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Theorem 4.12. If \( f \) attains at \( \bar{x} \) a local minimum, then one has \( 0 \in {\partial }_{F}f\left( \bar{x}\right) \) and \( 0 \in {\partial }_{D}f\left( \bar{x}\right) \) . | Proof. Let \( g \) be the constant function with value \( f\left( \bar{x}\right) \) . Then \( f \geq g \) with \( f\left( \bar{x}\right) = g\left( \bar{x}\right) \) and the preceding proposition applies. | No |
Proposition 4.14. The tangent cone at \( {\bar{x}}_{f} \mathrel{\text{:=}} \left( {\bar{x}, f\left( \bar{x}\right) }\right) \) to the epigraph \( {E}_{f} \) of \( f \) is the epigraph of the lower (or contingent) subderivate \( {f}^{D}\left( {\bar{x}, \cdot }\right) \) : | Proof. We have \( \left( {u, r}\right) \in {T}^{D}\left( {{E}_{f},{\bar{x}}_{f}}\right) \) iff there exist sequences \( \left( {t}_{n}\right) \rightarrow {0}_{ + },\left( {u}_{n}\right) \rightarrow u \) , \( \left( {r}_{n}\right) \rightarrow r \) such that \( \left( {\bar{x}, f\left( \bar{x}\right) }\right) + {t}_{n}\l... | Yes |
Corollary 4.15. The directional subdifferential \( {\partial }_{D}f\left( \bar{x}\right) \) of \( f \) at \( \bar{x} \in \operatorname{dom}f \) and the normal cone \( {N}_{D}\left( {{E}_{f},{\bar{x}}_{f}}\right) \) to the epigraph \( {E}_{f} \) of \( f \) at \( {\bar{x}}_{f} \mathrel{\text{:=}} \left( {\bar{x}, f\left(... | Proof. Relation (4.10) follows from the previous characterizations:\n\n\[{\bar{x}}^{ * } \in {\partial }_{D}f\left( \bar{x}\right) \Leftrightarrow \forall u \in X,\;\left\langle {{\bar{x}}^{ * }, u}\right\rangle \leq {f}^{D}\left( {\bar{x}, u}\right)\]\n\n\[\Leftrightarrow \forall \left( {u, r}\right) \in {T}^{D}\left(... | Yes |
Corollary 4.17. Let \( f \) be finite at \( \bar{x} \) and let \( \partial \) (resp. \( N \) ) stand either for \( {\partial }_{D} \) or \( {\partial }_{F} \) (resp. \( {N}_{D} \) or \( {N}_{F} \) ). If \( \partial f\left( \bar{x}\right) \) is nonempty, then with the preceding notation, \[ N\left( {{E}_{f},{\bar{x}}_{f... | Proof. Since \( N\left( {{E}_{f},{\bar{x}}_{f}}\right) \) is a closed convex cone, the inclusion \( \operatorname{cl}\left( {{\mathbb{R}}_{ + }(\partial f\left( \bar{x}\right) \times }\right. \) \( \{ - 1\} )) \subset N\left( {{E}_{f},{\bar{x}}_{f}}\right) \) stems from the previous corollary and proposition. Let us pr... | Yes |
Proposition 4.18. Let \( N = {N}_{D} \) (resp. \( N = {N}_{F} \) ) and \( \partial = {\partial }_{D} \) (resp. \( \partial = {\partial }_{F} \) ). If \( f \) : \( X \rightarrow \overline{\mathbb{R}} \) is finite at \( \bar{x} \) one has \( N\left( {{E}_{f},{\bar{x}}_{f}}\right) = {\mathbb{R}}_{ + }\left( {\partial f\le... | Proof. Since \( {\bar{x}}_{f} + {\mathbb{R}}_{ + }\left( {0,1}\right) \) is contained in \( {E}_{f} \), one has \( \{ 0\} \times {\mathbb{R}}_{ + } \subset {T}^{D}\left( {{E}_{f},{\bar{x}}_{f}}\right) \) , hence \( {N}_{F}\left( {{E}_{f},{\bar{x}}_{f}}\right) \subset {N}_{D}\left( {{E}_{f},{\bar{x}}_{f}}\right) \subset... | Yes |
Lemma 4.19. Let \( f : X \rightarrow \overline{\mathbb{R}} \) be finite at \( \bar{x} \), let \( {\bar{x}}_{f} \mathrel{\text{:=}} \left( {\bar{x}, f\left( \bar{x}\right) }\right), w \mathrel{\text{:=}} \left( {\bar{x}, r}\right), z \mathrel{\text{:=}} \left( {\bar{x}, s}\right) \) be in the epigraph \( {E}_{f} \) of \... | Proof. The first inclusion entails the second one. It follows from the relations \( {E}_{f} + z - w \subset {E}_{f} \) and \( N\left( {{E}_{f} + z - w, z}\right) = N\left( {{E}_{f}, w}\right) \) . Since \( {\bar{x}}_{f} + {\mathbb{R}}_{ + }\left( {0,1}\right) \subset {E}_{f} \) , \( z + \{ 0\} \times \left( {r - s, s -... | Yes |
Lemma 4.20. If \( f : X \rightarrow \overline{\mathbb{R}} \) is quiet at \( \bar{x} \), then for every \( \left( {{x}^{ * },{r}^{ * }}\right) \in {N}_{D}\left( {{E}_{f},{\bar{x}}_{f}}\right) \smallsetminus \) \( \{ \left( {0,0}\right) \} \) one has \( {r}^{ * } < 0 \) and \( {\left( -{r}^{ * }\right) }^{-1}{x}^{ * } \i... | Proof. If \( f \) is quiet at \( \bar{x} \) with rate \( c > 0 \) in the sense that \( f\left( x\right) - f\left( \bar{x}\right) \leq c\parallel x - \bar{x}\parallel \) for all \( x \) near \( \bar{x} \), then for all \( u \in X \) one has \( {f}^{D}\left( {\bar{x}, u}\right) \leq c\parallel u\parallel \), whence for a... | Yes |
For a subset \( E \) of \( X \), its distance function \( {d}_{E} \), and \( w \in \operatorname{cl}E \) one has\n\n\[{\partial }_{F}{d}_{E}\left( w\right) = {N}_{F}\left( {E, w}\right) \cap {B}_{{X}^{ * }}\]\n\n(4.15)\n\n\[{N}_{F}\left( {E, w}\right) = {\mathbb{R}}_{ + }{\partial }_{F}{d}_{E}\left( w\right)\]\n\n(4.16... | Proof. Since \( {d}_{E} \) is Lipschitzian with rate 1, one has \( {\partial }_{F}{d}_{E}\left( w\right) \subset {B}_{{X}^{ * }} \) . Moreover, as already observed in Proposition 4.13, one has \( {\partial }_{F}{d}_{E}\left( w\right) \subset {N}_{F}\left( {E, w}\right) \) . Conversely, given \( {w}^{ * } \in {N}_{F}\le... | Yes |
Proposition 4.22 (Borwein and Giles). Let \( E \) be a nonempty closed subset of a normed space \( X \) and let \( w \in X \smallsetminus E \) . Then for all \( {w}^{ * } \in {\partial }_{F}{d}_{E}\left( w\right) \) one has \( \begin{Vmatrix}{w}^{ * }\end{Vmatrix} = 1 \) . If \( x \in E \) is such that \( \parallel x -... | Proof. Let \( {w}^{ * } \in {\partial }_{F}{d}_{E}\left( w\right) \) and let \( \varepsilon > 0 \) be given. For a given sequence \( \left( {t}_{n}\right) \rightarrow {0}_{ + } \) in \( \left( {0,1}\right) \), let \( {x}_{n} \in E \) be such that \( \begin{Vmatrix}{{x}_{n} - w}\end{Vmatrix} \leq {d}_{E}\left( w\right) ... | Yes |
Proposition 4.25 (Scalarization). For every map \( g : X \rightarrow Y \) between two normed spaces and for every \( \bar{x} \in X,{y}^{ * } \in {Y}^{ * } \) one has the following inclusions. The first one is an equality if \( g \) is tangentially compact at \( \bar{x} \) ; the second one is an equality if \( g \) is s... | Proof. Let \( h \mathrel{\text{:=}} {y}^{ * } \circ g \), let \( {x}^{ * } \in {\partial }_{D}h\left( \bar{x}\right) \), and let \( G \) be the graph of \( g \) . Then for every \( \left( {u, v}\right) \in {T}^{D}\left( {G,\left( {\bar{x},\bar{y}}\right) }\right) \), where \( \bar{y} \mathrel{\text{:=}} g\left( \bar{x}... | Yes |
Proposition 4.26. Let \( V, W \) be open subsets of normed spaces \( Y \) and \( Z \) respectively and let \( M : Y \rightrightarrows Z \) be a multimap that is pseudo-Lipschitzian on \( V \times W \) with rate \( c \) in the sense that\n\n\[ \forall v,{v}^{\prime } \in V, w \in W \cap M\left( v\right) ,\;d\left( {w, M... | Proof. Let \( \left( {v, w}\right) \in V \times W,{z}^{ * } \in {Z}^{ * },{y}^{ * } \in {D}_{F}^{ * }M\left( {v, w}\right) \left( {z}^{ * }\right) \) and let \( r \) be a remainder such that\n\n\[ \left( {y, z}\right) \in M - \left( {v, w}\right) \Rightarrow \left\langle {{y}^{ * }, y}\right\rangle + \left\langle {-{z}... | Yes |
Proposition 4.29. For every closed subset of a Hilbert space \( X \) and every \( x \in S \), the set \( {N}_{P}\left( {S, x}\right) \) of proximal normals to \( S \) at \( x \) is convex. | Proof. Let \( {v}_{0},{v}_{1} \in {N}_{P}\left( {S, x}\right) \) . The preceding remark shows that for all \( r > 0 \) small enough, \( x \) is the projection of \( x + r{v}_{i}\left( {i = 0,1}\right) \) in \( S \), or equivalently,\n\n\[ \parallel x - s{\parallel }^{2} \geq {2r}\left( {{v}_{i} \mid s - x}\right) \;\fo... | Yes |
Proposition 4.30. For every closed subset \( S \) of a Hilbert space \( X \) and every \( x \in S \) , one has \( {N}_{P}\left( {S, x}\right) \subset {N}_{F}\left( {S, x}\right) \subset {N}_{D}\left( {S, x}\right) \) . | Proof. Let \( v \in {N}_{P}\left( {S, x}\right) \), so that for some \( r > 0 \), one has \( \parallel x - s{\parallel }^{2} \geq {2r}\left( {v \mid s - x}\right) \) for all \( s \in S \) . Then, given \( \varepsilon > 0 \), taking \( \delta \in \left( {0,{2r\varepsilon }}\right) \), for all \( s \in S \cap B\left( {x,... | Yes |
Lemma 4.32. Suppose \( f \) is Lipschitzian with rate \( c \) around some \( \bar{x} \in X \) and \( X \times \mathbb{R} \) is endowed with the norm given by \( \parallel \left( {x, r}\right) \parallel = c\parallel x\parallel + \left| r\right| \) . Then for \( \left( {u, r}\right) \) near \( \left( {\bar{x}, f\left( \b... | Proof. Since \( \left( {u, f\left( u\right) }\right) \in E \), the inequality \( {d}_{E}\left( {u, r}\right) \leq {\left( f\left( u\right) - r\right) }^{ + } \) holds for every function \( f \) . When \( f \) is Lipschitzian with rate \( c \) on a ball \( B\left( {\bar{x},\rho }\right) \) and \( \sigma \in \left( {0,\r... | Yes |
Proposition 4.33. If \( \partial \) is the subdifferential \( {\partial }_{\mathcal{D}} \) associated with a bornology \( \mathcal{B} \) , then for every function \( f \) on \( X \) finite at \( \bar{x} \), the implications (a) \( \Rightarrow \left( b\right) \Leftrightarrow \left( c\right) \Rightarrow \left( d\right) \... | Proof. (a) \( \Rightarrow \) (b) follows from the fact that for all \( \lambda \in {\mathbb{R}}_{ + } \), one has \( \lambda {d}_{E} \leq {\iota }_{E} \) , \( \lambda {d}_{E}\left( {\bar{x}}_{f}\right) = {\iota }_{E}\left( {\bar{x}}_{f}\right) \)\n\n(b) \( \Rightarrow \) (c) Let us prove that if \( {\bar{x}}^{ * } \not... | Yes |
Proposition 4.36. Suppose \( X = Y \times Z, g\left( x\right) = {g}_{1}\left( y\right), h\left( x\right) = {h}_{2}\left( z\right) \) for \( x \mathrel{\text{:=}} \left( {y, z}\right) \) and some functions \( {g}_{1} : Y \rightarrow \overline{\mathbb{R}},{h}_{2} : Z \rightarrow \overline{\mathbb{R}} \) . Then for \( \pa... | Proof. Let \( \bar{x} \mathrel{\text{:=}} \left( {\bar{y},\bar{z}}\right) \in Y \times Z \) . For \( f \mathrel{\text{:=}} g + h \) the inequality \( {g}_{1}^{D}\left( {\bar{y}, u}\right) + {h}_{2}^{D}\left( {\bar{z}, v}\right) \leq \) \( {f}^{D}\left( {\bar{x},\left( {u, v}\right) }\right) \) for all \( \left( {u, v}\... | No |
Corollary 4.38 (Fermat’s rule). If \( f \) attains on a subset \( F \) of \( X \) a local minimum at \( \bar{x} \in F \) and if \( f \) is \( F \) -differentiable, respectively \( H \) -differentiable, at \( \bar{x} \) then we have respectively\n\n\[ \n- {f}^{\prime }\left( \bar{x}\right) \in {N}_{F}\left( {F,\bar{x}}\... | Proof. Setting \( {f}_{F} \mathrel{\text{:=}} f + {\iota }_{F} \), where \( {\iota }_{F} \) is the indicator function of \( F \), applying the preceding two propositions and the definitions of normal cones, we get the result. | No |
Proposition 4.39. Suppose \( f = h \circ g \), where \( g : X \rightarrow \overline{\mathbb{R}} \) and \( h : \overline{\mathbb{R}} \rightarrow \overline{\mathbb{R}} \) is a nondecreasing function. If \( g\left( \bar{x}\right) \) and \( h\left( {g\left( \bar{x}\right) }\right) \) are finite, then \[ {\partial }_{D}h\le... | Proof. Let \( {\bar{r}}^{ * } \in {\partial }_{D}h\left( \bar{r}\right) ,{\bar{y}}^{ * } \in {\partial }_{D}g\left( \bar{x}\right) \) . There exist maps \( \varphi : X \rightarrow \overline{\mathbb{R}},\psi : \overline{\mathbb{R}} \rightarrow \overline{\mathbb{R}} \) such that \( \varphi \leq g,\varphi \left( \bar{x}\r... | Yes |
Proposition 4.40. Let \( X, Y \) be normed spaces, and let \( f = h \circ g \), where \( g : X \rightarrow Y \) is Hadamard, respectively Fréchet, differentiable at \( \bar{x} \) and \( h : Y \rightarrow \overline{\mathbb{R}} \) is finite at \( \bar{y} \mathrel{\text{:=}} g\left( \bar{x}\right) \) . Then we have respec... | Proof. Let us first consider the Fréchet case. Given \( {\bar{y}}^{ * } \in {\partial }_{F}h\left( \bar{y}\right) \), there exists some function \( \psi : Y \rightarrow \overline{\mathbb{R}} \) that is Fréchet differentiable at \( \bar{y} \) and such that \( \psi \leq h,\psi \left( \bar{x}\right) = \) \( h\left( \bar{x... | Yes |
Corollary 4.41. Let \( F \mathrel{\text{:=}} H \circ G \) where \( G : X \rightrightarrows Y \) and \( H \mathrel{\text{:=}} \{ h\} \) is the multimap associated with a single-valued map \( h : Y \rightarrow Z \) that is Hadamard differentiable at \( \bar{y} \in G\left( \bar{x}\right) \), respectively Fréchet different... | Proof. We note that \( F = \{ \left( {x, h\left( y\right) }\right) : y \in G\left( x\right) \} = \left( {{I}_{X} \times h}\right) \left( G\right) \) . Using Proposition 2.108, we see that for all \( \left( {{x}^{ * }, - {z}^{ * }}\right) \in {N}_{D}\left( {F,\left( {\bar{x},\bar{z}}\right) }\right) \) we have \( \left(... | Yes |
Proposition 4.42. Let \( X, Y \) be Banach spaces and let \( f = h \circ g \) be as in the preceding proposition, with \( {g}^{\prime }\left( \bar{x}\right) \left( X\right) = Y \) . Then if \( g \) is (strictly or) circa-differentiable at \( \bar{x} \), respectively if \( g \) is Hadamard differentiable and \( Y \) is ... | Proof. Let \( \widehat{g} \) be given by \( \widehat{g}\left( {x, r}\right) \mathrel{\text{:=}} \left( {g\left( x\right), r}\right) \) . Since \( {E}_{f} = {\widehat{g}}^{-1}\left( {E}_{h}\right) \), the result for the Fréchet (resp. Hadamard) case follows from the calculus rule for the normal cone to an inverse image ... | Yes |
Proposition 4.44. Let \( f = h \circ g \), where \( g : X \rightarrow Y \) and \( h : Y \rightarrow \overline{\mathbb{R}} \) . If \( g \) is stable at \( \bar{x} \in \bar{X} \) and if \( h \) is finite at \( \bar{y} \mathrel{\text{:=}} g\left( \bar{x}\right) \), then for all \( {y}^{ * } \in {\widetilde{\partial }}_{F}... | Proof. Let \( {x}^{ * } \in {\partial }_{F}f\left( \bar{x}\right) ,{y}^{ * } \in {\widetilde{\partial }}_{F}h\left( \bar{y}\right) \) and let \( c \in {\mathbb{R}}_{ + },\rho > 0 \) be such that \( \parallel g\left( x\right) - g\left( \bar{x}\right) \parallel \) \( \leq c\parallel x - \bar{x}\parallel \) for every \( x... | No |
Proposition 4.45. Given \( f, g : X \rightarrow {\mathbb{R}}_{\infty } \) lower semicontinuous and finite at \( \bar{x} \in X \) , let \( p = f \cdot g \) and let \( {\bar{x}}^{ * } \in {\partial }_{D}f\left( \bar{x}\right) ,{\bar{y}}^{ * } \in {\partial }_{D}g\left( \bar{x}\right) \) . If \( f\left( \bar{x}\right) > 0... | Proof. Let \( U \) be a neighborhood of \( \bar{x} \) on which \( f \) and \( g \) are positive, and let \( h \mathrel{\text{:=}} \log \circ f \mid U, k \mathrel{\text{:=}} \log \circ g \mid U \), setting \( \log \left( \infty \right) \mathrel{\text{:=}} \infty \) . Then Proposition 4.39 yields \( \left( {1/f\left( \ba... | Yes |
Theorem 4.47. Let \( X, Y \) be normed spaces, let \( f : X \times Y \rightarrow \overline{\mathbb{R}} \) be finite at \( \left( {\bar{x},\bar{y}}\right) \) and such that \( f\left( {\bar{x},\bar{y}}\right) = p\left( \bar{x}\right) \), where \( p\left( x\right) \mathrel{\text{:=}} \mathop{\inf }\limits_{{y \in Y}}f\lef... | \[ {\bar{x}}^{ * } \in \partial p\left( \bar{x}\right) \Rightarrow \left( {{\bar{x}}^{ * },0}\right) \in \partial f\left( {\bar{x},\bar{y}}\right) \] | Yes |
Proposition 4.48. Let \( f \mathrel{\text{:=}} h \circ g \), where \( g \mathrel{\text{:=}} \left( {{g}_{1},\ldots ,{g}_{m}}\right) : X \rightarrow {\mathbb{R}}^{m}, h : {\mathbb{R}}^{m} \rightarrow \mathbb{R} \) is of class \( {C}^{1} \) around \( \bar{y} \mathrel{\text{:=}} g\left( \bar{x}\right) \) and nondecreasing... | Proof. We give the proof for the firm subdifferential, the proof for the directional subdifferential being similar. We use the fact that for some map \( v \mathrel{\text{:=}} \left( {{v}_{1},\ldots ,{v}_{m}}\right) \) : \( {\mathbb{R}}^{m} \times {\mathbb{R}}^{m} \rightarrow {\mathbb{R}}_{ + }^{m} \) continuous around ... | Yes |
Proposition 4.49. Let \( X \) and \( Y \) be normed spaces, let \( f : X \rightarrow \overline{\mathbb{R}} \) be a lower semicontinuous function, let \( g : Y \rightarrow \mathbb{R} \) be Gâteaux differentiable at some \( \bar{y} \in Y \) with \( {g}^{\prime }\left( \bar{y}\right) \neq 0 \) . Let \( h : X \times Y \rig... | Proof. Without loss of generality we assume that \( \bar{x} = 0,\bar{y} = 0, f\left( \bar{x}\right) = g\left( \bar{y}\right) = 0 \) . Let \( \left( {{\bar{x}}^{ * },{\bar{y}}^{ * }}\right) \in \partial h\left( {\bar{x},\bar{y}}\right) \) with \( {\bar{y}}^{ * } \neq {g}^{\prime }\left( \bar{y}\right) \neq 0 \) . Let \(... | Yes |
Proposition 4.51. If \( S \) is compact, under assumptions (P1),(P2), one has\n\n\[ \n{\partial }_{F}m\left( \bar{x}\right) = \overline{\operatorname{co}}\left\{ {D{f}_{s}\left( \bar{x}\right) : s \in M\left( \bar{x}\right) }\right\} \n\]\n\n\[ \n{\partial }_{F}p\left( \bar{x}\right) = \mathop{\bigcap }\limits_{{s \in ... | The proof is left as an exercise that the reader can tackle while reading Sect. 4.7.1 | No |
Proposition 4.53. Let \( X \) be a normed space. There exists a Lipschitzian smooth bump function on \( X \) if and only if the following condition is satisfied:\n\nH) for all \( c > 1 \), there exists a function \( j : X \rightarrow \mathbb{R} \) that is smooth on \( X \smallsetminus \{ 0\} \), with a derivative that ... | Proof. Let us first observe that condition \( \left( \mathrm{H}\right) \) ensures the existence of a Lipschitzian smooth bump function: it suffices to take \( b \mathrel{\text{:=}} k \circ {j}^{2} \), where \( k : \mathbb{R} \rightarrow \mathbb{R} \) is a Lipschitzian smooth function satisfying \( k\left( 0\right) = 1 ... | Yes |
Lemma 4.55. For \( a > 0 \), let \( r : \left\lbrack {0, a}\right\rbrack \rightarrow {\mathbb{R}}_{ + } \) be a remainder, i.e., a function with a right derivative at 0 and such that \( r\left( 0\right) = 0,{r}_{ + }^{\prime }\left( 0\right) = 0 \) . Suppose \( b \mathrel{\text{:=}} \sup r\left( \left\lbrack {0, a}\rig... | Proof. Let \( {a}_{0} = a,{b}_{0} \mathrel{\text{:=}} b,{a}_{n} \mathrel{\text{:=}} {2}^{-n}a,{b}_{n} \mathrel{\text{:=}} \sup r\left( \left\lbrack {0,{a}_{n - 1}}\right\rbrack \right) \) for \( n \geq 1 \), so that \( \left( {b}_{n}\right) \) is nonincreasing and \( \left( {{b}_{n}/{a}_{n - 1}}\right) \rightarrow 0 \)... | Yes |
Theorem 4.56. Let \( X \) be a normed space satisfying condition \( \left( {H}_{F}\right) \) . Then for every lower semicontinuous function \( f \) on \( \bar{X},{\partial }_{F}^{V}f\left( \bar{x}\right) \), the viscosity Fréchet subdifferential of \( f \) at \( \bar{x} \), coincides with \( {\partial }_{F}f\left( \bar... | Proof. Without loss of generality we suppose \( \bar{x} = 0 \) . Clearly, \( {\partial }_{F}^{V}f\left( 0\right) \subset {\partial }_{F}f\left( 0\right) \) . Given \( {\bar{x}}^{ * } \in {\partial }_{F}f\left( 0\right) \), consider the remainder\n\n\[ r\left( t\right) \mathrel{\text{:=}} \sup \left\{ {f\left( 0\right) ... | Yes |
Proposition 4.57. Let \( E \) be a closed subset of a Banach space \( X \) and let \( \bar{x} \in E \) . For both viscosity subdifferentials \( \partial = {\partial }_{H},{\partial }_{F}^{V} \), one has \( N\left( {E,\bar{x}}\right) = {\mathbb{R}}_{ + }\partial {d}_{E}\left( \bar{x}\right) \) . | Proof. Since for every \( r \in {\mathbb{R}}_{ + } \) and every smooth function \( \varphi \) satisfying \( \varphi \leq r{d}_{E} \) around \( \bar{x},\varphi \left( \bar{x}\right) = r{d}_{E}\left( \bar{x}\right) \) one has \( \varphi \leq {\iota }_{E} \) near \( \bar{x} \), we get the inclusion \( {\mathbb{R}}_{ + }\p... | Yes |
Theorem 4.63. Let \( X \) and \( Y \) be smooth Banach spaces, let \( g : X \rightarrow Y \) be smooth around \( \bar{x} \in X \), and let \( f : X \rightarrow {\mathbb{R}}_{\infty }, h : Y \rightarrow {\mathbb{R}}_{\infty } \) be lower semicontinuous functions finite at \( \bar{x} \) and \( \bar{y} \mathrel{\text{:=}}... | Proof. In the case that \( X \) and \( Y \) are finite-dimensional and endowed with Euclidean norms, \( h \) is Lipschitzian with rate \( \ell \) around \( \bar{y} \), and \( g \) is linear and continuous. Let us identify \( {X}^{ * } \) with \( X \) and \( {Y}^{ * } \) with \( Y \), and let us define a decoupling (or ... | Yes |
Theorem 4.65 (Ekeland-Lebourg). Let \( f : X \rightarrow {\mathbb{R}}_{\infty } \) be a lower semicontinuous function on a smooth Banach space \( X \) . Then the set \( {G}_{\partial } \mathrel{\text{:=}} \{ \left( {x, r}\right) \in X \times \mathbb{R} : r = \) \( f\left( x\right) ,\partial f\left( x\right) \neq \varno... | Proof. Let \( \left( {\bar{x},\bar{r}}\right) \in G \) and let \( \varepsilon > 0 \) be given. Since \( f \) is lower semicontinuous, there exists \( \rho \in (0,\varepsilon \rbrack \) such that \( f\left( x\right) > f\left( \bar{x}\right) - \varepsilon \) for all \( x \in B \mathrel{\text{:=}} B\left\lbrack {\bar{x},\... | Yes |
Corollary 4.66. Let \( g : W \rightarrow \mathbb{R} \) be a continuous convex function on an open convex subset of an \( F \) -smooth (resp. H-smooth) Banach space \( X \) . Then the set \( D \) of points of W at which \( g \) is Fréchet (resp. Hadamard) differentiable is dense in W. In particular, F-smooth Banach spac... | Proof. This follows from Corollary 4.65 applied to \( f \mathrel{\text{:=}} - g \) (extended by \( + \infty \) outside some closed ball \( B \subset W \) ), since a concave function \( f \) is Fréchet (resp. Hadamard) differentiable at \( x \) whenever \( {\partial }_{F}f\left( x\right) \) (resp. \( {\partial }_{D}f\le... | Yes |
Theorem 4.69. Let \( \left( {{f}_{1},\ldots ,{f}_{k}}\right) \) be a family of lower semicontinuous functions on a smooth Banach space \( X \) . Suppose \( \left( {{f}_{1},\ldots ,{f}_{k}}\right) \) is quasicoherent around \( \bar{x} \in \operatorname{dom}f \) for \( f \mathrel{\text{:=}} {f}_{1} + \cdots + {f}_{k} \) ... | Proof in the Hadamard viscosity case. Let \( \varphi \) be a smooth function such that \( \varphi \leq f \) around \( \bar{x},\varphi \left( \bar{x}\right) = f\left( \bar{x}\right) \), and \( {\varphi }^{\prime }\left( \bar{x}\right) = {\bar{x}}^{ * } \) . Since \( \varphi \) is of class \( {D}^{1} \), there exist some... | Yes |
Theorem 4.70. Let \( X \) and \( Y \) be smooth Banach spaces, let \( g : X \rightarrow Y \) with closed graph \( G \), and let \( h : Y \rightarrow {\mathbb{R}}_{\infty } \) be uniformly continuous around \( \bar{y} \mathrel{\text{:=}} g\left( \bar{x}\right) \) or lower semicontinuous and inf-compact on the image unde... | Proof. Let \( {f}_{1},{f}_{2} : X \times Y \rightarrow {\mathbb{R}}_{\infty } \) be given by \( {f}_{1}\left( {x, y}\right) \mathrel{\text{:=}} {\iota }_{G}\left( {x, y}\right) ,{f}_{2}\left( {x, y}\right) \mathrel{\text{:=}} h\left( y\right) \) , so that \( h\left( {g\left( x\right) }\right) = \inf \left\{ {\left( {{f... | Yes |
Theorem 4.71. Let \( V \) and \( W \) be Banach spaces, \( V \) being smooth, let \( f : V \rightarrow {\mathbb{R}}_{\infty } \) be a lower semicontinuous function, let \( A : V \rightarrow W \) be a surjective continuous linear map, and let \( p : W \rightarrow \overline{\mathbb{R}} \) be the performance function give... | Proof. Let us first consider the Hadamard viscosity case. Let \( c \mathrel{\text{:=}} \max \left( {\parallel A\parallel ,1}\right) \) and let \( \psi \) be a function of class \( {D}^{1} \) such that \( \psi \left( \bar{w}\right) = p\left( \bar{w}\right) ,{\psi }^{\prime }\left( \bar{w}\right) = {\bar{w}}^{ * },\psi \... | No |
Corollary 4.72. Let \( W \) and \( X \) be smooth Banach spaces, let \( f : W \times X \rightarrow {\mathbb{R}}_{\infty } \) be a lower semicontinuous function, and let \( p : W \rightarrow {\mathbb{R}}_{\infty } \) be the performance function given by\n\n\[ p\left( w\right) \mathrel{\text{:=}} \inf \{ f\left( {w, x}\r... | If \( \partial = {\partial }_{F} \) and if \( W \) and \( X \) are \( F \) -smooth, one can take \( {w}^{ * } \in B\left( {{\bar{w}}^{ * },\varepsilon }\right) \) . | No |
Corollary 4.73. Let \( W \) and \( X \) be \( H \) -smooth Banach spaces, let \( j : W \times X \rightarrow {\mathbb{R}}_{\infty } \) be a locally Lipschitzian function, let \( G : W \rightrightarrows X \) be a multimap with closed graph, and let \( p : W \rightarrow {\mathbb{R}}_{\infty } \) be the performance functio... | Proof. In the preceding corollaries, set \( f \mathrel{\text{:=}} j + {\iota }_{G} \) . Applying the fuzzy sum rule would just give \( \begin{Vmatrix}{{x}^{ * } - {v}^{ * }}\end{Vmatrix} \leq m \) . Thus one returns to the proof of Theorem 4.71, in which was obtained a minimizer \( \bar{u} \in B\left\lbrack {\bar{v},\d... | Yes |
Theorem 4.74 (Approximate projection theorem). Let \( X \) be an H-smooth Banach space, let \( E \) be a closed subset of \( X \), and let \( \bar{w} \in X \smallsetminus E,{\bar{w}}^{ * } \in {\partial }_{H}{d}_{E}\left( \bar{w}\right) \) . Then for every \( \varepsilon > 0 \) and every compact subset \( K \) of \( X ... | Proof. In the preceding corollary we take \( W \mathrel{\text{:=}} X, G \) being the multimap with graph \( W \times E, j \) being given by \( j\left( {w, x}\right) \mathrel{\text{:=}} \parallel w - x\parallel \), so that \( p = {d}_{E} \) . Given \( {\bar{w}}^{ * } \in {\partial }_{H}{d}_{E}\left( \bar{w}\right) \) , ... | Yes |
Theorem 4.75 (Normal cone to an intersection). Let \( \\left( {{S}_{1},\\ldots ,{S}_{k}}\\right) \) be a family of subsets of a smooth Banach space satisfying the following linear coherence condition at \( \\bar{x} \\in S \\mathrel{\\text{:=}} {S}_{1} \\cap \\cdots \\cap {S}_{k} \) : for some \( c > 0, r > 0 \) ,\n\n\[... | When \( {\\bar{x}}^{ * } \\in {N}_{H}\\left( {S,\\bar{x}}\\right) \), there exists some \( c\\left( \\bar{x}\\right) > 0 \) such that for every \( \\varepsilon > 0 \) and every compact subset \( K \) of \( X \) one can find \( {x}_{i} \\in {S}_{i} \\cap B\\left( {\\bar{x},\\varepsilon }\\right) \) and \( {x}_{i}^{ * } ... | Yes |
Theorem 4.76. Let \( \left( {{f}_{1},\ldots ,{f}_{k}}\right) \) be a family of lower semicontinuous functions on a smooth Banach space \( X \) and let \( f \mathrel{\text{:=}} \max \left( {{f}_{1},\ldots ,{f}_{k}}\right) \) be finite at \( \bar{x} \in X \) . Let \( {S}_{i} \) be the epigraph of \( {f}_{i} \) . Suppose ... | Proof. When deducing this rule from Theorem 4.75 for \( {S}_{i} \mathrel{\text{:=}} \operatorname{epi}{f}_{i} \), we take into account Proposition 4.58 and the lower semicontinuity of \( {f}_{i} \) to ensure that for some \( \rho \in (0,\varepsilon \rbrack \) such that \( {f}_{i}\left( x\right) \geq {f}_{i}\left( \bar{... | Yes |
Theorem 4.77. Suppose \( X, Y \) are smooth, \( g \) is smooth, and the pair \( \left( {H, g}\right) \) is linearly coherent at \( \bar{x} \in F \) and \( {\bar{x}}^{ * } \in N\left( {F,\bar{x}}\right) \) . Then, when \( N = {N}_{F} \), for all \( \varepsilon > 0 \) there exist \( x \in B\left( {\bar{x},\varepsilon }\r... | Proof. We may suppose \( r \mathrel{\text{:=}} \begin{Vmatrix}{\bar{x}}^{ * }\end{Vmatrix} > 0 \) and even that \( r = 1 \) by homogeneity. Let \( \bar{y} \mathrel{\text{:=}} g\left( \bar{x}\right) \) . The linear coherence condition and Proposition 4.59 ensure that \( {\bar{x}}^{ * } \in \) \( c\partial \left( {{d}_{H... | Yes |
Theorem 4.78. Suppose \( g : X \rightarrow Y \) is smooth, \( f \mathrel{\text{:=}} h \circ g \) for some lower semicontinuous function \( h \) on \( Y \) with epigraph \( H \), and the pair \( \left( {H, g \times {I}_{\mathbb{R}}}\right) \) is linearly coherent at \( {\bar{x}}_{f} \mathrel{\text{:=}} \left( {\bar{x}, ... | Proof. Since the epigraph \( F \) of \( f \) satisfies \( F = {\left( g \times {I}_{\mathbb{R}}\right) }^{-1}\left( H\right) \) and since \( \left( {{\bar{x}}^{ * }, - 1}\right) \in \) \( N\left( {F,{\bar{x}}_{f}}\right) \), taking \( {\varepsilon }^{\prime } \in \left( {0,1/2}\right) \), the preceding result provides ... | No |
Theorem 4.79 (Ioffe). Let \( \left( {{f}_{1},\ldots ,{f}_{k}}\right) \) be a family of lower semicontinuous functions on \( X \) and let \( {\bar{x}}^{ * } \in \partial f\left( \bar{x}\right) \) for \( f \mathrel{\text{:=}} {f}_{1} + \cdots + {f}_{k} \) . If \( \left( {{f}_{1},\ldots ,{f}_{k}}\right) \) is linearly coh... | Proof. Let \( h : {X}^{k} \rightarrow {\mathbb{R}}_{\infty } \) be given by \( h\left( x\right) = {f}_{1}\left( {x}_{1}\right) + \cdots + {f}_{k}\left( {x}_{k}\right) \) for \( x \mathrel{\text{:=}} \left( {{x}_{1},\ldots ,{x}_{k}}\right) \) and let \( H \) be its epigraph. Denoting by \( g : X \rightarrow {X}^{k} \) t... | Yes |
Theorem 4.80. Let \( X \) be an F-smooth (resp. H-smooth) Banach space. Then for \( f \) in the set \( \mathcal{F}\left( X\right) \) of lower semicontinuous proper functions on \( X \), the function \( {\delta }_{f}\left( x\right) \mathrel{\text{:=}} \inf \left\{ {\begin{Vmatrix}{x}^{ * }\end{Vmatrix} : {x}^{ * } \in \... | Proof. Given \( f \in \mathcal{F}\left( X\right), x \in X, r, c > 0 \) such that \( f\left( x\right) < \inf f\left( {B\left( {x, r}\right) }\right) + {cr} \), we will find some \( u \in B\left( {x, r}\right) \) such that \( {\delta }_{f}\left( u\right) < c \) . Let \( {r}^{\prime } \in \left( {0, r}\right) ,{c}^{\prime... | Yes |
Proposition 4.81 (Fuzzy qualification condition). Let \( X \) be a F-smooth space.\n\n(a) Let \( \left( {{S}_{1},\ldots ,{S}_{k}}\right) \) be a family of subsets of \( X \) and \( \bar{x} \in S \mathrel{\text{:=}} {S}_{1} \cap \cdots \cap {S}_{k} \). Condition (4.37) is satisfied whenever the following alliedness cond... | Proof. (a) Let \( f \) be given by \( f\left( x\right) \mathrel{\text{:=}} d\left( {x,{S}_{1}}\right) + \cdots + d\left( {x,{S}_{k}}\right) \). By the local decrease principle, it suffices to find some \( \rho > 0, c > 0 \) such that for all \( w \in B\left( {\bar{x},{2\rho }}\right) \smallsetminus S \) and all \( {w}^... | Yes |
Theorem 4.83. Let \( {f}_{1},\ldots ,{f}_{k} \) be lower semicontinuous functions on a smooth Banach space \( X \) and let \( {\bar{x}}^{ * } \in {\partial }_{D}\left( {{f}_{1} + \cdots + {f}_{k}}\right) \left( \bar{x}\right) \) for some \( \bar{x} \in X \) . Then for every \( \varepsilon > 0 \) and every weak* neighbo... | Proof. Without loss of generality, we suppose \( \bar{x} = 0 \) . Given \( \varepsilon > 0 \) and a weak* neighborhood \( V \) of 0 in \( {X}^{ * } \), there exist \( r > 0 \) and a finite-dimensional subspace \( L \) of \( X \) such that \( {L}^{ \bot } + r{B}_{{X}^{ * }} \subset V \) . Let \( {\varepsilon }^{\prime }... | Yes |
Corollary 4.84. Let \( f \) be a lower semicontinuous function on a smooth space \( X \) and let \( {\bar{x}}^{ * } \in {\partial }_{D}f\left( \bar{x}\right) \) . Then for every \( \varepsilon > 0 \) and every weak* neighborhood \( V \) of 0 in \( {X}^{ * } \) there exist \( x \in B\left( {\bar{x},\varepsilon, f}\right... | The corresponding result for composition is as follows. It can be deduced from the preceding theorem by a proof similar to that of Theorem 4.70. | No |
Theorem 4.85. Let \( X \) and \( Y \) be smooth Banach spaces, let \( g : X \rightarrow Y \) with closed graph, and let \( h : Y \rightarrow {\mathbb{R}}_{\infty } \) be lower semicontinuous. Then for every \( {\bar{x}}^{ * } \in {\partial }_{D}\left( {h \circ g}\right) \left( \bar{x}\right) \) , every \( \varepsilon >... | \[ {x}^{ * } - {\bar{x}}^{ * } \in V,\;{y}^{ * } - {v}^{ * } \in W \] (4.43) \[ \left( {\begin{Vmatrix}{x}^{ * }\end{Vmatrix} + \begin{Vmatrix}{y}^{ * }\end{Vmatrix}}\right) \cdot \parallel g\left( x\right) - y\parallel < \varepsilon . \] (4.44) | Yes |
Theorem 4.88 (Fuzzy Rolle’s theorem). Let \( f \in \mathcal{F}\left( X\right) \) be finite at \( \bar{x} \in X \) and let \( \bar{y} \in X \smallsetminus \{ \bar{x}\} \) be such that \( f\left( \bar{y}\right) \geq f\left( \bar{x}\right) \) . Then there exist \( u \in \lbrack \bar{x},\bar{y}) \mathrel{\text{:=}} \left\l... | Proof. Let \( u \) be a minimizer of \( f \) on the compact set \( S \mathrel{\text{:=}} \left\lbrack {\bar{x},\bar{y}}\right\rbrack \) . Since \( f\left( \bar{y}\right) \geq f\left( \bar{x}\right) \) , we may suppose \( u \neq \bar{y} \) . Since \( g \mathrel{\text{:=}} {\iota }_{S} \) is inf-compact, for every \( \va... | Yes |
Theorem 4.89 (Fuzzy mean value theorem). Let \( f \in \mathcal{F}\left( X\right) \) be finite at \( \bar{x} \in X \) . Then for every \( \bar{y} \in X \smallsetminus \{ \bar{x}\} \) and for every \( r \in \mathbb{R} \) such that \( r \leq f\left( \bar{y}\right) \), there exist \( u \in \lbrack \bar{x},\bar{y}) \) and s... | Proof. Let \( {e}^{ * } \in {X}^{ * } \) be such that \( \left\langle {{e}^{ * },\bar{y} - \bar{x}}\right\rangle = f\left( \bar{x}\right) - r \) . Setting \( h \mathrel{\text{:=}} f + {e}^{ * } \), we see that \( h\left( \bar{y}\right) \geq h\left( \bar{x}\right) \), so that we can apply the Rolle’s theorem to \( h \) ... | Yes |
Theorem 4.90 (Multidirectional Rolle’s theorem). Let \( Y \) be a closed convex subset of \( X \) and let \( \bar{x} \in X \smallsetminus Y, D \mathrel{\text{:=}} \left\lbrack {\bar{x}, Y}\right\rbrack \) . Suppose \( f \in \mathcal{F}\left( X\right) \) is lower semicontinuous, finite at \( \bar{x} \), and bounded belo... | Proof. By assumption, \( \ell \mathrel{\text{:=}} { \land }_{D}f \mathrel{\text{:=}} \mathop{\sup }\limits_{{r > 0}}\inf f\left( {D + r{B}_{X}}\right) \) is finite (and \( \inf f(D + \) \( \left. {\sigma {B}_{X}}\right) \leq \ell \leq f\left( \bar{x}\right) < \infty ) \) . Taking \( \alpha \in \left( {0,{ \land }_{Y}f ... | Yes |
Theorem 4.91 (Multidirectional mean value theorem). Let \( Y \) be a closed convex subset of a smooth Banach space \( X \) and let \( \bar{x} \in X, D \mathrel{\text{:=}} \left\lbrack {\bar{x}, Y}\right\rbrack \) . Suppose \( f \in \mathcal{F}\left( X\right) \) is finite at \( \bar{x} \) and bounded below on \( D + \si... | Proof. We may suppose \( \bar{x} = 0, f\left( \bar{x}\right) = 0 \) . Let \( {q}_{n} \in \left( {r - f\left( \bar{x}\right) - {\varepsilon }_{n}, r - f\left( \bar{x}\right) }\right) \), where \( \left( {\varepsilon }_{n}\right) \rightarrow {0}_{ + } \), and let us set \( {\bar{x}}_{1} \mathrel{\text{:=}} \left( {\bar{x... | Yes |
Theorem 4.92 (Approximation of superdifferentials). Let \( f : X \rightarrow \mathbb{R} \) be a lower semicontinuous function. If \( X \) is an \( F \) -smooth Banach space, then for all \( \varepsilon > 0,\bar{x} \in X \) one has\n\n\[ \n{\widetilde{\partial }}_{F}f\left( \bar{x}\right) \subset {\overline{\operatornam... | Proof. Suppose, to the contrary, that there exist some \( \varepsilon > 0,\bar{y} \in X \), and \( {\bar{y}}^{ * } \in \) \( {\widetilde{\partial }}_{F}f\left( \bar{y}\right) \) such that \( {\bar{y}}^{ * } \notin C + \varepsilon {B}_{{X}^{ * }} \), where \( C \) denotes the weak* closed convex hull of \( {\partial }_{... | Yes |
Theorem 4.93 (Subbotin). Let \( C \) be a compact convex subset of an \( F \) -smooth Banach space \( X \) and let \( f : X \rightarrow {\mathbb{R}}_{\infty } \) be lower semicontinuous, finite at \( \bar{x} \in X \) . Let \( s < \inf \left\{ {{f}^{D}\left( {\bar{x}, v}\right) : v \in C}\right\} \) . Then for every \( ... | Proof. We first note that there exists some \( \tau > 0 \) such that for \( t \in (0,\tau \rbrack \) we have\n\n\[ \inf \left\{ {f\left( {\bar{x} + {tv} + {t}^{2}u}\right) - f\left( \bar{x}\right) : u \in {B}_{X}, v \in C}\right\} > {st} + {t}^{2}. \]\n\nOtherwise, there would be sequences \( \left( {t}_{n}\right) \rig... | Yes |
Corollary 4.94. Let \( X \) be an \( F \) -smooth Banach space and let \( f : X \rightarrow {\mathbb{R}}_{\infty } \) be finite at \( \bar{x} \in X \) and lower semicontinuous. Then for every \( \varepsilon > 0 \) one has \[ {\partial }_{D}f\left( \bar{x}\right) \subset {\overline{\operatorname{co}}}^{ * }\left( {{\par... | Proof. Let \( \varepsilon > 0 \) be given. Suppose the announced inclusion does not hold. Let \( {\bar{x}}^{ * } \in {\partial }_{D}f\left( \bar{x}\right) \smallsetminus {\overline{\operatorname{co}}}^{ * }\left( {{\partial }_{F}f\left( {B\left( {\bar{x},\varepsilon }\right) }\right) }\right) \) . Applying the Hahn-Ban... | Yes |
Theorem 4.95. Let \( f : W \rightarrow \mathbb{R} \) be a lower semicontinuous function on an open convex subset \( W \) of a smooth Banach space \( X \) . Then \( f \) is Lipschitzian with rate \( r \) if and only if for all \( x \in W \) and \( {x}^{ * } \in \partial f\left( x\right) \) one has \( \begin{Vmatrix}{x}^... | Proof. Necessity was given in Corollary 4.5. Let us prove sufficiency. Given \( \bar{x},\bar{y} \in \) \( W \), Theorem 4.89 yields \( u \in \left\lbrack {\bar{x},\bar{y}}\right\rbrack \) and sequences \( \left( {u}_{n}\right) \rightarrow u,\left( {u}_{n}^{ * }\right) \) such that \( f\left( \bar{y}\right) - \) \( f\le... | Yes |
Theorem 4.97. Let \( f : X \rightarrow {\mathbb{R}}_{\infty } \) be lower semicontinuous on a smooth Banach space \( X \) preordered by a closed convex cone \( P \) . Suppose that one has \( \partial f\left( x\right) \subset {P}^{0} \) (resp. \( \partial f\left( x\right) \subset - {P}^{0} \) ) for all \( x \in X \) . T... | Proof. Suppose, to the contrary, that there exist \( x, y \in X \) satisfying \( x \leq y \) and \( f\left( x\right) < \) \( f\left( y\right) \) . Let \( r \in \left( {f\left( x\right), f\left( y\right) }\right) \) . Then Theorem 4.89 yields \( u \) near \( \left\lbrack {x, y}\right\rbrack \) and \( {u}^{ * } \in \part... | Yes |
Proposition 4.100. Let \( X \) be a \( {\partial }_{F} \) -subdifferentiability space and let \( f : X \rightarrow {\mathbb{R}}_{\infty } \) be \( F \) -soft (resp. H-soft) at \( \bar{x} \) and Lipschitzian around \( \bar{x} \) . Then \( {\partial }_{D}f\left( \bar{x}\right) = {\mathrm{{cl}}}^{ * }\left( {{\partial }_{... | Proof. Since the set \( {\partial }_{D}f\left( \bar{x}\right) \) is weak* closed and contains \( {\partial }_{F}f\left( \bar{x}\right) \) and \( {\partial }_{H}f\left( \bar{x}\right) \), the inclusions \( {\operatorname{cl}}^{ * }\left( {{\partial }_{F}f\left( \bar{x}\right) }\right) \subset {\operatorname{cl}}^{ * }\l... | Yes |
If \( f \) is of class \( {C}^{1} \) at \( \bar{x} \), then \( f \) is F-soft at \( \bar{x} \). | If \( f \) is of class \( {D}^{1} \) at \( \bar{x} \), then \( f \) is H-soft at \( \bar{x} \), since \( \left( {{f}^{\prime }\left( {x}_{n}\right) }\right) \rightarrow {f}^{\prime }\left( \bar{x}\right) \) for the weak* topology whenever \( \left( {x}_{n}\right) \rightarrow \bar{x} \) | No |
If \( f \) is convex, then \( f \) is D-soft and F-soft on its domain. | Here we use the fact that \( {\partial }_{D}f \) and \( {\partial }_{F}f \) coincide with the subdifferential \( \partial f \) of convex analysis, so that when \( \left( {x}_{n}\right) { \rightarrow }_{f}x \) and \( {x}^{ * } \) is a weak* cluster point of a bounded sequence \( \left( {x}_{n}^{ * }\right) \) satisfying... | Yes |
Theorem 4.101. Let \( X \) and \( Y \) be smooth Banach spaces, let \( g : X \rightarrow Y \) be smooth around \( \bar{x} \in X \), and let \( f \in \mathcal{F}\left( X\right) \) be soft at \( \bar{x}, h \in \mathcal{F}\left( Y\right) \) soft at \( \bar{y} \mathrel{\text{:=}} g\left( \bar{x}\right) \) and Lipschitzian ... | Proof. We just treat the case \( \partial = {\partial }_{F} \) . Let \( \left( \left( {{w}_{n},{w}_{n}^{ * }}\right) \right) \) be a sequence in the graph of \( \partial k \) such that \( \left( {w}_{n}\right) { \rightarrow }_{k}\bar{x} \) and \( \left( {w}_{n}^{ * }\right) \) is bounded and has a weak* limit point \( ... | Yes |
Proposition 4.102. (a) If \( f \) is a separable function on \( X \mathrel{\text{:=}} {X}_{1} \times \cdots \times {X}_{k} \), i.e., if \( f\left( x\right) \mathrel{\text{:=}} {f}_{1}\left( {x}_{1}\right) + \cdots + {f}_{k}\left( {x}_{k}\right) \) for \( x \mathrel{\text{:=}} \left( {{x}_{1},\ldots ,{x}_{k}}\right) \),... | Proof. (a) The assertion stems from the relation \( \partial f\left( x\right) = \partial {f}_{1}\left( {x}_{1}\right) \times \cdots \times \partial f\left( {x}_{k}\right) \). | Yes |
Theorem 4.103. Let \( \left( {{f}_{1},\ldots ,{f}_{k}}\right) \) be a family of continuous functions on a smooth Banach space \( X \) . Let \( f \mathrel{\text{:=}} \max \left( {{f}_{1},\ldots ,{f}_{k}}\right) \) and let \( \bar{x} \in X, I \mathrel{\text{:=}} \left\{ {i \in {\mathbb{N}}_{k} : {f}_{i}\left( \bar{x}\rig... | Proof. By continuity, we have \( f = \mathop{\max }\limits_{{i \in I}}{f}_{i} \) around \( \bar{x} \), so that we may suppose \( I = {\mathbb{N}}_{k} \) . By Proposition 1.129, the family \( {\left( {f}_{i}\right) }_{i \in I} \) satisfies (4.39) at each point of a neighborhood of \( \bar{x} \) . By Theorem 4.76, given ... | Yes |
Corollary 4.104. Suppose \( f \) is lower semicontinuous on \( X \) and locally Lipschitzian on a segment \( \left\lbrack {\bar{x},\bar{y}}\right\rbrack \) of \( X \) and soft on it. Then there exist \( u \in \lbrack \bar{x},\bar{y}) \) and \( {u}^{ * } \in \partial f\left( u\right) \) such that \( f\left( u\right) \le... | Proof. Let \( \left( \left( {{u}_{n},{u}_{n}^{ * }}\right) \right) \) be the sequence of \( \partial f \) provided by Theorem 4.89. Since \( \left( {u}_{n}\right) \) converges to some point \( u \in \lbrack \bar{x},\bar{y}) \), the sequence \( \left( {u}_{n}^{ * }\right) \) is bounded. Since \( f \) is soft at \( u \) ... | Yes |
Lemma 4.105. Let \( W \) be a linear subspace of a normed space \( X \) and let \( g : X \rightarrow \mathbb{R} \) be a convex continuous function. Given \( z \in W \) and \( {z}_{W}^{ * } \in \partial \left( {\left. g\right| }_{W}\right) \left( z\right) \), there exists some \( {z}^{ * } \in \partial g\left( z\right) ... | Proof. By the Hahn-Banach theorem, there exists some \( {y}^{ * } \in {X}^{ * } \) that extends \( {z}_{W}^{ * } \) . Let \( {\iota }_{W} \) be the indicator function of \( W \) . Then we clearly have \( {y}^{ * } \in \partial \left( {g + {\iota }_{W}}\right) \left( z\right) \) . Now, the definition of the Moreau-Rocka... | Yes |
Lemma 4.106. Given an arbitrary function \( f : X \rightarrow {\mathbb{R}}_{\infty },\bar{x} \in \operatorname{dom}f \), and \( \varepsilon > 0 \), the simplified \( \varepsilon \) -subdifferential \( {\partial }^{\varepsilon }f\left( \bar{x}\right) \) of \( f \) at \( \bar{x} \) contains some element of norm at most \... | Proof. Let \( {\bar{x}}^{ * } \in {\partial }^{\varepsilon }f\left( \bar{x}\right) \) and let \( \rho > 0 \) be as in the definition of this set. Then for \( m \geq 1 \) and \( \left( {{t}_{1},\ldots ,{t}_{m}}\right) \in {\Delta }_{m},{x}_{1},\ldots ,{x}_{m} \in {\rho B} \) we have\n\n\[ f\left( {\bar{x} + {x}_{i}}\rig... | Yes |
Lemma 4.107. Given \( c \in {\mathbb{R}}_{ + },\left( {\varepsilon }_{n}\right) \rightarrow {0}_{ + } \), and an arbitrary function \( f : X \rightarrow \overline{\mathbb{R}} \) finite at \( \bar{x} \), one has \( {\partial }_{F}f\left( \bar{x}\right) \cap c{B}_{{X}^{ * }} \neq \varnothing \) if and only if there exist... | Proof. Let \( {\bar{x}}^{ * } \in {\partial }_{F}f\left( \bar{x}\right) \cap c{B}_{{X}^{ * }} \) . Let \( {\rho }_{n} > 0 \) be such that\n\n\[ \n\forall x \in {\rho }_{n}B,\;f\left( {\bar{x} + x}\right) - f\left( \bar{x}\right) - \left\langle {{\bar{x}}^{ * }, x}\right\rangle \geq - {\varepsilon }_{n}\parallel x\paral... | Yes |
Theorem 4.109. Let \( {W}_{0} \) be a separable subspace of a Banach space \( X \), let \( f \in \) \( \mathcal{F}\left( X\right) \), and let \( g : X \rightarrow \mathbb{R} \) be convex continuous. Then there exists a separable subspace \( W \) of \( X \) containing \( {W}_{0} \) such that for all \( w, z \in W \) the... | Proof. Let us first consider the case that \( g \) is a continuous linear form. Then since \( {\partial }_{F}\left( {f + g}\right) \left( w\right) = {\partial }_{F}f\left( w\right) + g \), with a similar relation for the restrictions to \( W \), the result follows from Theorem 4.108 applied to \( f + g \) . An examinat... | Yes |
Theorem 4.110. For a Banach space \( X \), the following properties are equivalent:\n\n(a) \( X \) is an Asplund space\n\n(b) For all \( n \in \mathbb{N} \smallsetminus \{ 0\} ,{X}^{n} \) is a reliable space for the Fréchet subdifferential \( {\partial }_{F} \)\n\n(c) For all \( n \in \mathbb{N} \smallsetminus \{ 0\} ,... | Proof. (a) \( \Rightarrow \) (b) Since \( {X}^{n} \) is an Asplund space by Corollary 3.101, it suffices to show that \( X \) is reliable whenever \( X \) is Asplund. Let \( \varepsilon > 0, f \in \mathcal{F}\left( X\right) \), and a convex continuous function \( g : X \rightarrow \mathbb{R} \) be given such that \( f ... | Yes |
Proposition 4.112. The following condition on \( {u}^{ * } \in {W}^{ * } \) ensures that \( {u}^{ * } \in {\widetilde{\partial }}_{F}p\left( u\right) \) :\n\n\( \left( {p}^{ + }\right) \) for every \( \varepsilon > 0 \) there exists \( \delta > 0 \) such that for all \( v \in B\left( {u,\delta }\right) ,\alpha > 0 \), ... | Proof. Given \( \varepsilon > 0 \), let \( \delta > 0 \) be as in condition \( \left( {\mathrm{p}}^{ + }\right) \) . Then for every \( v \in B\left( {u,\delta }\right) \) and every \( \alpha > 0 \), we pick \( x \in S\left( {u,\alpha }\right) \) and \( {w}^{ * } \in B\left( {{u}^{ * },\varepsilon }\right) \) such that ... | Yes |
Corollary 4.113. Suppose the following conditions bearing on some \( M \in {\mathcal{M}}_{u} \) hold: \( \left( {a}^{ + }\right) \) For all \( x \in M \) the function \( {F}_{x} \) is superdifferentiable at \( u \)\n\n\( \left( {b}^{ + }\right) \mathop{\limsup }\limits_{{\alpha \rightarrow {0}_{ + }}}{D}_{\alpha }^{ + ... | Proof. Given \( \varepsilon > 0 \), we take \( \delta > 0,\alpha > 0 \) as in condition \( \left( {\mathrm{e}}^{ + }\right) \) . Then we use condition \( \left( {\mathrm{b}}^{ + }\right) \) to pick \( {u}^{ * } \in \lim \mathop{\sup }\limits_{{\alpha \rightarrow {0}_{ + }}}{D}_{\alpha }^{ + } \), so that there exist \(... | Yes |
Corollary 4.116. Suppose that for all \( x \in X \) the function \( {F}_{x} \) is lower semicontinuous and bounded below on \( W \) and there exist some \( \varepsilon > 0,\lambda > p\left( u\right), c > 0 \), and \( V \in \) \( \mathcal{N}\left( u\right) \) such that for all \( v \in V \) and all \( x \in \left\lbrack... | Proof. For all \( v \in V, x \in \left\lbrack {{F}_{v} \leq \lambda }\right\rbrack \), picking \( {v}^{ * } \in {\widetilde{\partial }}_{F}^{\varepsilon }{F}_{x}\left( v\right) \cap B\left( {0, c}\right) \), one can find \( \rho > 0 \) such that\n\n\[ \forall w \in B\left( {v,\rho }\right) ,\;{F}_{x}\left( w\right) \le... | Yes |
Proposition 4.117. If the following conditions hold for some \( {u}^{ * } \in {W}^{ * }, M \subset X \), then \( p \) is subdifferentiable at \( u \) and \( {u}^{ * } \in {\partial }_{F}p\left( u\right) \) . \n\n\( \left( {a}^{ - }\right) {F}_{x} \) is subdifferentiable at \( u \) for all \( x \in M \)\n\n\( \left( {b}... | Proof. Given \( \varepsilon > 0 \), condition \( \left( {\mathrm{e}}^{ - }\right) \) yields some \( \alpha ,\delta > 0 \) such that (4.72) holds for all \( x \in M,{w}^{ * } \in {\partial }_{F}{F}_{x}\left( u\right) \) . Taking a smaller \( \delta \) if necessary, we may assume \( V = B\left( {u,\delta }\right) \) in \... | Yes |
Proposition 4.118. Suppose that for some \( M \in {\mathcal{M}}_{u} \) the conditions \( \left( {a}^{ - }\right) ,\left( {e}^{ - }\right) \) of Proposition 4.117 hold and\n\n(d) The perturbation \( F \) is docile with respect to \( M \)\n\n\( \left( {d}^{ - }\right) \mathop{\liminf }\limits_{{e\left( x\right) \rightarr... | Proof. Given \( {u}^{ * } \in \mathop{\liminf }\limits_{{e\left( x\right) \rightarrow 0}}{D}^{ - }\left( x\right) \), let us prove that condition \( \left( {\mathrm{b}}^{ - }\right) \) of Proposition 4.117 is satisfied. Given \( \alpha ,\varepsilon > 0 \), condition \( \left( {\mathrm{d}}^{ - }\right) \) yields some \(... | Yes |
Proposition 4.119. Suppose that for some \( M \subset X \) the conditions \( \left( {a}^{ - }\right) ,\left( {e}^{ - }\right) \) of Proposition 4.117 hold and\n\n(c) The perturbation \( F \) is compliant with respect to \( M \)\n\n\( \left( {c}^{ - }\right) \) The set \( A \mathrel{\text{:=}} \mathop{\liminf }\limits_{... | Proof. Given \( {u}^{ * } \in A \), let us prove that condition \( \left( {\mathrm{b}}^{ - }\right) \) of Proposition 4.117 is satisfied. Given \( \alpha ,\varepsilon > 0 \), the compliance assumption yields some \( \beta > 0 \) and \( V \in \mathcal{N}\left( u\right) \) such that \( S\left( {v,\beta }\right) \cap M \s... | Yes |
Proposition 4.120. Suppose the following conditions hold for some \( {u}^{ * } \in {W}^{ * } \) and \( M \in {\mathcal{M}}_{u} : \n\n(a) For all \( x \in M,{F}_{x} \) is differentiable at \( u \) and \( D{F}_{x} \rightarrow {u}^{ * } \) as \( F\left( {u, x}\right) { \rightarrow }_{M}p\left( u\right) \)\n\n(d) \( F \) i... | Proof. We note that for \( {D}_{\alpha } \mathrel{\text{:=}} \left\{ {D{F}_{x}\left( u\right) : x \in S\left( {\alpha, u}\right) \cap M}\right\} \) ,(a) implies that \( \left\{ {u}^{ * }\right\} = \) \( \mathop{\lim }\limits_{{\alpha \rightarrow {0}_{ + }}}{D}_{\alpha } \) in the sense that for every \( \varepsilon > 0... | Yes |
Lemma 4.122. The following assumptions ensure that condition \( \left( {e}_{C}\right) \) holds:\n\n( \( {a}^{\prime } \) ) There exists \( \theta > 0 \) such that for all \( x \in S\left( {u,\theta }\right) ,{F}_{x} \) is differentiable on \( B\left( {u,\theta }\right) \) \( \left( {e}_{C}^{\prime }\right) \) For every... | Proof. Given \( \varepsilon > 0 \), we take \( \alpha ,\delta > 0 \) as in \( \left( {\mathrm{e}}_{C}^{\prime }\right) \) . Then for \( x \in S\left( {u,\alpha }\right) \) one has\n\n\[ \forall v, w \in B\left( {u,\delta }\right) ,\;\left| {{F}_{x}\left( v\right) - {F}_{x}\left( w\right) - \left\langle {D{F}_{x}\left( ... | Yes |
For every solution \( \bar{x} \) of \( \left( \mathcal{P}\right) \), one has \( {\partial }_{D}p\left( 0\right) \subset K\left( \bar{x}\right) \) . | Let \( \bar{y} \in {\partial }_{D}p\left( 0\right) ,\bar{z} \mathrel{\text{:=}} g\left( \bar{x}\right) \in C \) . Let us set \( F\left( z\right) \mathrel{\text{:=}} {g}^{-1}\left( {C - z}\right) \) for \( z \in Z \) . Let \( w \mathrel{\text{:=}} r\left( {z - \bar{z}}\right) \), with \( z \in C, r > 0 \) . For \( t \in... | Yes |
Lemma 4.127. Let \( {\bar{z}}^{ * } \in {Z}^{ * } \) with \( \begin{Vmatrix}{\bar{z}}^{ * }\end{Vmatrix} = 1,\gamma \in \left( {0,1}\right) \) and let \( C \) be the Bishop-Phelps cone given by \( C \mathrel{\text{:=}} \left\{ {z \in Z : \left\langle {{\bar{z}}^{ * }, z}\right\rangle \geq \gamma \parallel z\parallel }\... | Proof. The last relation is valid for every convex cone and is obtained by using the inclusion \( C + z \subset C \), implying that \( N\left( {C, z}\right) \subset N\left( {C,0}\right) = {C}^{0} \) and by observing that for \( y \in N\left( {C, z}\right) \) one has \( \langle y,{2z} - z\rangle \leq 0,\langle y,0 - z\r... | Yes |
Lemma 4.128. Let \( g \in \mathcal{F}\left( X\right) \) . For \( x \in \operatorname{dom}g \) one has \( {\mathbb{R}}_{ + }{\partial }_{D}g\left( x\right) \subset {N}_{D}\left( {{S}_{g}\left( x\right), x}\right) \) , \( {\mathbb{R}}_{ + }{\partial }_{F}g\left( x\right) \subset {N}_{F}\left( {{S}_{g}\left( x\right), x}\... | Proof. Given \( {x}^{ * } \in {\partial }_{D}g\left( x\right) \) and \( v \in T\left( {{S}_{g}\left( x\right), x}\right) \), taking sequences \( \left( {t}_{n}\right) \rightarrow {0}_{ + },\left( {v}_{n}\right) \rightarrow v \) such that \( x + {t}_{n}{v}_{n} \in {S}_{g}\left( x\right) \), we have \( {x}^{ * } \in {N}_... | Yes |
Corollary 4.130. Let \( f \) be a lower semicontinuous function on an F-smooth Banach space \( X \) and let \( \bar{x} \in \operatorname{dom}f,{\bar{x}}^{ * } \in {\partial }_{F}^{\infty }f\left( \bar{x}\right) \) . Then for every \( \varepsilon > 0 \), there exist \( x \in B\left( {\bar{x},\varepsilon, f}\right) ,{x}^... | Proof. The space \( X \times \mathbb{R} \) is \( \mathrm{F} \) -smooth and \( g : X \times \mathbb{R} \rightarrow {\mathbb{R}}_{\infty } \) given by \( g\left( {x, r}\right) \mathrel{\text{:=}} f\left( x\right) - r \) is lower semicontinuous with \( {\partial }_{F}g\left( {x, r}\right) \subset {\partial }_{F}f\left( x\... | Yes |
Lemma 4.131. Let \( G : X \rightrightarrows Y \) be a multimap with closed graph between two Asplund spaces and let \( f \mathrel{\text{:=}} d\left( {0, G\left( \cdot \right) }\right), S \mathrel{\text{:=}} {G}^{-1}\left( 0\right) \) . Given \( x \in X \smallsetminus S \) and \( {x}^{ * } \in \) \( {\partial }_{F}f\lef... | Proof. It suffices to apply Corollary 4.73 with \( \left( {W, X,\varepsilon }\right) \) changed into \( \left( {X, Y,{\varepsilon }_{n}}\right) \) , where \( \left( {\varepsilon }_{n}\right) \rightarrow {0}_{ + } \) and \( j\left( {x, v}\right) \mathrel{\text{:=}} \parallel v\parallel \) . Then in place of \( \left( {u... | Yes |
Proposition 4.132. Let \( \bar{x} \in S \mathrel{\text{:=}} {G}^{-1}\left( 0\right) \), where \( G : X \rightrightarrows Y \) is a multimap with closed graph between two Asplund spaces. Suppose that for some \( c, r > 0 \) and for all \( x \in B\left( {\bar{x}, r}\right) \smallsetminus S \) with \( f\left( x\right) \ma... | Proof. It suffices to prove that for every \( b \in \left( {0,{c}^{-1}}\right) \), every \( x \in B\left( {\bar{x}, r}\right) \smallsetminus S \) with \( f\left( x\right) \mathrel{\text{:=}} d\left( {0, G\left( x\right) }\right) < {cr} \) and every \( {x}^{ * } \in {\partial }_{F}f\left( x\right) \) one has \( \begin{V... | Yes |
Theorem 4.133. Let \( \bar{x} \in S \mathrel{\text{:=}} {G}^{-1}\left( 0\right) \), where \( G : X \rightrightarrows Y \) is a multimap with closed graph between two Asplund spaces. Suppose that for some \( c > 0 \) and some open neighborhoods \( U \) of \( \bar{x}, V \) of 0 and for all \( u \in U \smallsetminus S, v ... | Proof. Let \( r > 0 \) be such that \( B\left( {\bar{x}, r}\right) \subset U, B\left( {0,{cr}}\right) \subset V \) . Given \( x \in B\left( {\bar{x}, r}\right) \smallsetminus S \) with \( f\left( x\right) \mathrel{\text{:=}} d\left( {0, G\left( x\right) }\right) \geq {cr} \), relation (4.76) obviously holds. When \( f\... | Yes |
Theorem 4.134. Let \( X \) and \( Y \) be Asplund spaces, let \( U \) and \( V \) be open subsets of \( X \) and \( Y \) respectively, let \( F : X \rightrightarrows Y \) be a multimap with closed graph, and let \( c > 0 \). (a) \( F \) is metrically regular on \( U \times V \) with rate \( c \) if and only if for all ... | Proof. Assuming that\n\n\[ \inf \left\{ {\begin{Vmatrix}{u}^{ * }\end{Vmatrix} : {u}^{ * } \in {D}_{F}^{ * }F\left( {u, v}\right) \left( {v}^{ * }\right), u \in U, v \in F\left( u\right) \cap V,{v}^{ * } \in {S}_{{Y}^{ * }}}\right\} \geq c, \]\n\nlet us prove that for all \( \left( {x, y}\right) \in U \times V \) we ha... | Yes |
Proposition 5.3. Let \( f \in \mathcal{L}\left( W\right) \) and let \( x \in W \) . Then\n\n(a) The set \( {\partial }_{C}f\left( x\right) \) is a nonempty weak* compact convex subset of \( {X}^{ * } \) ;\n\n(b) If \( f \) is Lipschitzian with rate \( k \) near \( x \) then \( \begin{Vmatrix}{x}^{ * }\end{Vmatrix} \leq... | Proof. (a),(b) The nonemptiness of \( {\partial }_{C}f\left( x\right) \mathrel{\text{:=}} \partial {f}^{C}\left( {x, \cdot }\right) \left( 0\right) \) results from the Hahn-Banach theorem and the fact that \( {f}^{C}\left( {x, \cdot }\right) \) is sublinear and continuous. If \( f \) is Lipschitzian with rate \( k \) n... | Yes |
Corollary 5.4. If \( f \in \mathcal{L}\left( W\right) \), then the multimap \( {\partial }_{C}f\left( \text{.}\right) {isuppersemicontinuouson} \) \( W \) for the weak* topology on \( {X}^{ * } \) . | Proof. Suppose, to the contrary, that for some weak* open subset \( V \) of \( {X}^{ * } \) containing \( {\partial }_{C}f\left( x\right) \) there exist sequences \( \left( {x}_{n}\right) \rightarrow x \) and \( \left( {x}_{n}^{ * }\right) \) with \( {x}_{n}^{ * } \in {\partial }_{C}f\left( {x}_{n}\right) \smallsetminu... | Yes |
Proposition 5.6. Let \( f : W \rightarrow \mathbb{R} \) be a locally Lipschitzian function. Then\n\n\[ \forall \left( {x, v}\right) \in W \times X,\;{f}^{C}\left( {x, v}\right) \geq {f}^{D}\left( {x, v}\right) ,\;{\partial }_{D}f\left( x\right) \subset {\partial }_{C}f\left( x\right) . \]\n\nIn particular, if \( f \in ... | Proof. The inequalities \( {f}^{C}\left( {x, v}\right) \geq \mathop{\limsup }\limits_{{t \rightarrow {0}_{ + }}}{t}^{-1}\left( {f\left( {x + {tv}}\right) - f\left( x\right) }\right) \geq {f}^{D}\left( {x, v}\right) \) are obvious and yield the announced inclusion. If \( f \) is Hadamard differentiable at \( x \), in pa... | Yes |
Corollary 5.8. If a locally Lipschitzian function \( f : W \rightarrow \mathbb{R} \) attains its minimum on \( W \) at \( x \), then \( 0 \in {\partial }_{C}f\left( x\right) \) . | Proof. In such a case one has \( {f}^{D}\left( {x, \cdot }\right) \geq 0,0 \in {\partial }_{D}f\left( x\right) \), hence \( 0 \in {\partial }_{C}f\left( x\right) \) . | Yes |
Theorem 5.10. Let \( f, g : W \rightarrow \mathbb{R} \) be locally Lipschitz functions. Then\n\n\[ \n{\partial }_{C}\left( {f + g}\right) \left( x\right) \subset {\partial }_{C}f\left( x\right) + {\partial }_{C}g\left( x\right) \]\n\n\[ \n{\partial }_{C}\left( {f \vee g}\right) \left( x\right) \subset \operatorname{co}... | Proof. The first relation is a consequence (via the equality \( {\partial }_{C}f\left( x\right) = \partial {f}^{C}\left( {x, \cdot }\right) \left( 0\right) \) and the corresponding rule of convex analysis) of the inequality\n\n\[ \n{\left( f + g\right) }^{C}\left( {x, \cdot }\right) \leq {f}^{C}\left( {x, \cdot }\right... | Yes |
Lemma 5.11. Suppose \( f : W \rightarrow \mathbb{R} \) is locally Lipschitzian and \( W \) contains the segment \( \left\lbrack {x, y}\right\rbrack \) . Then the function \( h \) given by \( h\left( r\right) \mathrel{\text{:=}} f\left( {x}_{r}\right) \) with \( {x}_{r} \mathrel{\text{:=}} x + r\left( {y - x}\right) \) ... | Proof. The fact that \( h \) is Lipschitzian stems from the compactness of \( \left\lbrack {0,1}\right\rbrack \) . Since the two closed convex sets appearing in relation (5.1) are compact intervals of \( \mathbb{R} \) , it suffices to prove that for \( v = - 1, + 1 \) one has\n\n\[ \max \left\{ {{r}^{ * }v : {r}^{ * } ... | Yes |
Theorem 5.12 (Lebourg’s mean value theorem [621]). Let \( f : W \rightarrow \mathbb{R} \) be locally Lipschitzian on an open subset \( W \) of \( X \) containing \( \left\lbrack {x, y}\right\rbrack \) . Then there exist some \( w \in \) \( \rbrack x, y\left\lbrack { \mathrel{\text{:=}} \{ \left( {1 - t}\right) x + {ty}... | Proof. Let \( h : \mathbb{R} \rightarrow \mathbb{R} \) be given by \( h\left( r\right) \mathrel{\text{:=}} f\left( {x}_{r}\right) \) for \( {x}_{r} \mathrel{\text{:=}} x + r\left( {y - x}\right) \) and let \( k \) be given by \( k\left( r\right) = h\left( r\right) + r\left\lbrack {f\left( x\right) - f\left( y\right) }\... | Yes |
Theorem 5.13 (Chain rule). Let \( X \) and \( Y \) be Banach spaces, let \( W \) (resp. \( Z \) ) be an open subset of \( X \) (resp. \( Y \) ), let \( g : W \rightarrow Y, h : Z \rightarrow \mathbb{R} \) be locally Lipschitz maps such that \( g\left( W\right) \subset Z \), and let \( f \mathrel{\text{:=}} h \circ g \)... | Proof. The local Lipschitz property of \( f \) is straightforward. Let \( x \in W, y \mathrel{\text{:=}} g\left( x\right) \) . (a) The inclusion \( {\partial }_{C}f\left( x\right) \subset {\overline{\operatorname{co}}}^{ * }\left( A\right) \), with \( A \mathrel{\text{:=}} \left\{ {{\partial }_{C}\left( {{y}^{ * } \cir... | Yes |
Proposition 5.14. Let \( V \) and \( W \) be two Banach spaces, let \( A \in L\left( {V, W}\right) \) with \( W = \) \( A\left( V\right) ,\bar{v} \in V,\bar{w} \mathrel{\text{:=}} A\bar{v} \) and let \( j \) and \( p \) be locally Lipschitzian functions on \( V \) and \( W \) respectively such that \( p \circ A \leq j ... | Proof. Let \( {\bar{w}}^{ * } \in {\partial }_{C}p\left( \bar{w}\right) \) and let \( v \in V, w \mathrel{\text{:=}} {Av} \) . Let us pick sequences \( \left( {t}_{n}\right) \rightarrow {0}_{ + } \) , \( \left( {w}_{n}\right) \rightarrow \bar{w} \) such that \( \left( {1/{t}_{n}}\right) \left( {p\left( {{w}_{n} + {t}_{... | Yes |
Theorem 5.16. Let \( f : W \rightarrow \mathbb{R} \) be a Lipschitzian function on an open subset \( W \) of \( {\mathbb{R}}^{d} \). Let \( N \) be a set of measure zero in \( W \) and let \( {N}_{f} \) be the set of points of \( W \) at which \( f \) is not differentiable. Then, for all \( \bar{x} \in W,{\partial }_{C... | Proof. Let us first observe that the set \( A\left( \bar{x}\right) \) between curly braces in the above formula is nonempty, since \( Z \) is dense in \( W \) and \( {f}^{\prime } \) is bounded on \( Z \). Thus \( A\left( \bar{x}\right) \) is compact and (by Carathéodory’s theorem) \( C\left( \bar{x}\right) \) is compa... | Yes |
Proposition 5.18 (Vectorial mean value theorem). Let \( g : W \rightarrow {\mathbb{R}}^{p} \) be locally Lipschitzian, where \( W \) is an open convex subset of \( {\mathbb{R}}^{n} \) . Then for all \( x, y \in W \) one has\n\n\[ g\left( y\right) - g\left( x\right) \in \operatorname{co}\left( {{\partial }_{C}g\left( \l... | Proof. Let us first consider the case in which \( \left\lbrack {x, y}\right\rbrack \cap {N}_{g} \) is a set of one-dimensional measure zero. Then since \( {\partial }_{C}g\left( \left\lbrack {x, y}\right\rbrack \right) \) is compact,\n\n\[ g\left( y\right) - g\left( x\right) = {\int }_{0}^{1}{g}^{\prime }\left( {x + t\... | Yes |
Theorem 5.19 ([214, Theorem 2.6.6]). Let \( f \mathrel{\text{:=}} h \circ g \), where \( g : W \rightarrow {\mathbb{R}}^{p} \) and \( h \) : \( {\mathbb{R}}^{p} \rightarrow \mathbb{R} \) are locally Lipschitzian, \( W \) being an open subset of \( {\mathbb{R}}^{n} \) . Then | \[ \forall x \in W,\;{\partial }_{C}f\left( x\right) \subset \operatorname{co}\left( {{\partial }_{C}h\left( {g\left( x\right) }\right) \circ {\partial }_{C}g\left( x\right) }\right) . \] | Yes |
Corollary 5.20. Let \( g : W \rightarrow Y \mathrel{\text{:=}} {\mathbb{R}}^{p} \) be locally Lipschitzian, \( W \) being an open subset of \( {\mathbb{R}}^{n} \). Then\n\n\[ \forall x \in W,{y}^{ * } \in {Y}^{ * },\;{\Delta }_{C}g\left( x\right) \left( {y}^{ * }\right) = {y}^{ * } \circ \left( {{\partial }_{C}g\left( ... | Proof. For every \( x \in W,{y}^{ * } \in {Y}^{ * } \), the preceding theorem with \( h \mathrel{\text{:=}} {y}^{ * } \) ensures that\n\n\[ {\Delta }_{C}g\left( x\right) \left( {y}^{ * }\right) \mathrel{\text{:=}} {\partial }_{C}\left( {{y}^{ * } \circ g}\right) \left( x\right) \subset \operatorname{co}\left( {{y}^{ * ... | Yes |
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