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Ample parameterization of variational inclusions. (English) Zbl 1008.49009
Consider the parametrized variational inclusion $$0\in f(w,x)+F(x),$$ where $$w\in\mathbb{R} ^{d}$$ is the parameter, $$x\in\mathbb{R} ^{n}$$ is the solution, $$f:\mathbb{R} ^{d}\times\mathbb{R} ^{n}\rightarrow\mathbb{R} ^{m}$$ is a $${\mathcal C}^{1}$$ function and $$F:\mathbb{R} ^{n}\rightrightarrows \mathbb{R} ^{m}$$ is a set-valued mapping with closed graph. The aim of this paper is to study local properties of the solution mapping $$S:\mathbb{R} ^{d}\rightrightarrows\mathbb{R} ^{n},$$ $$S(w):=\{x\in\mathbb{R} ^{n}\mid 0\in f(w,x)+F(x)\}.$$ One says that the variational inclusion above is amply parametrized at $$(w_{\ast},x_{\ast})\in\text{gph}S$$ if $$\text{ rank}\nabla_{w}f(w_{\ast},x_{\ast})=m.$$ Under the hypothesis the variational inclusion above is amply parametrized the authors investigate Lipschitz-type properties of $$S$$ like calmness, Aubin continuity, Lipschitzian localization, as well as graphical properties related to generalized differentiation.

##### MSC:
 49J40 Variational inequalities 49J52 Nonsmooth analysis 49K40 Sensitivity, stability, well-posedness 90C31 Sensitivity, stability, parametric optimization 49J53 Set-valued and variational analysis
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