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Smoothing a nondifferentiable convex function: The technique of the rolling ball. (English) Zbl 0921.49008

Let \(X\) be a locally convex topological linear space, \(f:X \to {\mathbb R} \cup \{+\infty\}\) a proper convex function and \(A\) a closed subset of \(X \times {\mathbb R}\).
The author studies the properties of the \(A\)-regularization \(f_{A}:X \to \overline{{\mathbb R}}\) of \(f\) defined by \[ f_{A}(x) = \inf \left\{ \lambda + \Psi_{\text{epi}(f) + A}(x,\lambda): \lambda \in {\mathbb R} \right\} , \] where \(\Psi_{B}\) is the indicator of \(B\) and \(\text{ epi}(f)\) is the epigraph of \(f\). This notion extends the classical Moreau-Yosida regularization and the Lipschitz regularization of J.-B. Hiriart-Urruty [Math. Scand. 47, 123-134 (1980; Zbl 0436.26005)].

MSC:

49J52 Nonsmooth analysis
58C20 Differentiation theory (Gateaux, Fréchet, etc.) on manifolds

Citations:

Zbl 0436.26005
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