Variational pairs and applications to stability in nonsmooth analysis.
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.
scientific article; zbMATH DE number 1751625
scientific article