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Strongly differentiable multifunctions and directional differentiability of marginal functions. (English) Zbl 0979.49019

Demyanov, V. (ed.) et al., Quasidifferentiability and related topics. Dedicated to Prof. Franco Giannessi on his 65th birthday and to Prof. Diethard Pallaschke on his 60th birthday. Dordrecht: Kluwer Academic Publishers. Nonconvex Optim. Appl. 43, 163-171 (2000).
Let \(F: \mathbb{R}^n\Rightarrow \mathbb{R}^m\) be a (convex-) compact-valued multifunction. \(F\) is called strongly differentiable at a point \(x\in \mathbb{R}^n\) in a direction \(\overline x\in \mathbb{R}^n\) if there exist two nonempty (convex) compact sets \(G^+(x,\overline x)\) and \(G^-(x,\overline x)\) of \(\mathbb{R}^m\) such that \[ \rho_H(F(x+ t\overline x)+ tG^-(x,\overline x), F(x)+ tG^+(x,\overline x))= o(t). \] Here \(\rho_H(A, B)\) is the Hausdorff distance between the sets \(A\) and \(B\). The Dini lower derivative \(DF(x,y)(\overline x)\) of \(F\) at the point \((x,y)\in \text{gr}(F)\) in the direction \(\overline x\) is defined by \[ DF(x, y)(\overline x)= \{\overline y\in \mathbb{R}^m\mid y+ t\overline y+ o(t)\in F(x+ t\overline x) \forall t\geq 0\}. \] In the first part of the paper it is shown that for a strongly differentiable convex-compact-valued multifunction \(F\) the relation \[ DF(x, y)(\overline x)+ G^-(x,\overline x)= G^+(x,\overline x)+DF(x, y)(0) \] holds. For the proof the authors discuss the directional differentiability (with respect to \(x\)) of the support function \(S_F(x,p):= S_{F(x)}(p)\) of the set \(F(x)\) and they point out the following relations: \[ S_F'(x,p;\overline x)= S_{G^+(x,\overline x)}(p)- S_{G^-(x,\overline x)}(p). \]
\[ DF (x, y)(\overline x)= \{\overline y\in \mathbb{R}^m\mid\langle p,\overline y\rangle\leq S_F'(x, p;\overline x) \forall p\in (F(x)- y)^*\}. \] In the second part of the paper, the directional differentiability of a marginal function according to \[ \varphi(x)= \max\{f(x, y)\mid y\in F(x)\} \] is discussed where \(F: \mathbb{R}^n\Rightarrow \mathbb{R}^m\) is a strongly differentiable compact-valued multifunction and \(f: \mathbb{R}^n\times \mathbb{R}^m\to \mathbb{R}\) is a smooth function. Using the set \(\omega(x)= \{y\in F(x)\mid f(x,y)= \varphi(x)\}\), the following relation is proved: \[ \varphi'(x;\overline x)= \max_{y\in \omega(x)} \max_{y^+\in G^+(x,\overline x)} \max_{y^-\in G^-(x,\overline x)} \{\langle\nabla_x f(x,y), \overline x\rangle+ \langle\nabla_y f(x,y), y^+- y^-\rangle\}. \] If \(F\) is convex-compact-valued, then this relations reduces to \[ \varphi'(x;\overline x)= \max_{y\in \omega(x)} \max_{\overline y\in DF(x,y)(\overline x)} \langle\nabla f(x,y),(\overline x,\overline y)\rangle. \]
For the entire collection see [Zbl 0949.00047].

MSC:

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