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Geometric approximation on proximal normals. (English) Zbl 0908.49016
In the paper, some new results of proximal analysis are presented.
Let $$S$$ be a closed nonempty subset of the Hilbert space $$H$$. Regarding the projection of the point $$x\not\in S$$ onto $$S$$ according to $\text{proj}_S(x):= \{s\in S\mid\| x-s\|= d_S(x)\}$ (where $$d_S$$ is the distance function), the proximal normal cone to $$S$$ at the point $$s\in S$$ according to $N^P_S(s):= \{\lambda(x- s)\mid \lambda>0,\;x\not\in S\text{ such that }s\in \text{proj}_S(x)\},$ and the proximal subdifferential of a lower-semicontinuous function $$f: H\to\mathbb{R}$$ according to $\partial_Pf(x):= \{\zeta\mid(\zeta, -1)\in N^P_{\text{epi}(f)}(x, f(x))\}$ then it is known from former papers that the proximal subdifferential $$\partial_Pd_S(x)$$ of the distance function $$d_S$$ is closely connected with the proximal normals of the set $$S$$. So in case of $$x\not\in S$$ and $$\partial_Pd_S(x)\neq\emptyset$$, both sets $$\text{proj}_S(x)$$ and $$\partial_Pd_S(x)$$ are singletons: $$\{s\}$$ and $$\{\zeta\}$$, and it is $\zeta= {x-s\over\| x-s\|} (\in N^P_S(s)).$ The converse result, however, is not true in general: even if the set $$\text{proj}_S(x)$$ is singleton it can be $$\partial_Pd_S(x)= \emptyset$$.
In the present paper, the authors give some approximation results using the more general $$\delta$$-projection $\text{proj}^\delta_S(x):= \{s\in S\mid\| x-s\|^2\leq d_S(x)^2+\delta^2\}.$ It is shown that for $$x\not\in S$$, for small $$\delta>0$$ and for $$s\in\text{proj}^\delta_S(x)$$ there exists a point $$\overline x$$ near $$x$$ and a subgradient $$\zeta\in\partial_P d_S(\overline x)$$ such that the difference $$\left\|{x- s\over\| x-s\|}- \zeta\right\|$$ will be sufficiently small. Hence, the vector $${x-s\over\| x-s\|}$$ can be approximated by some nearby proximal subgradients.
In the second main theorem, the authors present a new result concerning the approximation of singular proximal subgradients of a l.s.c. function, i.e., vectors $$\zeta$$ with $$(\zeta,0)\in N^P_{\text{epi}(f)}(x, f(x))$$, by proximal subgradients.

##### MSC:
 49J52 Nonsmooth analysis
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