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Optimization of discontinuous functions: A generalized theory of differentiation. (English) Zbl 1035.49017
The authors provide a general theory of differentiability for not necessary continuous functions. For this they extend formally the definitions of Clarke’s generalized directional derivative and Clarke’s generalized gradient (introduced originally for locally Lipschitz functions) to arbitrary functions.
Let \(f: \mathbb{R}^n\to\mathbb{R}\) be an arbitrary (not necessarily continuous) function and let \[ f^\circ(x, v)= \inf_{\varepsilon> 0}\;\sup_{\|\overline x- x\|<\varepsilon,\, 0< t<\varepsilon} {f(\overline x+ tv)- f(\overline x)\over t} \] be Clarke’s generalized directional derivative of \(f\).
In the paper, the semigradient and the singular semigradient of \(f\) are introduced by \[ \begin{aligned} {\mathbf S}{\mathbf G}f(x) &= \{\zeta\in(\mathbb{R}^n)\mid \zeta(v)\leq f^\circ(x, v)\quad \forall v\in \mathbb{R}^n\},\\ {\mathbf S}{\mathbf G}^\infty f(x) &= \{\zeta\in(\mathbb{R}^n)^*\mid \zeta(v)\leq 0\quad \forall v\in \text{dom\,} f^\circ(x,\cdot)\}.\end{aligned} \] Clearly, for locally Lipschitz functions these notions coincide with Clarke’s generalized gradient and Clarke’s singular generalized gradient, respectively. For more general functions however, it is \({\mathbf S}{\mathbf G}f(x)\supseteq\partial f(x)\) and \({\mathbf S}{\mathbf G}^\infty f(x)\supseteq \partial^\infty f(x)\).
The authors present a sum rule for two arbitrary functions \(f,g:\mathbb{R}^n\to \mathbb{R}\), in which two cases are discussed:
(1) Good case: \[ \begin{aligned} {\mathbf S}{\mathbf G}(f+ g)(x) & \subseteq \text{cl}({\mathbf S}{\mathbf G}f(x)+{\mathbf S}{\mathbf G}g(x)),\\ {\mathbf S}{\mathbf G}^\infty(f+ g)(x) &\subseteq \text{cl}({\mathbf S}{\mathbf G}^\infty f(x)+{\mathbf S}{\mathbf G}^\infty g(x)).\end{aligned} \] (2) Second case: \[ \text{There exist a nonzero }\zeta\in{\mathbf S}{\mathbf G}^\infty f(x)\text{ and a nonzero \(\xi\in{\mathbf S}{\mathbf G}^\infty g(x)\) such that } \]
\[ \text{(a) }\zeta+\xi= 0\text{ and (b) }-\zeta\not\in{\mathbf S}{\mathbf G}^\infty f(x)\text{ or }-\xi\not\in{\mathbf S}{\mathbf G}^\infty g(x). \] The results are used for the study of some optimization and optimal control problems with not necessarily continuous objective functions.

49J52 Nonsmooth analysis
49J53 Set-valued and variational analysis
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