A generalized derivative for calm and stable functions
Let \(X\) be an open subset of a locally convex topological vector space \(E\) and \(f: X\to \mathbb{R}\) be a function. According to this paper, the radial strict derivative of \(f\) at \(x\in X\) in the direction \(y\in E\) is \[ f^ \square(x,y)= \sup_{w\in E}\limsup_{t\to 0_ +} t^{- 1}\bigl(f(x+ tw+ ty)- f(x+ tw)\bigr), \] and the radial strict subdifferential of \(f\) at \(x\) is the set \[ \partial^ \square f(x)= \bigl\{ x^*\in E^*: \forall y\in E,\;\langle x^*,y\rangle\leq f^ \square(x,y)\bigr\}. \] If \(f\) is differentiable at \(x\), then \(f^ \square(x,\cdot)= f'(x,\cdot)\) and \(\partial^ \square f(x)=\{ f'(x)\}\). This is a property which is not shared by Clarke's generalized derivative and gradient. If \(f\) attains at \(x\) a local minimum, then one has \(0\in \partial^ \square f(x)\) too as the case of Clarke's generalized gradient. In addition, for any continuous \(f\), \(x\in X\) and \(y\in E\), one has \[ f^ \circ(x,y)=\limsup_{u\to x} f^ \square(u,y), \] where \(f^ \circ\) is Clarke's generalized derivative of \(f\). The calmness of \(f\) can serve as a criterion for the nonemptiness of \(\partial^ \square f(x)\). \(f\) is said to be calm at \(x\) if there exist a continuous seminorm \(q\) on \(E\) and a neighborhood \(V\) of 0 in \(E\) such that \[ f(x+ v)\geq f(x)- q(v)\qquad\text{for all } v\in V. \] If \(f\) is calm at \(x\), then \(f^ \square (x,\cdot)\) is calm at 0 and \(\partial^ \square f(x)\) is nonempty. For \(f: E\to\overline{\mathbb{R}}=\overline{\mathbb{R}}\cup\{- \infty,+\infty\}\), the pseudo-strict derivative of \(f\) at \(x\in\text{dom }f\) is defined by \[ f^ \land(x,y)= \sup_{w\in E} e- \limsup_{\textstyle{{(t,r,v)\to (0,f(x),w)\atop t>0,r\geq f(x+ iv)}}} t^{-1}\bigl(f(x+ tv+ ty)- r\bigr), \] where \(e-\limsup\) is somewhat like epiconvergence, and \[ \partial^ \land f(x)= \bigl\{ x^*\in E^*: x^*\leq f^ \land(x,\cdot)\bigr\}. \] For this generalized derivative, a chain rule and a mean value theorem hold. Applications to optimization are pointed out.
scientific article; zbMATH DE number 90498
scientific article