Abstract

This paper considers the parameterized infinite dimensional optimization problem minimize{t≥0:  S∩{x+tF}≠∅},\hbox{minimize}\quad\bigl\{t\geq 0:\;S \cap\{x+tF\}\not= \emptyset\bigr\}, where SS is a nonempty closed subset of a Hilbert space HH and F⊆HF\subseteq H is closed convex satisfying 0∈int  F0\in \iint F. The optimal value T(x)T(x) depends on the parameter x∈Hx\in H, and the (possibly empty) set S∩(x+T(x)F)S\cap (x+T(x)F) of optimal solutions is the ``FF-projection'' of xx into SS. We first compute proximal and Fréchet subgradients of T(⋅)T(\cdot) in terms of normal vectors to level sets, and secondly, in terms of the FF-projection. Sufficient conditions are also obtained for the differentiability and semiconvexity of T(⋅)T(\cdot), results which extend the known case when FF is the unit ball

Author information

Contact details are reproduced from the original publication and may be historical.

Giovanni Colombo

Dip. di Matematica Pura e Applicata, Università di Padova, Via Belzoni 7, 35131 Padova, Italy

colombo@math.unipd.it

Peter R. Wolenski

Dept. of Mathematics, Louisiana State University, 326 Lockett Hall, Baton Rouge,
LA 70803-4918, U.S.A.

wolenski@math.lsu.edu

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Suggested citation

G. Colombo, P. R. Wolenski. “Variational Analysis for a Class of Minimal Time Functions in Hilbert Spaces.” Journal of Convex Analysis 11 (2004), No. 2, 335–361. https://doi.org/10.68381/jca11021

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Published by Heldermann Verlag, 2004. Rights now held by Banach Press LLC.