Proximal Normal Cone Analysis on Smooth Banach Spaces and Applications

Abstract

Unlike the Frechet, Penot, Clarke, and other “order one” notions of normal cones, the notion of proximal normal cone describes variational behavior of “order two,” and most of results on proximal normal cone are established in the Hilbert space framework. A possible reason is that in a general Banach space (such as in the classical Banach spaces $l^p$ and $L^p$ with $1 \leq p < 2$), the proximal normal cone of a “smooth” subset $A$ at every point of $A$ may consist of the zero element only. Nevertheless, on the other hand, we show in this paper that for a fairly big class of smooth Banach spaces (including the classical Banach spaces $l^p$ and $L^p$ with $2 \leq p <\infty$), the consideration of proximal normal cones provides a lot of useful information, especially for leading to satisfactory treatment for constrained optimization problems. Some of our results are new even in the Hilbert space case.