Variational Analysis for a Class of Minimal Time Functions in Hilbert Spaces
Giovanni Colombo, Peter R. Wolenski · Journal of convex analysis · 2004
This paper considers the parameterized infinite dimensional optimization problem \hbox{minimize}\quad\bigl\{t\geq 0:\;S \cap\{x+tF\} ot= \emptyset\bigr\}, minimize { t ≥ 0 : S ∩ { x + t F } ≠ ∅ } , where S S is a nonempty closed subset of a Hilbert space H H and F\subseteq H F ⊆ H is closed convex satisfying 0\in {{\hbox{int}\;}}F 0 ∈ int F . The optimal value T(x) T ( x ) depends on the parameter x\in H x ∈ H , and the (possibly empty) set S\cap (x+T(x)F) S ∩ ( x + T ( x ) F ) of optimal solutions is the “ F F -projection” of x x into S S . We first compute proximal and Fréchet subgradients of T(\cdot) T ( ⋅ ) in terms of normal vectors to level sets, and secondly, in terms of the F F -projection. Sufficient conditions are also obtained for the differentiability and semiconvexity of <