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Variational pairs and applications to stability in nonsmooth analysis. (English) Zbl 1035.49014
In the paper, it is pointed out that many of the basic results of nonsmooth analysis (e.g., subdifferential calculus, mean-value inequalities and optimality conditions) can be derived directly by general variational principles. In this manner, the authors provide a general definition of a variational pair $$(X,\delta)$$ where $$X$$ is a complete metric space and $$\delta$$ associates to any lower semicontinuous function $$f:X\to \mathbb{R}\cup \{+\infty\}$$ and any point $$x\in X$$, a nonnegative extended real number $$\delta f(x)$$. The main property which is required on $$\delta$$ is that for every $$\overline{x}\in X$$ and for every numbers $$\sigma,r>0$$ with $f(\overline{x})< \inf_{d(x,\overline{x})< r}f(x)+ \sigma r$ one can find some $$x\in X$$ with $$d(x,\overline{x})< r$$ and $$\delta f(x)< \sigma$$. It is shown that with $\delta_1f(x)= \begin{cases} 0, &\text{ if $$x$$ is a local minimizer of $$f$$},\\ \limsup_{y\to x} \frac{f(y)-f(x)} {y-x}, &\text{ otherwise}, \end{cases}$ and $\delta_2f(x)= d(0,\partial f(x))$ (where $$\partial$$ is a suitable subdifferential operator), $$(X,\delta_1)$$ and $$(X,\delta_2)$$ are examples of a variational pair.
The both main theorems of the paper are consequences of the notion of variational pair. The first one can be seen as a general (global) metric regularity result for a lower semicontinuous function. The second one is basically a local and parametric refinement of the previous one, dealing with the local solvability of an inequality of the type $$f(x,p)\leq 0$$ with parameter $$p$$ of a topological space.
The results are applied for the characterization of so-called asymptotically well-behaved functions, for the discussion of metric regularity properties for parametrized families of multifunctions and for the derivation of a stability assertion for parametrized optimization problems assuming an extended Mangasarian-Fromowitz type constraint qualification.

##### MSC:
 49J52 Nonsmooth analysis 49J40 Variational inequalities
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