New constructions for local approximation of Lipschitz functions. I. (Q1395854)

The author presents a new generalized subdifferential notion for Lipschitz functions: \(\mathbb{R}^n\to \mathbb{R}\). Based on the set \(\eta(_0)\) of smooth curves \(r(x_0, \alpha, g)\) which converges (for \(\alpha\to +0\)) against \(x_0\) from the direction \(g\) and on the averages of the gradients \(\nabla f(r(\cdot))\) along these curves he introduces the set \[ \begin{gathered} Df(x_0)= \text{conv}\Biggl\{v\in \mathbb{R}^n\mid\exists g\in \mathbb{R}^n,\,\| g\|= 1,\,\exists r(x_0,\cdot, g)\in \eta(x_0),\, \exists\alpha_k\to +0:\\ v= \lim_{k\to \infty}\,{1\over\alpha_k} \int^{\alpha_k}_0\nabla f(r(x_0, \alpha, g))\,d\alpha\Biggr\}.\end{gathered} \] It is shown that \(Df(x_0)\) is a closed bounded set which is contained in the Clarke subdifferential \(\partial_{Cl}f(x_0)\) and it fulfills the relation \[ \sup_{\overline g\to g,r(x_0, \alpha,\overline g)\in \eta(x_0), \alpha\to +0} {f(r(x_0, \alpha,\overline g))- f(x_0)\over \alpha}= \max_{v\in Df(x_0)}\, (v,g)\;\;\forall g\in\mathbb{R}^n. \] Moreover, if \(f\) is differentiable then \(Df(x_0)= \{\nabla f(x_0)\}\) and if \(f\) is a d.c.-function (i.e., \(f\) is represented as the difference of two convex functions \(f_1\) and \(f_2\)) then \[ \partial f_1(x_0)- \partial f_2(x_0)\subset Df(x_0)= \partial_{Cl}f(x_0). \] In comparison with these notions it is shown that \[ \begin{gathered} \partial_{Cl}f(x_0)= \text{conv}\Biggl\{v\in \mathbb{R}^n\mid\exists g\in \mathbb{R}^n,\,\| g\|= 1,\,\exists r(x_0,\cdot, g)\in \eta(x_0),\,\exists\alpha_k\to +0,\\ \exists h_m\to 0: v= \lim_{k,m\to \infty}\, {1\over\alpha_k} \int^{\alpha_k}_0\nabla f(r(x_0+ h_m, \alpha, g))\,d\alpha\Biggr\}.\end{gathered} \]