Higher Order Conditions with and without Lagrange Multipliers

Abstract

Let Q be a convex subset of a vector space, $\mathcal{U} \subset Q$, $\mathcal{S}$ a topological vector space, C a convex subset of $\mathcal{Z}$ with a nonempty interior, $\phi = (\phi _1 ,\phi _2 ):Q \to \mathbb{R}^m \times \mathcal{Z}$, $\bar q \in Q$ and $\phi _2 (\bar q) \in C$. We assume that $\phi $ has a pth order Taylor approximation at $\bar q$ when it is restricted to an arbitrary finite-dimensional simplex in Q with a vertex at $\bar q$. In the case when $\mathcal{U}$ is a proper subset of Q we also assume that Qis a uniform space, $\phi $ continuous and $\mathcal{U}$ “abundant.” We establish a number of higher order sufficient conditions, not involving any Lagrange multipliers, for the existence of neighborhoods $G_1 $ and $G_2 $ of the origins such that $\phi _1 (\bar q) + G_1 \subset \{ {\phi _1 (u)} |u \in \mathcal{U},\phi _2 (u) + G_2 \subset C\} $. These sufficient conditions, involving nonconvex sets of Variations, are shown by an example to be stronger than those in the literature. We ...