Generic Fréchet-differentiability and perturbed optimization problems in Banach spaces

Abstract:

We define a function F on a Banach space V to be locally $\varepsilon$-supported by ${u^\ast } \in {V^\ast }$ at $u \in V$ if there exists an $\eta > 0$ such that $\left \| {v - u} \right \| \leqslant \eta \Rightarrow F(v) \geqslant F(u) + \langle {u^\ast },v - u\rangle - \varepsilon \left \| {v - u} \right \|$. We prove that if the Banach space V admits a nonnegative Fréchet-differentiable function with bounded nonempty support, then, for any $> 0$ and every lower semicontinuous function F, there is a dense set of points $u \in V$ at which F is locally $\varepsilon$-supported. The applications are twofold. First, to the study of functions defined as pointwise infima; we prove for instance that every concave continuous function defined on a Banach space with Fréchet-differentiable norm is Fréchet-differentiable generically (i.e. on a countable intersection of open dense subsets). Then, to the study of optimization problems depending on a parameter $u \in V$; we give general conditions, mainly in the framework of uniformly convex Banach spaces with uniformly convex dual, under which such problems generically have a single optimal solution, depending continuously on the parameter and satisfying a first-order necessary condition.