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The support functions of Clarke’s generalized Jacobian matrix and of its plenary hull. (Les fonctions d’appui de la jacobienne généralisée de Clarke et de son enveloppe plénière.) (French) Zbl 0940.49017

Given a locally Lipschitz function \(F:{\mathcal O}\subset{\mathbb R}^n\to {\mathbb R^m}\), its generalized Jacobian matrix \({\mathcal J}F(x)\) is defined by \[ {\mathcal J}F(x):=\text{co}\{\lim JF(x_k): x_k\to x, \;x_k\in D_F\} \] for \(x\in{\mathcal O}\), where \(D_F\) denotes the set of those points where \(F\) is differentiable and \(JF(x_k)\) denotes the Jacobian matrix of \(F\) at \(x_k\).
The main results of the paper offer formulae for the support function of \({\mathcal J}F(x)\), that is for \[ \sigma_{{\mathcal J}F(x)}(M) :=\max\{\text{tr}(X^TM): X\in{\mathcal J}F(x)\} \quad(M\in{\mathbb R}^{n\times m}), \] and also for the support function of the plenary hull of \({\mathcal J}F(x)\) defined by \[ \text{plen}({\mathcal J}F(x)) :=\{M\in{\mathbb R}^{n\times m}: M(u)\in {\mathcal J}F(x)(u)\;(\forall u\in {\mathbb R}^n)\}. \] Consequences of these results are discussed as well.

MSC:

49J52 Nonsmooth analysis
26B10 Implicit function theorems, Jacobians, transformations with several variables
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