Extremum Problems with Inequalities as Subsidiary Conditions

Abstract

This paper deals with an extension of Lagrange’s multiplier rule to the case, where the subsidiary conditions are inequalities instead of equations. Only extrema of differentiable functions of a finite number of variables will be considered. There may however be an infinite number of inequalities prescribed. Lagrange’s rule for the situation considered here differs from the ordinary one, in that the multipliers may always be assumed to be positive. This makes it possible to obtain sufficient conditions for the occurence or a minimum in terms of the first derivatives only.