WHY LAGRANGE MULTIPLIERS WITH EXTREME MAGNITUDESGIVE EXTREMA OF DEFINITE HERMITIAN FORMS ONQUADRIC SURFACES

摘要

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 positivedefinite Hermitian form on a level set of any nonnecessarily definite Hermitian polynomial correspondsto the Lagrange multiplier with the smallest (respectively, largest) absolute value. Locally, the law ofsines for the triangle with vertices at the center and at two stationary points reveals that the objectivevalues are in the same order as the magnitudes of the Lagrange multipiers. Global geometry explainsthe same results for global minima and global maxima by showing that the constraining quadricsurface 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 differencebetween two stationary points with the same objective value is a generalized eigenvector.