Generalized Hessian for C1,1 functions in infinite dimensional normed spaces
Mathematical Programming: Series A and B
Approximate Jacobian Matrices for Nonsmooth Continuous Maps and C1-Optimization
SIAM Journal on Control and Optimization
Nonlinear Analysis: Theory, Methods & Applications
Support functions of the Clarke generalized Jacobian and of its plenary hull
Nonlinear Analysis: Theory, Methods & Applications
Semismooth Matrix-Valued Functions
Mathematics of Operations Research
Mathematics of Operations Research
Mathematical Programming: Series A and B
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Given a $C^{1,1}$-function $f:U\rightarrow R$ (where $U\subset R^{n}$ open), we deal with the question of whether or not at a given point $x_{0}\in U$ there exists a local minorant $\varphi$ of $f$ of class $C^{2}$ that satisfies $\varphi(x_{0})=f(x_{0})$, $D\varphi(x_{0})=Df(x_{0})$, and $D^{2}\varphi(x_{0})\in\mathcal{H}f(x_{0})$ (the generalized Hessian of $f$ at $x_{0}$). This question is motivated by the second-order viscosity theory of the PDEs, since for nonsmooth functions, an analogous result between subgradients and first-order viscosity subjets is known to hold in every separable Asplund space. In this work we show that the aforementioned second-order result holds true whenever $\mathcal{H}f(x_{0})$ has a minimum with respect to the positive semidefinite cone (thus, in particular, in one dimension), but it fails in two dimensions even for piecewise polynomial functions. We extend this result by introducing a new notion of directional minimum of $\mathcal{H}f(x_{0})$.