Optimization of discontinuous functions: A generalized theory of differentiation (Q2706315)

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.NEWLINENEWLINE Let \(f: \mathbb{R}^n\to\mathbb{R}\) be an arbitrary (not necessarily continuous) function and let NEWLINE\[NEWLINEf^\circ(x, v)= \inf_{\varepsilon> 0}\;\sup_{\|\overline x- x\|<\varepsilon,\, 0< t<\varepsilon} {f(\overline x+ tv)- f(\overline x)\over t}NEWLINE\]NEWLINE be Clarke's generalized directional derivative of \(f\).NEWLINENEWLINE In the paper, the semigradient and the singular semigradient of \(f\) are introduced by NEWLINE\[NEWLINE\begin{aligned} {\mathbf S}{\mathbf G}f(x) &= \{\zeta\in(\mathbb{R}^n)\mid \zeta(v)\leq f^\circ(x, v)\quad \forall v\in \mathbb{R}^n\},\\ {\mathbf S}{\mathbf G}^\infty f(x) &= \{\zeta\in(\mathbb{R}^n)^*\mid \zeta(v)\leq 0\quad \forall v\in \text{dom\,} f^\circ(x,\cdot)\}.\end{aligned}NEWLINE\]NEWLINE 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 \({\mathbf S}{\mathbf G}^\infty f(x)\supseteq \partial^\infty f(x)\).NEWLINENEWLINE The authors present a sum rule for two arbitrary functions \(f,g:\mathbb{R}^n\to \mathbb{R}\), in which two cases are discussed:NEWLINENEWLINE (1) Good case: NEWLINE\[NEWLINE\begin{aligned} {\mathbf S}{\mathbf G}(f+ g)(x) & \subseteq \text{cl}({\mathbf S}{\mathbf G}f(x)+{\mathbf S}{\mathbf G}g(x)),\\ {\mathbf S}{\mathbf G}^\infty(f+ g)(x) &\subseteq \text{cl}({\mathbf S}{\mathbf G}^\infty f(x)+{\mathbf S}{\mathbf G}^\infty g(x)).\end{aligned}NEWLINE\]NEWLINE (2) Second case: NEWLINE\[NEWLINE\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 }NEWLINE\]NEWLINE NEWLINE\[NEWLINE\text{(a) }\zeta+\xi= 0\text{ and (b) }-\zeta\not\in{\mathbf S}{\mathbf G}^\infty f(x)\text{ or }-\xi\not\in{\mathbf S}{\mathbf G}^\infty g(x).NEWLINE\]NEWLINE The results are used for the study of some optimization and optimal control problems with not necessarily continuous objective functions.