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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.

49J52 Nonsmooth analysis
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