Higher Order Conditions with and without Lagrange Multipliers
Jack Warga · SIAM Journal on Control and Optimization · 1986
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 also generalize prior “Lagrangian” conditions, more akin to the usual necessary conditions.