplier with the smallest (respectively, largest) absolute value. Locally, the law of sines for the triangle with vertices at the center and at two stationary points reveals that the objective values are in the same order as the magnitudes of the Lagrange multipiers. Global geometry explains the same results for global minima and global maxima by showing that the constraining quadric surface and the Lagrange multiplier form a set of geodetic coordinates for the entire ambient space. Duality and the symmetry of the constraining quadratic hypersurface also explain why the difference between two stationary points with the same objective value is a generalized eigenvector.