A new representation of a proximal subdifferential by employing a directional derivative
Let \(f:\mathbb R^n\to\mathbb R\) be lsc and let the directional derivative \(f'(x;v)\) of \(f\) at \(x\) exist for each \(v\in\mathbb R^n\). Then \(\partial_pf(x)=\{\xi\); \(f'(x;v)\geq \langle\xi,v\rangle\), \(\forall v\in\mathbb R^n\}\). The upper semicontinuity property of \(x\mapsto \partial_pf(x)\) is then established via this representation; as well as a proximal sum rule. Finally, given an lsc bounded from below function \(f:\mathbb R^n\to\mathbb R\), a constant \(\alpha> 0\), and a minimizing sequence \((x_i)\) for \(p(u):=\inf\{f(x)+\alpha\|x-u\|^2\); \(x\in\mathbb R^n\}\), \(u\in\mathbb R^n\), there exists \(\bar x\in\mathbb R^n\) with i) \(x_i\to \bar x\), ii) \(p(u)\) is attained uniquely at \(\bar x\), iii) \(2\alpha (u-\bar x)\in \partial_pf(\bar x)\).
scientific article; zbMATH DE number 5882872