Implicit Functions and Optimization Problems without Continuous Differentiability of the Data

Abstract

Let $\phi$ be a function from a normed linear space X into a finite-dimensional Euclidean space Y and let A be a continuous linear mapping from XontoY. We assume that $\phi$ is continuous in a neighborhood of some point $\hat x \in X$ and that $\phi$ admits A as its differential at the point $\hat x$. In this paper we prove that under those conditions there exist a neighborhood U of $\hat x $ and a mapping $\zeta $ from U into X such that $\phi (x + \zeta (x)) = \phi (\hat x) + A(x - \hat x)$ for all $x \in U$ and such that $\lim _{\eta \to 0 + } \sup _{|x - \hat x| \leqq \eta } |\zeta (x)|/\eta = 0$. Besides the above “correction function theorem” this paper contains related implicit and inverse function theorems. The correction function theorem is applied to the proof of the multiplier rule for mathematical programming problems with equality and inequality constraints without assuming the continuous differentiability of the data in a neighborhood of the optimal solution.