Minima of Functions of Several Variables with Inequalities as Side Conditions

Abstract

The problem of determining necessary conditions and sufficient conditions for a relative minimum of a function $f({x_1},{x_2},....,{x_n})$ in the class of points $x = ({x_1},{x_2},....,{x_n})$ Satisfying the equations $\rm {g_{\alpha}(X)= 0 (\alpha = 1, 2,....,m),}$ where the functions f and g_α have continuous derivatives of at least the second order, has been satisfactorily treated [1]^*. This paper proposes to take up the corresponding problem in the class of points x satisfying the inequalities $\begin{array}{clcclclclcl}\rm {g_{\alpha}(x)\geqq 0} & & & & & & \rm{\alpha = 1,2,...,m}\end{array}$ where m may be less than, equal to, or greater than n.