On continuous approximation of the maximum-function subdifferential (Q1333721)

The paper is concerned with the study of the subdifferential of the maximum-function \[ f(u)= \max\{\varphi(u,y)\mid y\in\Omega\}, \] where \(\Omega\) is a compact set in \(\mathbb{R}^ m\), the function \(\varphi: \mathbb{R}^ n\times \Omega\to \mathbb{R}\) is assumed continuous together with its partial gradient \(\nabla_ u\varphi (u,y)\). Along with the known formula \[ \partial f(u)= \text{co} \{\nabla_ u\varphi (u,y)\mid y\in R(u)\}, \] where \(R(u)= \{y\in\Omega\mid \varphi(u,y)= f(u)\}\), the author obtains another expression for the subdifferential \(\partial f(u)\) based on the concept of discrete gradient introduced in his thesis. The main result consists in the construction of locally Lipschitz multivalued mappings approximating the subdifferential \(\partial f(u)\) in the Hausdorff metric.