Ample parameterization of variational inclusions. (English) Zbl 1008.49009

Dontchev, A. L.; Rockafellar, R. T.

SIAM J. Optim. 12, No. 1, 170-187 (2001).

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.

Reviewer: Constantin Zălinescu (Iaşi)

MSC:

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

Keywords:

variational inequalities; calmness; Aubin continuity; Lipschitzian localizations; graphical derivatives; sensitivity of minimizers; variational analysis