Ample parameterization of variational inclusions (Q2784407)

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.