On Lagrange Multipliers and Inequalities

Abstract

Necessary and sufficient conditions for minima (maxima) of nonlinear functionals subjected to linear constraints are derived. Two classes of functionals are considered ( a ) convex (concave) functionals for which necessary and sufficient conditions for global minima (maxima) are obtained, and ( b ) more general functionals possessing continuous second derivatives for which necessary and sufficient conditions for local optima are obtained. In the first case the theorems presented here are special cases of the well-known Kuhn-Tucker theorems. Some simple examples are included.