A Necessary and Sufficient Condition for a Constrained Minimum

Abstract

Let U be an open subset of $\mathbb{R}^n $, X a compact semi-analytic subset of U, $(f_0 ,f):U \to \mathbb{R} \times \mathbb{R}^n $ analytic, and $0 \in f(X)$. It is proven that a point $x_0 \in X$ minimizes $f_0 (x)$ subject to $f(x) = 0$ if and only if $x_0 \in X$ minimizes $f_0 (x) + c | {f(x)} |^{1/N} $ for all sufficiently large c and N. This reduces the constrained minimization problem to a finite number of unconstrained problems.