Journal of Convex Analysis 11 (2004), No. 2, 335--361
Copyright Heldermann Verlag 2004
Giovanni Colombo
Dip. di Matematica Pura e Applicata, Universita 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
\def\iint{{\hbox{int}\;}} This paper considers the parameterized infinite dimensional optimization problem $$ \hbox{minimize}\quad\bigl\{t\geq 0:\;S \cap\{x+tF\}\not= \emptyset\bigr\}, $$ where $S$ is a nonempty closed subset of a Hilbert space $H$ and $F\subseteq H$ is closed convex satisfying $0\in \iint F$. The optimal value $T(x)$ depends on the parameter $x\in H$, and the (possibly empty) set $S\cap (x+T(x)F)$ of optimal solutions is the ``$F$-projection'' of $x$ into $S$. We first compute proximal and Fr\'echet subgradients of $T(\cdot)$ in terms of normal vectors to level sets, and secondly, in terms of the $F$-projection. Sufficient conditions are also obtained for the differentiability and semiconvexity of $T(\cdot)$, results which extend the known case when $F$ is the unit ball.
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