Metric regularity and subdifferential calculus in Banach spaces
Throughout the paper it is assumed that \(X\) and \(Y\) are reflexive Banach spaces. The authors consider the special multivalued mapping \(F(x)=- g(x)+D\), where \(g:X \to Y\) need not satisfy any differentiability assumptions and \(D \subset Y\). Let \(C \subset X\) and \(x_0\in C\). \(F\) is called metrically regular at \((x_0, y_0)\in \text{graph} F\) w.r.t. the set \(C\) iff there exist \(k \geq 0\) and \(\varepsilon > 0\) such that \(d(x,C \cap F^{-1} (y)) \leq k d(x, F(x))\) for all \(x\in (x_0+\varepsilon B_X) \cap C\) and \(y\in y_0+\varepsilon B_Y\), where \(d(x, F(x))\) is the distance of \(x\) to the set \(F(x)\) \((d (x, \emptyset) :=\infty)\). At first they give sufficient conditions for the above reglarity. Using the subdifferential calculus for limiting Fréchet subdifferentials \(\partial_F\) (sum of subdifferentials, chain rule) and the Fréchet coderivative \(D^*_F g(x_0)\) defined by \[ D^*_F g(x_0) (y^*) :=\bigl\{ x^*\in X^* \mid (x^*,-y^*)\in \mathbb{R}_+\partial_F d \bigl(\bigl( x_0, g(x_0) \bigr), \text{graph } g \bigr) \bigr\} \] they prove (Theorem 2.3) the metrical regularity of \(F\) in the above sense under the assumptions that \(C,D\) are closed, \(g\) is Lipschitzian around \(x_0\) with constant \(K_g\), \(D\) is epi-Lipschitz at \(g (x_0)\) and that the regularity condition \[ y^*\in \partial_Fd \bigl( g(x_0), D \bigr) \wedge 0\in D^*_F g \bigl( x_0 (y^*) \bigr)+K_g \partial_F d(x_0, C) \Rightarrow y^*=0 \] is satisfied. In their work they do not need the epi-Lipschitz property of the set \(C\). As application of this theorem they give conditions for the metrical regularity in the special cases \(F(x)=- x+D\) (Cor. 2.4), \(D=X \times \text{epi} f\) (Cor. 2.5), \(g\) is strictly differentiable at \(x_0\) (Cor. 2.6). Using the above results they can prove the more general chain rule for the limiting Fréchet subdifferential \[ \partial_F (f \circ g) (x_0) \subset \bigcup_{y^*\in \partial_F f(y_0)} D^*_F g(x_0) (y^*), \] where \(g:X \to Y\) is continuous around \(x_0\), \(f:Y \to \overline \mathbb{R}\) is l.s.c around \(y_0=(x_0)\), \(|(f \circ g) (x_0) |<\infty\) and \(F (x,y,r)=- (x,y,r)+X \times \text{epi} f\) is metrically regular at \(((x_0, g(x_0), f(g(x_0))), 0)\) w.r.t. the set \(\{(x,y)\mid g(x)=y \} \times \mathbb{R}\). Furthermore, formulas for the subdifferentials of the indicator function of \(C \cap g^{-1} (D)\) at \(x_0\) and for the sum \(f_1+f_2\) (both l.s.c., only one of them directionally Lipschitz at \(x_0)\) are given under the above general assumptions.
scientific article; zbMATH DE number 773791
scientific article