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A generalized derivative for calm and stable functions. (English) Zbl 0787.49007
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.
Reviewer: S.Shih (Tianjin)

##### MSC:
 49J52 Nonsmooth analysis 49K27 Optimality conditions for problems in abstract spaces 26B05 Continuity and differentiation questions 26B12 Calculus of vector functions