Controllability and a Multiplier Rule for Nondifferentiable Optimization Problems

Let K be a compact subset of a normed vector space $\mathcal{X}$, C a convex body in a Banach space $\mathcal{Y}$, $k_0 \in K$, $(\phi,\Phi ):K \to \mathbb{R}^m \times \mathcal{Y}$ continuous, and $\Phi (k_0 ) \in C$. We introduce the concept of a “directional derivate container” $\Lambda ^\varepsilon (\phi,\Phi )(k_0 )$ for $(\phi,\Phi )$ at $k_0 $ whose definition is equivalent to that of a directional derivative in the special case where $(\phi,\Phi )$ is “finitely $C^1 $,” that is, all the restrictions of $(\phi,\Phi )$ to finite-dimensional convex subsets of K are continuously differentiable. In general, $(\phi,\Phi )$ admits a (nonunique) directional derivate container at $k_0 $ if it can be uniformly approximated by finitely $C^1 $ functions $(\phi _i,\Phi _i )$ whose directional derivatives in some “neighborhood” of $k_0 $ in K, viewed as functions on K to $\mathbb{R}^m \times \mathcal{Y}$, form a bounded and equicontinuous subset of $C(K,\mathbb{R}^m \times \mathcal{Y})$. For a given $\Lambda ^\varepsilon (\phi,\Phi )(k_0 )$ we define the corresponding “scalar directional derivate container” $\mathcal{L}\Lambda (\phi,\Phi,C)(k_0 )$ as the collection of all $(l,\lambda )$ such that $l = (l_1,l_2 )$ is a weak star limit of $l^i = (l_1^i,l_2^i ) \in \mathbb{R}^m \times \mathcal{Y}^ * $, $|l_1^i | + |l_2^i | = 1$, $l_2^i y \leqq 0$ if the closed ball in $\mathcal{Y}$ of center y and radius ${1 / i}$ is contained in $C - \Phi (k_0 )$, $l \ne 0$, and $\lambda $ is a pointwise limit of functions $l^i \circ M^i :K \to \mathbb{R}$ with $M^i \in \Lambda ^{{1 / i}} (\phi,\Phi )(k_0 )$. We then prove a “controllability-multiplier rule” alternative which states (defining $S^F (0,\kappa )$ as the closed ball of center 0 and radius $\kappa $) that either there exist $\kappa > 0$ and a finite-dimensional subset $K^ * $ of K such that $k_0 \in K^ * $ and $\{ {\phi (k)\mid k \in K^ *,\Phi (k) + S^F (0,\kappa ) \subset C} \}$ contains a neighborhood of $\phi (k_0 )$ in $\mathbb{R}^m $ or there exists $(l_1,l_2,\lambda ) \in \mathcal{L}\Lambda (\phi,\Phi,C)(k_0 )$ such that $\lambda k_0 = {\operatorname{Min}}_{k \in K} \lambda k$, $l_2 \Phi (k_0 ) = {\operatorname{Max}}_{c \in C} l_2 c$. These results will be used elsewhere to study optimal control problems defined by hereditary functional-integral equations involving nondifferentiable functions of state variables.