Second-Order Optimality Conditions in Sets of ${\sf L}^\infty$ Functions with Range in a Polyhedron

Formal extensions of the general second-order necessary conditions and sufficient conditions for local optimality in a polyhedral convex set $U\subset {\Bbb R}^m$ are established for ${\sf L}^\infty$-local optimality and ${\sf L}^2$-local optimality in the infinite-dimensional nonpolyhedral convex set $\Omega$ of ${\sf L}^\infty$ functions $u(\cdot ): [0,1]\rightarrow U$. A more refined analysis for nonconvex cost functions with specially structured differentials yields optimality conditions that apply to an important class of constrained input Bolza optimal control problems. The gap between the necessary conditions and sufficient conditions in this setting is uncharacteristically small for infinite-dimensional problems. In the control problem context, the ${\sf L}^\infty$-local optimality conditions and ${\sf L}^2$-local optimality conditions entail a mild strengthening of a pointwise strict complementarity condition and variants of the Legendre--Clebsch condition and the Pontryagin minimum principle. In related recent studies, similar second-order sufficient conditions for the special case $U=[0,\infty )$ are the key hypotheses in corresponding local convergence theories for iterative constrained minimization algorithms.