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Lemma 5.22 (Hiriart-Urruty [481]). A vector \( v \) belongs to \( {T}^{C}\left( {E, a}\right) \) if and only if \( {d}_{E}^{C}\left( {a, v}\right) \leq 0 \), where \( {d}_{E} \) is the distance function associated with \( E \) . | Proof. Let \( v \in {T}^{C}\left( {E, a}\right) \) and let \( \left( {{t}_{n},{a}_{n}}\right) \rightarrow \left( {{0}_{ + }, a}\right) \) be such that\n\n\[ {d}_{E}^{C}\left( {a, v}\right) = \mathop{\lim }\limits_{n}\frac{1}{{t}_{n}}\left( {{d}_{E}\left( {{a}_{n} + {t}_{n}v}\right) - {d}_{E}\left( {a}_{n}\right) }\righ... | Yes |
Proposition 5.23. The Clarke tangent cone \( {T}^{C}\left( {E, a}\right) \) to \( E \) at \( a \in \operatorname{cl}\left( E\right) \) is a closed convex cone contained in the tangent cone \( T\left( {E, a}\right) \) to \( E \) at \( a \) (and is even contained in the incident tangent cone \( {T}^{I}\left( {E, a}\right... | Proof. The closedness of \( {T}^{C}\left( {E, a}\right) \) is a general property of inner limits. The stability of \( {T}^{C}\left( {E, a}\right) \) by homotheties is obvious. The stability of \( {T}^{C}\left( {E, a}\right) \) under addition stems from the preceding lemma and the sublinearity of \( {d}_{E}^{C}\left( {a... | Yes |
Let \( X, Y \) be normed spaces, let \( W \) be an open subset of \( X \), and let \( g : W \rightarrow Y \) be a Lipschitzian map with rate \( c \) . Then the Clarke tangent cone \( {T}^{C}\left( {G,\left( {a, b}\right) }\right) \) to the graph \( G \) of \( g \) at \( \left( {a, b}\right) \in G \) is not just a conve... | In order to prove these assertions, let \( \left( {u, v}\right) \in {T}^{C}\left( {G,\left( {a, b}\right) }\right) \) , so that for all sequences \( \left( {a}_{n}\right) \rightarrow a,\left( {t}_{n}\right) \rightarrow {0}_{ + } \), since \( \left( {b}_{n}\right) \mathrel{\text{:=}} \left( {g\left( {a}_{n}\right) }\rig... | Yes |
Proposition 5.25. (a) The Clarke normal cone \( {N}_{C}\left( {E, a}\right) \) to a subset \( E \) of \( X \) at \( a \in \) \( \operatorname{cl}\left( E\right) \) is a weak* closed convex cone. The Clarke tangent cone \( {T}^{C}\left( {E, a}\right) \) is in turn the polar cone to \( {N}_{C}\left( {E, a}\right) \) . | Proof. It remains to prove assertion (b). Let \( r \in {\mathbb{R}}_{ + } \) and \( {x}^{ * } \in {\partial }_{C}{d}_{E}\left( a\right) \) . Lemma 5.22 asserts that for every \( v \in {T}^{C}\left( {E, a}\right) \) one has \( {d}_{E}^{C}\left( {a, v}\right) \leq 0 \), hence \( \left\langle {r{x}^{ * }, v}\right\rangle ... | Yes |
Proposition 5.26. (a) If \( E \) is a convex subset of \( X \) and \( a \in \operatorname{cl}\left( E\right) \), then \( {T}^{C}\left( {E, a}\right) = T\left( {E, a}\right) \) and \( {N}_{C}\left( {E, a}\right) = N\left( {E, a}\right) = \left\{ {{x}^{ * } \in {X}^{ * } : \forall x \in E\left\langle {{x}^{ * }, x - a}\r... | Proof. (a) Let \( v \in {\mathbb{R}}_{ + }\left( {E - a}\right), v = r\left( {e - a}\right) \) with \( r \in {\mathbb{R}}_{ + }, e \in E \) . For all sequences \( \left( {t}_{n}\right) \rightarrow {0}_{ + },\left( {e}_{n}\right) \rightarrow a \) with \( {e}_{n} \in E \) for all \( n \in \mathbb{N} \), we have \( {e}_{n... | Yes |
Proposition 5.27. Let \( X, Y \) be normed spaces, let \( W \) be an open subset of \( X \), and let \( h : W \rightarrow Y \) be a mapping that is circa-differentiable at some point \( e \) of a subset \( E \) of \( W \) . Let \( F \) be a subset of \( Y \) such that \( h\left( E\right) \subset F \) . Suppose \( h \) ... | Proof. Let \( u \in {T}^{C}\left( {E, e}\right) \), let \( v \mathrel{\text{:=}} {h}^{\prime }\left( e\right) \left( u\right) \), and let \( \left( \left( {{t}_{n},{y}_{n}}\right) \right) \) be a sequence of \( \mathbb{P} \times F \) with limit \( \left( {0, h\left( e\right) }\right) \) . Since \( h \) is open at \( e ... | Yes |
Proposition 5.28. Let \( X, Y \) be normed spaces, let \( A \subset X, B \subset Y \), and let \( \left( {x, y}\right) \in \) \( A \times B \) . Then\n\n\[ \n{T}^{C}\left( {A \times B,\left( {x, y}\right) }\right) = {T}^{C}\left( {A, x}\right) \times {T}^{C}\left( {B, y}\right) ,\n\]\n\n\[ \n{N}_{C}\left( {A \times B,\... | Proof. Since the projections \( {p}_{X} : X \times Y \rightarrow X,{p}_{Y} : X \times Y \rightarrow Y \) are continuous and open, the inclusion \( {T}^{C}\left( {A \times B,\left( {x, y}\right) }\right) \subset {T}^{C}\left( {A, x}\right) \times {T}^{C}\left( {B, y}\right) \) is a consequence of the preceding propositi... | Yes |
Theorem 5.29. Given a function \( f : X \rightarrow \overline{\mathbb{R}} \) finite at \( x \in X \) and Lipschitzian around \( x \), let \( E \mathrel{\text{:=}} \operatorname{epi}f, e \mathrel{\text{:=}} \left( {x, f\left( x\right) }\right) \) . Then\n\n\[ \n{T}^{C}\left( {E, e}\right) = \operatorname{epi}{f}^{C}\lef... | Proof. Let \( \left( {u, r}\right) \in {T}^{C}\left( {E, e}\right) \) . Let \( {\left( \left( {x}_{n},{t}_{n}\right) \right) }_{n} \rightarrow \left( {x,{0}_{ + }}\right) \) be such that\n\n\[ \n{f}^{C}\left( {x, u}\right) = \lim \left( {1/{t}_{n}}\right) \left( {f\left( {{x}_{n} + {t}_{n}u}\right) - f\left( {x}_{n}\ri... | Yes |
Proposition 5.31. Let \( S \mathrel{\text{:=}} \{ x \in X : g\left( x\right) \leq g\left( \bar{x}\right) \} \) . If \( 0 \notin {\partial }_{C}g\left( \bar{x}\right) \), then one has\n\n\[ \left\{ {v : {g}^{C}\left( {\bar{x}, v}\right) \leq 0}\right\} \subset {T}^{C}\left( {S,\bar{x}}\right) \]\n\n(5.9)\n\n\[ {N}_{C}\l... | Proof. Since \( {g}^{C}\left( {\bar{x}, \cdot }\right) \) is the support function of the weak* closed convex set \( {\partial }_{C}g\left( \bar{x}\right) \) , which does not contain 0, there exists some \( u \in X \) such that \( {g}^{C}\left( {\bar{x}, u}\right) < 0 \) . Let us show first that such a vector belongs to... | Yes |
Proposition 5.33. For a function \( f : X \rightarrow \overline{\mathbb{R}} \) finite at \( x \in X \), the following assertions are equivalent:\n\n(a) \( {f}^{C}\left( {x,0}\right) > - \infty \) ;\n\n(b) \( {f}^{C}\left( {x,0}\right) = 0 \) ;\n\n(c) \( {\partial }_{C}f\left( x\right) \neq \varnothing \) .\n\nMoreover,... | Proof. Let \( E \mathrel{\text{:=}} \) epi \( f \) and let \( e \mathrel{\text{:=}} \left( {x, f\left( x\right) }\right) \) . Since \( {T}^{C}\left( {E, e}\right) \) is the epigraph of \( {f}^{C}\left( {x, \cdot }\right) \) and \( \left( {0,0}\right) \in {T}^{C}\left( {E, e}\right) \), one has \( {f}^{C}\left( {x,0}\ri... | Yes |
Proposition 5.34. If \( f : X \rightarrow \overline{\mathbb{R}} \) is convex and finite at \( x \in X \), then \( {\partial }_{C}f\left( x\right) = \) \( {\partial }_{MR}f\left( x\right) \), where \( {\partial }_{MR}f\left( x\right) \) is the subdifferential of \( f \) at \( x \) in the sense of convex analysis. | Proof. In view of Definition 5.32, for \( E \mathrel{\text{:=}} \operatorname{epi}f,{x}^{ * } \in {\partial }_{C}f\left( x\right) \) if and only if \( \left( {{x}^{ * }, - 1}\right) \in {N}_{C}\left( {E, e}\right) = N\left( {E, e}\right) \) (Proposition 5.26), if and only if \( {x}^{ * } \in {\partial }_{MR}f\left( x\r... | Yes |
If \( X, Y \) are normed spaces, if \( f \in \mathcal{F}\left( X\right), g \in \mathcal{F}\left( Y\right) \), and if \( h \) is defined by \( h\left( {x, y}\right) \mathrel{\text{:=}} f\left( x\right) + g\left( y\right) \), then for all \( \left( {x, y}\right) \in \operatorname{dom}h,\left( {u, v}\right) \in X \times Y... | Proof. Let \( q : X \times \mathbb{R} \times Y \times \mathbb{R} \rightarrow X \times Y \times \mathbb{R} \) be given by \( q\left( {{x}^{\prime },{r}^{\prime },{y}^{\prime },{s}^{\prime }}\right) = \left( {{x}^{\prime },{y}^{\prime },{r}^{\prime } + {s}^{\prime }}\right) \) and let \( {x}_{f} \mathrel{\text{:=}} \left... | Yes |
Proposition 5.36. Let \( f : X \rightarrow \overline{\mathbb{R}} \) be finite at \( x \in X \) and let \( e \mathrel{\text{:=}} \left( {x, f\left( x\right) }\right), E \mathrel{\text{:=}} \operatorname{epi}f \) . Then \( {\partial }_{C}^{\infty }f\left( x\right) \) is a weak* closed convex cone and one has the decompos... | Proof. The right-hand side of relation (5.19) is clearly contained in the left-hand side. Let \( \left( {{x}^{ * },{r}^{ * }}\right) \in {N}_{C}\left( {E, e}\right) \) . Then for every \( \left( {v, r}\right) \in {T}^{C}\left( {E, e}\right) \), one has \( \left\langle {{x}^{ * }, v}\right\rangle + {r}^{ * }r \leq \) 0 ... | Yes |
Corollary 5.37. If \( f : X \rightarrow \mathbb{R} \) is Lipschitzian around \( x \), then \( {\partial }_{C}^{\infty }f\left( x\right) = \{ 0\} \) . | Proof. When \( f \) is Lipschitzian around \( x \in X \), the set \( {\partial }_{C}f\left( x\right) \) is bounded and nonempty, so that its recession cone \( {\partial }_{C}^{\infty }f\left( x\right) \) is \( \{ 0\} \) . | Yes |
Let \( E \) be the epigraph of a Lipschitzian function \( f : W \rightarrow \mathbb{R} \) on some open subset \( W \) of a normed space \( X \). Then \( u \mathrel{\text{:=}} \left( {0,1}\right) \in H\left( {E, a}\right) \) for all \( a \in E \) | as is easily checked. | No |
Let \( E \) be a convex subset with nonempty interior. Then \( E \) has the cone property around all \( a \in \operatorname{cl}\left( E\right) \) . | In fact, given \( a \in \operatorname{cl}\left( E\right), u \in X \) such that \( a + u \in \) int \( E \), one can find \( \varepsilon \in \left( {0,1}\right) \) such that \( B\left( {a + u,{2\varepsilon }}\right) \subset E \), whence for \( e \in E \cap B\left( {a,\varepsilon }\right) \) , \( t \in \left( {0,\varepsi... | Yes |
Theorem 5.39. The set \( H\left( {E, a}\right) \) of hypertangent vectors to a subset \( E \) of \( X \) at \( a \in \) \( \operatorname{cl}\left( E\right) \) is an open convex cone contained in \( {T}^{C}\left( {E, a}\right) \) . Moreover one has\n\n\[ \n{T}^{C}\left( {E, a}\right) + H\left( {E, a}\right) = H\left( {E... | Proof. Since \( 0 \in {T}^{C}\left( {E, a}\right) \), the inclusion \( H\left( {E, a}\right) \subset {T}^{C}\left( {E, a}\right) + H\left( {E, a}\right) \) holds. Let \( u \in H\left( {E, a}\right) \) and \( v \in {T}^{C}\left( {E, a}\right) \) . Given sequences \( \left( {e}_{n}\right) { \rightarrow }_{E}a,\left( {t}_... | Yes |
Corollary 5.40. Suppose that \( E \) has the cone property around \( a \in \operatorname{cl}\left( E\right) \) . Then the multimap \( {N}_{C}\left( {E, \cdot }\right) \) is closed at a on \( \operatorname{cl}\left( E\right) \) : if \( \left( {x}_{n}\right) \rightarrow a \) in \( \operatorname{cl}\left( E\right) ,\left(... | Proof. Since \( E \) has the cone property around \( a \), one has \( {T}^{C}\left( {E, a}\right) = \operatorname{cl}\left( {H\left( {E, a}\right) }\right) \), so that it suffices to show that \( \left\langle {{x}^{ * }, u}\right\rangle \leq 0 \) for all \( u \in H\left( {E, a}\right) \) when \( {x}^{ * } \) is a weak*... | Yes |
Proposition 5.41. Let \( E \) and \( F \) be two subsets of \( X \) and let \( a \in \operatorname{cl}\left( {E \cap F}\right) \) be such that \( {T}^{C}\left( {E, a}\right) \cap H\left( {F, a}\right) \neq \varnothing \) . Then \( {T}^{C}\left( {E, a}\right) \cap {T}^{C}\left( {F, a}\right) \subset {T}^{C}\left( {E \ca... | Proof. Let \( u \in {T}^{C}\left( {E, a}\right) \cap H\left( {F, a}\right) \) . For all sequences \( \left( {a}_{n}\right) { \rightarrow }_{E \cap F}a,\left( {t}_{n}\right) \rightarrow {0}_{ + } \) one can find a sequence \( \left( {u}_{n}\right) \rightarrow u \) such that \( {a}_{n} + {t}_{n}{u}_{n} \in E \) for all \... | Yes |
Proposition 5.43. Under each of the following conditions \( f \) has the cone property around \( x \in {f}^{-1}\left( \mathbb{R}\right) \) : (a) \( f \) is Lipschitzian around \( x \) . (b) \( f \) is convex and bounded above on some neighborhood of some point \( y \) . (c) \( f \) is the indicator function \( {\iota }... | Proof. (a) If \( f \) is Lipschitzian around \( x \), then one has \( \left( {0,1}\right) \in H\left( {\operatorname{epi}f,\left( {x, f\left( x\right) }\right) }\right) \) . (b) Suppose \( f \) is convex and bounded above by \( m \) on some neighborhood of some point \( y \) . Then \( \left( {y, m + 1}\right) \) belong... | Yes |
Proposition 5.44 ([875]). If the function \( f : X \rightarrow \overline{\mathbb{R}} \) is finite at \( x \in X \) and has the cone property around \( x \), then \( {f}^{0}\left( {x, \cdot }\right) = {f}^{C}\left( {x, \cdot }\right) \) on \( \operatorname{dom}{f}^{0}\left( {x, \cdot }\right) \), and \( {f}^{0}\left( {x... | Proof. Suppose \( f \) has the cone property around \( x \) . Let \( E \) be its epigraph and \( e \mathrel{\text{:=}} \left( {x, f\left( x\right) }\right) \) . Then \( \operatorname{dom}{f}^{0}\left( {x, \cdot }\right) \) is the projection \( {p}_{X}\left( {H\left( {E, e}\right) }\right) \) of \( H\left( {E, e}\right)... | Yes |
Corollary 5.45. If \( f : X \rightarrow \overline{\mathbb{R}} \) is finite and continuous at \( x \in X \) and has the cone property around \( x \), then \( - f \) has the cone property and\n\n\[{\partial }_{C}\left( {-f}\right) \left( x\right) = - {\partial }_{C}f\left( x\right)\] | Proof. Let \( E \) (resp. \( F \) ) be the epigraph of \( f \) (resp. \( - f \) ) and let \( e \mathrel{\text{:=}} \left( {x, f\left( x\right) }\right) \) , \( {e}^{\prime } \mathrel{\text{:=}} \left( {x, - f\left( x\right) }\right) \) . Given \( {x}^{ * } \in {\partial }_{C}\left( {-f}\right) \left( x\right) \), let u... | Yes |
Proposition 5.47. A set \( E \) is regular at \( a \in \operatorname{cl}\left( E\right) \) if and only if \( {N}_{C}\left( {E, a}\right) = {N}_{D}\left( {E, a}\right) \). A function \( f : X \rightarrow \overline{\mathbb{R}} \) is regular at \( x \in {f}^{-1}\left( \mathbb{R}\right) \) if and only if \( {\partial }_{C}... | Proof. If \( {T}^{C}\left( {E, a}\right) = {T}^{D}\left( {E, a}\right) \), then \( {N}_{C}\left( {E, a}\right) \mathrel{\text{:=}} {\left( {T}^{C}\left( E, a\right) \right) }^{0} = {\left( {T}^{D}\left( E, a\right) \right) }^{0} \mathrel{\text{:=}} \) \( {N}_{D}\left( {E, a}\right) \). Conversely, suppose \( {N}_{C}\le... | Yes |
Proposition 5.48. Let \( g \) be Lipschitzian around \( \bar{x} \) and let \( S \mathrel{\text{:=}} \{ x \in X : g\left( x\right) \leq g\left( \bar{x}\right) \} \) . If \( 0 \notin {\partial }_{C}g\left( \bar{x}\right) \) and if \( g \) is regular at \( \bar{x} \), then \( S \) is regular at \( \bar{x} \) and one has \... | Proof. Let \( v \in {T}^{D}\left( {S,\bar{x}}\right) \) : there exist sequences \( \left( {t}_{n}\right) \rightarrow {0}_{ + },\left( {v}_{n}\right) \rightarrow v \) such that \( \bar{x} + {t}_{n}{v}_{n} \in S \) or \( g\left( {\bar{x} + {t}_{n}{v}_{n}}\right) \leq g\left( \bar{x}\right) \) for all \( n \in \mathbb{N} ... | Yes |
Proposition 5.49. Let \( X, Y \) be normed spaces, let \( W \) be an open subset of \( X \), let \( F \) (resp. \( G \) ) be a subset of \( X \) (resp. \( Y \) ), and let \( g : W \rightarrow Y \) be a mapping that is circa-differentiable at some point a of \( E \mathrel{\text{:=}} F \cap {g}^{-1}\left( G\right) \) . S... | Proof. Let us first show that if \( u \in {T}^{C}\left( {F, a}\right) \cap {A}^{-1}\left( {H\left( {G, b}\right) }\right) \), then \( u \in {T}^{C}\left( {E, a}\right) \) . Let \( \left( {t}_{n}\right) \rightarrow {0}_{ + },\left( {a}_{n}\right) { \rightarrow }_{E}a \) . Since \( u \in {T}^{C}\left( {F, a}\right) \), t... | Yes |
Theorem 5.50. Let \( X, Y \) be normed spaces, let \( W \) be an open subset of \( X \), let \( g \) : \( W \rightarrow Y \) be circa-differentiable at \( \bar{x} \in X \) of \( X \), and let \( h : Y \rightarrow \overline{\mathbb{R}} \) be finite at \( \bar{y} \mathrel{\text{:=}} g\left( \bar{x}\right) \) . Let \( f \... | Proof. Let \( k : X \times \mathbb{R} \rightarrow Y \times \mathbb{R} \) be given by \( k\left( {x, r}\right) = \left( {g\left( x\right), r}\right) \) and let \( E \) (resp. \( G \) ) be the epigraph of \( f \) (resp. \( h \) ), so that \( E = {k}^{-1}\left( G\right) \) . Let \( a \mathrel{\text{:=}} \left( {\bar{x}, f... | Yes |
Corollary 5.52. Let \( f : X \rightarrow \overline{\mathbb{R}} \) be lower semicontinuous and finite at \( \bar{x} \in X \) and let \( g \) be locally Lipschitzian around \( \bar{x} \) . Then relations (5.26),(5.27) hold. | Proof. This follows from the relations \( \operatorname{dom}{g}^{0}\left( {\bar{x}, \cdot }\right) = X,0 \in \operatorname{dom}{f}^{C}\left( {\bar{x}, \cdot }\right) \) . | No |
Corollary 5.53. Let \( S \) be a closed subset of \( X \) and let \( \bar{x} \in S \) be a local minimizer of a locally Lipschitzian function \( j \) . Then \( 0 \in {\partial }_{C}j\left( \bar{x}\right) + {N}_{C}\left( {S,\bar{x}}\right) \) . | Proof. Taking for \( f \) the indicator function of \( S \) and \( g = j \), the result is a consequence of the preceding corollary and of the rule \( 0 \in {\partial }_{C}\left( {j + {\iota }_{S}}\right) \left( \bar{x}\right) \) . | Yes |
Corollary 5.54. Let \( f,{g}_{1},\ldots ,{g}_{k} \) be locally Lipschitzian functions and let \( \bar{x} \) be a minimizer of \( f \) on the set \( S \mathrel{\text{:=}} \left\{ {x \in X : {g}_{i}\left( x\right) \leq 0\bar{i} \in {\mathbb{N}}_{k}}\right\} \) . Let \( I \mathrel{\text{:=}} \left\{ {i \in {\mathbb{N}}_{k... | Proof. Since \( {\partial }_{C}{g}_{i}\left( \bar{x}\right) \) is nonempty for all \( i \in {\mathbb{N}}_{k} \) and \( {y}_{i} = 0 \) for \( i \in {\mathbb{N}}_{k} \smallsetminus I \), we may suppose \( I = {\mathbb{N}}_{k} \) . Let \( g \mathrel{\text{:=}} \mathop{\max }\limits_{{i \in I}}{g}_{i} \) . We cannot have \... | Yes |
Proposition 5.55. Let \( X \) and \( Y \) be normed spaces and let \( g : X \rightarrow \overline{\mathbb{R}}, h : Y \rightarrow \overline{\mathbb{R}} \) be finite and lower semicontinuous at \( \bar{x} \in X \) and \( \bar{y} \in Y \) respectively. Then for \( f \) given by \( f\left( {x, y}\right) = g\left( x\right) ... | Proof. The roles of \( g \) and \( h \) being symmetric, it suffices to show that for every \( \left( {{x}^{ * },{y}^{ * }}\right) \in {\partial }_{C}f\left( {\bar{x},\bar{y}}\right) \) one has \( {x}^{ * } \in {\partial }_{C}g\left( \bar{x}\right) \) . This will be a consequence of the fact that setting \( \bar{p} \ma... | Yes |
Theorem 5.56. Let \( f, g : X \rightarrow \overline{\mathbb{R}} \) be two functions finite and lower semicontinuous at \( \bar{x} \in X \) and such that \( \operatorname{dom}{f}^{C}\left( {\bar{x}, \cdot }\right) \cap \operatorname{dom}{g}^{0}\left( {\bar{x}, \cdot }\right) \neq \varnothing \) and \( f\left( \bar{x}\ri... | Proof. Let \( F \mathrel{\text{:=}} \operatorname{epi}f, G \mathrel{\text{:=}} \operatorname{epi}g \) and let \( e \mathrel{\text{:=}} \left( {\bar{x}, h\left( \bar{x}\right) }\right) \in H \mathrel{\text{:=}} \operatorname{epi}h \) . Since \( H = F \cap \) \( G \), Proposition 5.49 shows that \( {T}^{C}\left( {F, e}\r... | Yes |
Proposition 5.57. Let \( f = h \circ g \), where \( g : W \rightarrow \mathbb{R} \) is lower semicontinuous on an open subset \( W \) of \( X \) and \( h : S \rightarrow \mathbb{R} \) is defined and continuous on an open interval \( S \) of \( \mathbb{R} \) containing \( g\left( W\right) \) . If \( h \) is circa-differ... | Proof. The inverse function theorem ensures that there exist open intervals \( J \subset S \) , \( I \subset \mathbb{R} \) containing \( \bar{s} \) and \( \bar{r} \mathrel{\text{:=}} h\left( \bar{s}\right) \) respectively such that \( h \) induces a homeomorphism from \( J \) onto \( I \) . Then, denoting by \( F \) (r... | Yes |
Proposition 5.58. Let \( A \in L\left( {V, W}\right) \) be a surjective, continuous, linear map between two normed spaces, let \( j : V \rightarrow \overline{\mathbb{R}}, p : W \rightarrow \mathbb{R} \) be lower semicontinuous and such that \( p \circ A \leq j \) . Suppose that for some \( \bar{v} \in V \) and all sequ... | Proof. Let us denote by \( E \) (resp. \( P \) ) the epigraph of \( j \) (resp. \( p \) ), let us set \( \bar{x} \mathrel{\text{:=}} \left( {\bar{v}, j\left( \bar{v}\right) }\right) ,\bar{y} \mathrel{\text{:=}} \left( {\bar{w}, p\left( \bar{w}\right) }\right) \), and let us show that for all \( \left( {u, r}\right) \in... | Yes |
Proposition 5.59. Let \( f : X \rightarrow \overline{\mathbb{R}}, g : Y \rightarrow \overline{\mathbb{R}} \) be lower semicontinuous functions on normed spaces \( X \) and \( Y \) respectively, and let \( h : X \times Y \rightarrow \overline{\mathbb{R}} \) be given by \( h\left( {x, y}\right) \mathrel{\text{:=}} \) \( ... | Proof. Using support functions, it suffices to show that for all \( \left( {u, v}\right) \in X \times Y \) one has\n\n\[{h}^{C}\left( {\left( {\bar{x},\bar{y}}\right) ,\left( {u, v}\right) }\right) \leq \max \left( {{f}^{C}\left( {\bar{x}, u}\right) ,{g}^{C}\left( {\bar{y}, v}\right) }\right) .\n\nBy definition of \( {... | Yes |
Theorem 5.61 ([784]). For every subset \( E \) of a Banach space \( X \) and every \( a \in \) \( \operatorname{cl}\left( E\right) \) one has\n\n\[ \mathop{\liminf }\limits_{{x \rightarrow {Ea}}}{T}^{D}\left( {E, x}\right) \subset {T}^{C}\left( {E, a}\right) . \] | Proof. Given \( v \in X \smallsetminus {T}^{C}\left( {E, a}\right) \), let us show that there exists \( \alpha > 0 \) such that\n\n\[ \forall \delta > 0\exists e \in B\left( {a,\delta }\right) \cap E : \;B\left( {v,\alpha }\right) \cap {T}^{D}\left( {E, e}\right) = \varnothing , \]\n\ni.e., \( v \notin \mathop{\liminf ... | Yes |
Corollary 5.62. If a subset \( E \) of \( X \) is sleek at \( a \in \operatorname{cl}\left( E\right) \), then it is regular at \( a \) . | Proof. Since sleekness at \( a \) means that \( {T}^{D}\left( {E, a}\right) \subset \mathop{\liminf }\limits_{{x \rightarrow E}}{T}^{D}\left( {E, x}\right) \), the regularity of \( E \) at \( a \) stems from the inclusions (5.32) and \( {T}^{C}\left( {E, a}\right) \subset {T}^{D}\left( {E, a}\right) \) . | Yes |
Theorem 5.64. Let \( E \) be a subset of a Banach space \( X \) and let \( a \in \operatorname{cl}\left( E\right) \) . Denote by \( {N}_{L}^{\mathrm{{cl}}}\left( {E, a}\right) \) the set of weak* cluster points of bounded sequences \( \left( {x}_{n}^{ * }\right) \) such that for some sequence \( \left( {x}_{n}\right) {... | Proof. To prove (5.35) it remains to show that given \( v \in {T}^{C}\left( {E, a}\right) \) and \( \varepsilon > 0 \), one can find \( \delta > 0 \) such that for all \( x \in E \cap B\left( {a,\delta }\right) \) one has \( \left( {v + \varepsilon {B}_{{X}^{* * }}}\right) \cap {T}^{* * }\left( {E, x}\right) \neq \varn... | Yes |
Corollary 5.66. For a closed subset \( E \) of a reflexive Banach space \( X \) and \( a \in E \) , setting \( {N}_{L}\left( {E, a}\right) \mathrel{\text{:=}} {\mathrm{w}}^{ * } - \operatorname{seq} - \lim \mathop{\sup }\limits_{{x{ \rightarrow }_{E}a}}{N}_{F}\left( {E, x}\right) \), one has\n\n\[ \left. {{N}_{C}\left(... | Proof. Since by Exercise 7 of Sect.4.1.3, for all \( x \in E \) one has \( {N}_{F}\left( {E, x}\right) = \) \( {\left( {T}^{\sigma }\left( E, x\right) \right) }^{0} = {\left( \overline{\operatorname{co}}\left( {T}^{\sigma }\left( E, x\right) \right) \right) }^{0} \), taking polars in relation (5.37) and applying Theore... | No |
For every subset \( E \) of a normed space \( X \) and every \( a \in \operatorname{cl}E \) the set \( {T}^{M}\left( {E, a}\right) \) is a closed convex cone. Moreover, one has\n\n\[ \n{T}^{C}\left( {E, a}\right) \subset {T}^{M}\left( {E, a}\right) \subset {T}^{I}\left( {E, a}\right) \subset {T}^{D}\left( {E, a}\right)... | The set \( {T}^{M}\left( {E, a}\right) \) is closed, since it appears as the intersection of the family of sets \( \lim \mathop{\inf }\limits_{n}{t}_{n}^{-1}\left( {E - a - {t}_{n}{z}_{n}}\right) \) indexed by the triples \( \left( {\left( {t}_{n}\right), z,\left( {z}_{n}\right) }\right) \) satisfying \( a + {t}_{n}{z}... | Yes |
Proposition 5.71. Let \( X, Y \) be normed spaces and let \( E \) (resp. \( F \) ) be a subset of an open subset \( W \) of \( X \) (resp. of \( Y \) ). Let \( h : W \rightarrow Y \) be Hadamard differentiable at \( a \in \) \( \mathrm{{cl}}E \cap W \) and directionally open at a on \( E \) with respect to \( F \) in t... | Proof. Let \( u \in {T}^{M}\left( {E, a}\right) \) and let \( v \mathrel{\text{:=}} {h}^{\prime }\left( a\right) \left( u\right) \) . Given \( z \in {T}^{D}\left( {F, b}\right) \) and sequences \( \left( {z}_{n}\right) \rightarrow z,\left( {t}_{n}\right) \rightarrow {0}_{ + } \) with \( b + {t}_{n}{z}_{n} \in F \) for ... | Yes |
Proposition 5.73. If \( {H}^{M}\left( {E, a}\right) \) is nonempty, one has\n\n\[ \n{T}^{M}\left( {E, a}\right) = \operatorname{cl}\left( {{H}^{M}\left( {E, a}\right) }\right) ,\;\text{ int }{T}^{M}\left( {E, a}\right) \subset {H}^{M}\left( {E, a}\right) ,\;{N}_{M}\left( {E, a}\right) = {\left( {H}^{M}\left( E, a\right... | Proof. The inclusion \( \operatorname{cl}\left( {{H}^{M}\left( {E, a}\right) }\right) \subset {T}^{M}\left( {E, a}\right) \) stems from the closedness of \( {T}^{M}\left( {E, a}\right) \) . Let \( u \in {H}^{M}\left( {E, a}\right) \) . Then by (5.43), for every \( w \in {T}^{M}\left( {E, a}\right) \) and every \( t > 0... | Yes |
Proposition 5.75. For every function \( f \) finite at \( x,{f}^{M}\left( {x, \cdot }\right) \) is sublinear. Moreover, | \[ {x}^{ * } \in {\partial }_{M}f\left( x\right) \Leftrightarrow \left( {{x}^{ * }, - 1}\right) \in {N}_{M}\left( {\operatorname{epi}f,\left( {x, f\left( x\right) }\right) }\right) . \] This equivalence follows from the definitions, with \( E \mathrel{\text{:=}} \operatorname{epi}f, e \mathrel{\text{:=}} \left( {x, f\l... | No |
Proposition 5.76. Suppose \( f : X \rightarrow \overline{\mathbb{R}} \) is finite at \( x \) and epi-derivable at \( x \) in the sense that \( {T}^{D}\left( {E, e}\right) = {T}^{I}\left( {E, e}\right) \) for \( E \mathrel{\text{:=}} \operatorname{epi}f, e \mathrel{\text{:=}} \left( {x, f\left( x\right) }\right) \) . Th... | Proof. We have seen that \( {T}^{M}\left( {E, e}\right) = {T}^{D}\left( {E, e}\right) \boxminus {T}^{D}\left( {E, e}\right) \) when \( E \) is tangentable at \( e \) . Then it remains to check that when \( A \mathrel{\text{:=}} B \boxminus C \) and \( A, B, C \) are the epigraphs of functions \( g, h, k \) respectively... | Yes |
Proposition 5.79. Let \( f : X \rightarrow \overline{\mathbb{R}} \) be finite at \( x \in X \) and let \( e \mathrel{\text{:=}} \left( {x, f\left( x\right) }\right), E \mathrel{\text{:=}} \operatorname{epi}f \) . Then \( {\partial }_{M}^{\infty }f\left( x\right) \) is a weak* closed convex cone, and one has the decompo... | Since \( {N}_{M}\left( {E, e}\right) \subset {N}_{C}\left( {E, e}\right) \), hence \( {\partial }_{M}^{\infty }f\left( x\right) \subset {\partial }_{C}^{\infty }f\left( x\right) \), we get that \( {\partial }_{M}^{\infty }f\left( x\right) = \{ 0\} \) when \( f \) is Lipschitzian around \( x \) . | Yes |
Proposition 6.3. When \( X \) is an Asplund space, Definitions 6.1 and 6.2 coincide. | Proof. It suffices to prove that \( {\bar{x}}^{ * } \in {\partial }_{L}f\left( \bar{x}\right) \) in the sense of Definition 6.1 whenever \( {\bar{x}}^{ * } \) is a weak* cluster point of a bounded sequence \( \left( {x}_{n}^{ * }\right) \) such that \( {x}_{n}^{ * } \in {\partial }_{F}^{{\varepsilon }_{n}}f\left( {x}_{... | Yes |
Proposition 6.4 (Limiting subdifferentiability of Lipschitz functions). Let \( X\;{be} \) an Asplund space. Let \( f \) be a function on \( X \) that is Lipschitzian around \( \bar{x} \in {f}^{-1}\left( \mathbb{R}\right) \) . Then \( {\partial }_{L}f\left( \bar{x}\right) \neq \varnothing \) . Moreover, if \( c \) is th... | Proof. By Theorem 4.65 there is a sequence \( \left( {x}_{n}\right) \rightarrow \bar{x} \) such that \( {\partial }_{F}f\left( {x}_{n}\right) \neq \varnothing \) . Let \( {x}_{n}^{ * } \in {\partial }_{F}f\left( {x}_{n}\right) \) . Then for every \( {c}^{\prime } > c \), there is a number \( m \) such that \( \begin{Vm... | Yes |
Proposition 6.6. If \( C \) is a convex subset of an arbitrary normed space \( X \) and if \( \bar{x} \in C \), then one has \( {N}_{L}\left( {C,\bar{x}}\right) = N\left( {C,\bar{x}}\right) \), the normal cone in the sense of convex analysis: | Proof. Let us note that for every \( \varepsilon > 0 \) one has \( {x}^{ * } \in {N}_{F}^{\varepsilon }\left( {C,\bar{x}}\right) \) if and only if for all \( \gamma > \varepsilon \) there exists \( \delta > 0 \) such that \( \left\langle {{x}^{ * }, w - \bar{x}}\right\rangle \leq \gamma \parallel w - \bar{x}\parallel \... | Yes |
Lemma 6.7. Let \( S \) be a closed subset of an Asplund space \( X \) and let \( \bar{x} \in S \) . Then \( {\bar{x}}^{ * } \in {\partial }_{L}{d}_{S}\left( \bar{x}\right) \) if and only if there exist sequences \( \left( {x}_{n}\right) \rightarrow \bar{x},\left( {x}_{n}^{ * }\right) \overset{ * }{ \rightarrow }{\bar{x... | Proof. Let \( {\bar{x}}^{ * } \in {\partial }_{L}{d}_{S}\left( \bar{x}\right) \) . By definition, there are sequences \( \left( {w}_{n}\right) \rightarrow \bar{x},\left( {w}_{n}^{ * }\right) \overset{ * }{ \rightarrow }{\bar{x}}^{ * } \) such that \( {w}_{n}^{ * } \in {\partial }_{F}{d}_{S}\left( {w}_{n}\right) \) for ... | Yes |
Proposition 6.8. The limiting normal cone to a subset \( S \) of an Asplund space \( X \) satisfies \[ {N}_{L}\left( {S,\bar{x}}\right) = {\mathbb{R}}_{ + }{\partial }_{L}{d}_{S}\left( \bar{x}\right) ,\;\bar{x} \in S. \] | Proof. Let \( {\bar{x}}^{ * } \in {N}_{L}\left( {S,\bar{x}}\right) \) : there exists a sequence \( \left( \left( {{x}_{n},{x}_{n}^{ * }}\right) \right) \) in \( S \times {X}^{ * } \) such that \( \left( {x}_{n}\right) \rightarrow \bar{x},\left( {x}_{n}^{ * }\right) \overset{ * }{ \rightarrow }{\bar{x}}^{ * } \) and \( ... | Yes |
Proposition 6.9. Let \( f : X \rightarrow \overline{\mathbb{R}} \) be a lower semicontinuous function on an Asplund space and let \( x \in {f}^{-1}\left( \mathbb{R}\right) \) . Then, denoting by \( E \) the epigraph of \( f \) and setting \( e \mathrel{\text{:=}} \) \( \left( {x, f\left( x\right) }\right) \) one has\n\... | Proof. Given \( {x}^{ * } \in {\partial }_{L}f\left( x\right) \), let \( \left( {x}_{n}\right) { \rightarrow }_{f}x,\left( {x}_{n}^{ * }\right) \overset{ * }{ \rightarrow }{x}^{ * } \) such that \( {w}_{n}^{ * } \in {\partial }_{F}f\left( {x}_{n}\right) \) for all \( n \in \mathbb{N} \) . Then \( \left( {e}_{n}\right) ... | Yes |
Theorem 6.10. Let \( W \) be an open subset of an Asplund space \( X \) and let \( f : W \rightarrow \mathbb{R} \) be a locally Lipschitzian function. Then for all \( a \in W \) one has\n\n\[ \n{\partial }_{C}f\left( a\right) = {\overline{\operatorname{co}}}^{ * }\left( {{\partial }_{L}f\left( a\right) }\right) \n\] | Proof. It suffices to prove that the support functions of these sets coincide, i.e.,\n\n\[ \n{f}^{C}\left( {a, u}\right) = \sup \left\{ {\left\langle {{a}^{ * }, u}\right\rangle : \exists \left( {x}_{n}\right) \rightarrow a,\exists \left( {x}_{n}^{ * }\right) \overset{ * }{ \rightarrow }{a}^{ * },{x}_{n}^{ * } \in {\pa... | Yes |
Corollary 6.11. For a closed subset \( E \) of an Asplund space \( X \) and \( a \in E \) one has \[ \left. {{N}_{C}\left( {E, a}\right) = {\overline{\operatorname{co}}}^{ * }\left( {{N}_{L}\left( {E, a}\right) }\right) }\right) . | Proof. When \( X \) is an Asplund space one has \( {\partial }_{C}{d}_{E}\left( a\right) = {\overline{\mathrm{{co}}}}^{ * }\left( {{\partial }_{L}{d}_{E}\left( a\right) }\right) \), hence by Propositions 5.25, 4.13, using the notation lim sup for the sequential weak* lim sup, \[ {N}_{C}\left( {E, a}\right) = {\operator... | Yes |
Proposition 6.15. If \( f : X \rightarrow \overline{\mathbb{R}} \) is lower semicontinuous and if \( {E}_{f} : X \rightrightarrows \mathbb{R} \) is the multimap with graph \( \operatorname{epi}f \), the limiting subdifferential of \( f \) at \( \bar{x} \in {f}^{-1}\left( \mathbb{R}\right) \) satisfies\n\n\[{\partial }_... | Since \( {N}_{F}\left( {{E}_{f}, e}\right) \subset {X}^{ * } \times {\mathbb{R}}_{ - } \) for all \( e \in {E}_{f} \), we have \( {N}_{L}\left( {{E}_{f}, e}\right) \subset {X}^{ * } \times {\mathbb{R}}_{ - } \) . This inclusion incites us to introduce the singular limiting subdifferential of \( f \) at \( \bar{x} \) by... | Yes |
Proposition 6.16 (Scalarization). Let \( X \) and \( Y \) be Asplund spaces and let \( g : X \rightarrow \) \( Y \) be continuous at \( \bar{x} \in X \) . Then, for all \( {y}^{ * } \in {Y}^{ * } \), one has the following inclusion, which is an equality when \( g \) is Lipschitzian around \( \bar{x} \) : | Proof. Given \( {y}^{ * } \in {Y}^{ * } \), let us set \( f \mathrel{\text{:=}} {y}^{ * } \circ g \) . For all \( x \in X \), Proposition 4.25 shows that \( {\partial }_{F}f\left( x\right) \subset {D}_{F}^{ * }g\left( x\right) \left( {y}^{ * }\right) \) . Given \( {\bar{x}}^{ * } \in {\partial }_{L}f\left( \bar{x}\righ... | Yes |
Proposition 6.17. (a) If \( f, g \in \mathcal{F}\left( X\right) \) coincide around \( x \), then \( {\partial }_{L}f\left( x\right) = {\partial }_{L}g\left( x\right) \) . | Proof. (a) is obvious. | No |
Corollary 6.18. If \( A \) and \( B \) are closed subsets of Banach spaces \( X \) and \( Y \) respectively, and \( \left( {\bar{x},\bar{y}}\right) \in A \times B \), then \( {N}_{L}\left( {A \times B,\left( {\bar{x},\bar{y}}\right) }\right) = {N}_{L}\left( {A,\bar{x}}\right) \times {N}_{L}\left( {B,\bar{y}}\right) \) ... | Proof. Since \( {\iota }_{A \times B}\left( {x, y}\right) = {\iota }_{A}\left( x\right) + {\iota }_{B}\left( y\right) \), the result follows from Proposition 6.17 (e). | Yes |
Proposition 6.19. Let \( X \) be an Asplund space, let \( g \mathrel{\text{:=}} \left( {{g}_{1},\ldots ,{g}_{m}}\right) : X \rightarrow {\mathbb{R}}^{m} \) with \( {g}_{i} \in \mathcal{L}\left( X\right) \), and let \( h : {\mathbb{R}}^{m} \rightarrow \mathbb{R} \) be of class \( {C}^{1} \) around \( \bar{y} \mathrel{\t... | Proof. The result follows from Proposition 4.48 by a passage to the limit. | No |
Proposition 6.21. Let \( V, W \) be Banach spaces, let \( A \in L\left( {V, W}\right) \) with \( W = A\left( V\right) \) , \( \bar{w} \in W, B \subset {p}^{-1}\left( \bar{w}\right), j \in \mathcal{F}\left( V\right), p \in \mathcal{F}\left( W\right) \) be such that \( p \circ A \leq j \) .\n\n(a) Suppose that for every ... | Proof. (a) Let \( {\bar{w}}^{ * } \in {\partial }_{L}p\left( \bar{w}\right) \) . There exist sequences \( \left( {w}_{n}\right) { \rightarrow }_{p}\bar{w},\left( {w}_{n}^{ * }\right) { \rightarrow }^{ * }{\bar{w}}^{ * },\left( {\varepsilon }_{n}\right) \rightarrow \) \( {0}_{ + } \) such that \( {w}_{n}^{ * } \in {\par... | Yes |
Theorem 6.22 (Sum rule for limiting subdifferentials). Let \( X \) be an Asplund space and let \( f = {f}_{1} + \cdots + {f}_{k} \), where \( {f}_{1},\ldots ,{f}_{k - 1} \) are Lipschitzian around \( \bar{x} \) and \( {f}_{k} \) is lower semicontinuous on \( X \) and finite at \( \bar{x} \) . Then\n\n\[{\partial }_{L}f... | Proof. Let \( {\bar{x}}^{ * } \in {\partial }_{L}f\left( \bar{x}\right) \) and let \( \left( {x}_{n}\right) { \rightarrow }_{f}\bar{x},\left( {x}_{n}^{ * }\right) \overset{ * }{ \rightarrow }{\bar{x}}^{ * } \) with \( {x}_{n}^{ * } \in {\partial }_{F}f\left( {x}_{n}\right) \) for all \( n \) . Given a sequence \( \left... | Yes |
Theorem 6.23 (Chain rule for limiting subdifferentials). Let \( X \) and \( Y \) be As-plund spaces, let \( g : X \rightarrow Y \) be a map with closed graph that is continuous at \( \bar{x} \in X \) , and let \( h : Y \rightarrow {\mathbb{R}}_{\infty } \) be Lipschitzian around \( \bar{y} \mathrel{\text{:=}} g\left( \... | Proof. Let \( f \mathrel{\text{:=}} h \circ g \) and let \( {\bar{x}}^{ * } \in {\partial }_{L}f\left( \bar{x}\right) \) . There exists a sequence of pairs \( \left( {{x}_{n},{x}_{n}^{ * }}\right) \) in the graph of \( {\partial }_{F}f \) such that \( \left( {x}_{n}\right) \rightarrow \bar{x},\left( {x}_{n}^{ * }\right... | Yes |
Proposition 6.24. Let \( Q \) be a weak* locally compact cone of the dual \( {X}^{ * } \) of a Banach space. Then a net \( {\left( {x}_{i}^{ * }\right) }_{i \in I} \) of \( Q \), that weak* converges to 0 converges in norm to 0 . | Proof. Let \( V \) be a neighborhood of 0 in the weak* topology such that \( Q \cap V \) is weak* compact. Since weak* compact subsets are bounded, we can find \( s > 0 \) such that \( Q \cap V \subset s{B}_{{X}^{ * }} \) . Then for all \( \varepsilon > 0 \) there exists \( {i}_{\varepsilon } \in I \) such that \( {x}_... | Yes |
Lemma 6.25. If \( B, C \) are subsets of a normed space \( X \) and if \( P \) is a cone of \( X \) , one has the following implication; the reverse implication holds whenever \( C + P \) is closed, convex:\n\n\[ B \subset C + P \Rightarrow {P}^{0} \subset Q\left( {B, C}\right) \mathrel{\text{:=}} \left\{ {{x}^{ * } \i... | Proof. Let \( {x}^{ * } \in {P}^{0} \) . For all \( b \in B \) one can find \( c \in C, p \in P \) such that \( b = c + p \), so that \( \left\langle {{x}^{ * }, b - c}\right\rangle \leq 0 \) and \( \left\langle {{x}^{ * }, b}\right\rangle \leq \mathop{\sup }\limits_{{c \in C}}\left\langle {{x}^{ * }, c}\right\rangle =... | Yes |
Proposition 6.27. For a weak* closed and convex cone \( Q \) of \( {X}^{ * } \) the following assertions are equivalent:\n\n(a) \( Q \) is weak* locally compact;\n\n(b) There exists a weak* neighborhood \( V \) of 0 such that \( Q \cap V \) is bounded;\n\n(c) \( Q \) is contained in a Loewen cone.\n\n(d) If \( {\left( ... | Proof. (a) \( \Rightarrow \) (b) is obvious, since every weak* compact set is bounded.\n\n(b) \( \Rightarrow \) (c) We may suppose \( V = {F}^{0} \), where \( F \) is a finite set \( F \mathrel{\text{:=}} \left\{ {{x}_{1},\ldots ,{x}_{m}}\right\} \) . Let \( r > 0 \) be such that \( Q \cap V \subset r{B}_{{X}^{ * }} \)... | Yes |
Proposition 6.29. Suppose \( S \) has the cone property up to a compact set \( K \) around \( \bar{x} \), as in (6.5). Then there exist \( \gamma > 0 \) and a neighborhood \( V \) of \( \bar{x} \) such that\n\n\[ \forall x \in S \cap V,\;\gamma {B}_{X} \subset K + T\left( {S, x}\right) ,\]\n\n\[ \forall x \in S \cap V,... | Proof. Let \( \gamma ,\delta ,\tau > 0 \) be such that (6.5) holds for \( U \mathrel{\text{:=}} \gamma {B}_{X}, V \mathrel{\text{:=}} B\left( {\bar{x},\delta }\right) \) . Given \( x \in S \cap V, u \in U \), for \( t \in \left( {0,\tau }\right) \) let \( {z}_{t} \in K \) be such that \( x + {tu} - t{z}_{t} \in S \) . ... | Yes |
Proposition 6.32. Let \( S \) be a closed subset of a Banach space \( X \) such that \( {B}_{{X}^{ * }} \) is sequentially weak* compact and let \( \bar{x} \in S \) . Then each of the following conditions implies that \( S \) is normally compact at \( \bar{x} \) :\n\n(a) \( S \) is a closed convex set with nonempty int... | Proof. (a) The result has been proved in Lemma 3.71 (a).\n\n(b) It is a consequence of Propositions 6.24 and 6.27.\n\n(c) and (d) are a consequences of (b) and Proposition 6.29 (or of its proof).\n\n(e) is a special case of (d). | Yes |
Lemma 6.34. If \( F : X \rightrightarrows Y \) is pseudo-Lipschitzian around \( \left( {\bar{x},\bar{y}}\right) \), then \( F \) is coderivatively bounded at \( \left( {\bar{x},\bar{y}}\right) \), hence is coderivatively compact at \( \left( {\bar{x},\bar{y}}\right) \). | In fact, there exists \( c > 0 \) such that \( \begin{Vmatrix}{x}^{ * }\end{Vmatrix} \leq c\begin{Vmatrix}{y}^{ * }\end{Vmatrix} \) for all \( {y}^{ * } \in {Y}^{ * } \), all \( \left( {x, y}\right) \) near \( \left( {\bar{x},\bar{y}}\right) \), and all \( {x}^{ * } \in {D}_{F}^{ * }F\left( {x, y}\right) \left( {y}^{ *... | Yes |
Corollary 6.35. If \( M : W \rightrightarrows Z \) is metrically regular around \( \left( {\bar{w},\bar{z}}\right) \), then \( {M}^{-1} \) is coderivatively compact at \( \left( {\bar{z},\bar{w}}\right) \) . | Proof. The result stems from the fact that \( {M}^{-1} \) is pseudo-Lipschitzian around \( \left( {\bar{z},\bar{w}}\right) \) . | No |
Proposition 6.36. If \( F \) has the partial cone property up to a compact set around \( \left( {\bar{x},\bar{y}}\right) \) then there exist \( \alpha ,\beta > 0 \), a neighborhood \( W \) of \( \left( {\bar{x},\bar{y}}\right) \), and a compact subset \( K \) of \( X \) such that for all \( \left( {x, y}\right) \in F \... | Proof. Let \( \alpha ,\beta ,\tau > 0, W \in \mathcal{N}\left( {\bar{x},\bar{y}}\right) \), and let \( K \) be a compact subset of \( X \) such that (6.6) holds. Let \( \gamma > 0 \) be such that \( K \subset \gamma {B}_{X} \) . Then for all \( \left( {x, y}\right) \in F \cap W, u \in {B}_{X} \) , and all \( t \in \lef... | Yes |
Proposition 6.40. A lower semicontinuous function \( f : X \rightarrow \overline{\mathbb{R}} \) on an Asplund space is subdifferentially compact at a point \( \bar{x} \) where it is finite if and only if its epigraph multimap \( E \mathrel{\text{:=}} {E}_{f} \) is coderivatively compact at \( {\bar{x}}_{f} \mathrel{\te... | Proof. Suppose \( E \) is coderivatively compact at \( {\bar{x}}_{f} \) . Given sequences \( \left( {t}_{n}\right) \rightarrow {0}_{ + } \) , \( \left( {x}_{n}\right) { \rightarrow }_{f}\bar{x},\left( {w}_{n}^{ * }\right) \overset{ * }{ \rightarrow }0 \) such that \( {w}_{n}^{ * } \in {t}_{n}{\partial }_{F}f\left( {x}_... | Yes |
Theorem 6.41 (Normal cone to an intersection). Let \( \left( {{S}_{1},\ldots ,{S}_{k}}\right) \) be a family of closed subsets of an Asplund space satisfying the following linear coherence condition at \( \bar{x} \in S \mathrel{\text{:=}} {S}_{1} \cap \cdots \cap {S}_{k} \) : for some \( c > 0,\rho > 0 \) ,\n\n\[ \fora... | Proof. Let \( {\bar{x}}^{ * } \in {N}_{L}\left( {S,\bar{x}}\right) \), so that by Proposition 6.8, \( {\bar{x}}^{ * } = r{\bar{u}}^{ * } \) for some \( r \in {\mathbb{R}}_{ + } \) , \( {\bar{u}}^{ * } \in {\partial }_{L}{d}_{S}\left( \bar{x}\right) \) . Let \( f \mathrel{\text{:=}} {cd}\left( {\cdot ,{S}_{1}}\right) + ... | Yes |
Proposition 6.43. A finite family \( {\left( {S}_{i}\right) }_{i \in I}\left( {I \mathrel{\text{:=}} {\mathbb{N}}_{k}}\right) \) of closed subsets of a normed space \( X \) is allied at \( \bar{x} \in S \mathrel{\text{:=}} {S}_{1} \cap \cdots \cap {S}_{k} \) if and only if given \( {x}_{n, i} \in {S}_{i},{x}_{n, i}^{ *... | Proof. Since \( {\partial }_{F}{d}_{{S}_{i}}\left( {x}_{n, i}\right) \subset {N}_{F}\left( {{S}_{i},{x}_{n, i}}\right) \) for all \( {x}_{n, i} \in {S}_{i} \) and all \( \left( {n, i}\right) \), condition (6.12) follows from alliedness. Conversely, suppose condition (6.12) is satisfied. Let \( {\left( {x}_{n, i}\right)... | Yes |
Theorem 6.44. Let \( {\left( {S}_{i}\right) }_{i \in I}\left( {I \mathrel{\text{:=}} {\mathbb{N}}_{k}}\right) \) be a finite family of closed subsets of an Asplund space \( X \) that is allied at \( \bar{x} \in S \mathrel{\text{:=}} {S}_{1} \cap \cdots \cap {S}_{k} \) . Then there exist \( c,\rho > 0 \) such that the l... | Proof. Relation (6.11) yields some \( \gamma ,\rho \in \left( {0,1}\right) \) such that for all \( {x}_{i} \in {S}_{i} \cap B\left( {\bar{x},{5\rho }}\right) \) , \( {x}_{i}^{ * } \in {N}_{F}\left( {{S}_{i},{x}_{i}}\right) \) for \( i \in I \) satisfying \( \begin{Vmatrix}{{x}_{1}^{ * } + \cdots + {x}_{k}^{ * }}\end{Vm... | Yes |
Proposition 6.46. A finite family \( {\left( {S}_{i}\right) }_{i \in I}\left( {I \mathrel{\text{:=}} {\mathbb{N}}_{k}}\right) \) of closed subsets of an Asplund space \( X \) is allied at \( \bar{x} \in S \mathrel{\text{:=}} {S}_{1} \cap \cdots \cap {S}_{k} \) if and only if it is synergetic at \( \bar{x} \) and the no... | Proof. The necessity condition (\ | No |
Proposition 6.48. Let \( F, G : X \rightrightarrows Y \) and let \( \bar{z} \mathrel{\text{:=}} \left( {\bar{x},\bar{y}}\right) \in F \cap G \) . Then\n\n\[ \n{D}_{D}^{ * }F\left( {\bar{x},\bar{y}}\right) ▱{D}_{D}^{ * }G\left( {\bar{x},\bar{y}}\right) \subset {D}_{D}^{ * }\left( {F \cap G}\right) \left( {\bar{x},\bar{y... | Proof. The first assertion is an immediate consequence of an inclusion for the normal cone to an intersection; here one uses the facts that the normal cones are convex and that the passage to the normal cone is antitone. | No |
Corollary 6.51. Let \( X,{Y}_{1},{Y}_{2} \) be Asplund spaces and let the multimaps \( {F}_{1} : X \rightrightarrows {Y}_{1} \) , \( {F}_{2} : X \rightrightarrows {Y}_{2} \) have closed graphs. If they are cooperative at \( \left( {\bar{x},{\bar{y}}_{1},{\bar{y}}_{2}}\right) \), then for every \( \left( {{\bar{y}}_{1}^... | Proof. Let \( F \mathrel{\text{:=}} \left( {{F}_{1},{F}_{2}}\right) \) and let \( {M}_{1} \) and \( {M}_{2} \) be defined as above, so that \( F = {M}_{1} \cap {M}_{2} \) and one has the relations\n\n\[ \n{D}_{L}^{ * }{F}_{1}\left( {\bar{x},{\bar{y}}_{1}}\right) \left( {\bar{y}}_{1}^{ * }\right) = {D}_{L}^{ * }{M}_{1}\... | No |
Corollary 6.52. Let \( X \) be an Asplund space and let \( f \mathrel{\text{:=}} \left( {{f}_{1},\ldots ,{f}_{k}}\right) : X \rightarrow {\mathbb{R}}^{k} \) . Suppose \( \left( {\operatorname{epi}{f}_{1},\ldots ,\operatorname{epi}{f}_{k}}\right) \) is cooperative at \( \left( {\bar{x},\bar{y}}\right) \mathrel{\text{:=}... | The versatility of set-valued analysis can be experienced through the following statement whose proof consists in taking inverses. | No |
Corollary 6.53. Let \( {X}_{1},{X}_{2}, Y \) be Asplund spaces, let \( {G}_{1} : {X}_{1} \rightrightarrows Y,{G}_{2} : {X}_{2} \rightrightarrows Y \) be multimaps with closed graphs and let \( G : {X}_{1} \times {X}_{2} \rightrightarrows Y \) be defined by \( G\left( {{x}_{1},{x}_{2}}\right) \mathrel{\text{:=}} \) \( {... | Proof. One has \( y \in G\left( {{x}_{1},{x}_{2}}\right) \) if and only if \( \left( {{x}_{1},{x}_{2}}\right) \in {F}_{1}\left( y\right) \times {F}_{2}\left( y\right) \) for \( {F}_{1} \mathrel{\text{:=}} {G}_{1}^{-1} \) , \( {F}_{2} \mathrel{\text{:=}} {G}_{2}^{-1} \) . Thus, the result stems from Corollary 6.51 when ... | No |
Proposition 6.54. Let \( U, V, W \) be normed spaces, \( W \) being an Asplund space, let \( C \subset V, E \subset W \), and let \( p : V \rightarrow W \) be linear and continuous and such that \( p\left( C\right) \subset E \) . Let \( e \in E, A \subset {p}^{-1}\left( e\right) \cap C \) . (a) If \( p{ \mid }_{C} \) i... | Proof. (a) Let us first recall from Proposition 2.108 that for \( {e}^{ * } \in {N}_{F}\left( {E, e}\right) \) we have \( {p}^{\top }\left( {e}^{ * }\right) \in {N}_{F}\left( {C, c}\right) \) for all \( c \in {p}^{-1}\left( e\right) \) . Here, since \( p \) is linear and continuous, a direct proof is even easier than i... | Yes |
Proposition 6.55. Let \( F : X \rightrightarrows Y \) be a multimap with closed graph between two Asplund spaces and let \( C \) be a closed subset of \( X,\bar{y} \in E \mathrel{\text{:=}} F\left( C\right), B \subset {F}^{-1}\left( \bar{y}\right) \cap C \) . Suppose that the multimap \( y \mapsto {F}^{-1}\left( y\righ... | Proof. (a) Since for all \( \left( {x, y}\right) \in X \times Y \) one has \( d\left( {x, C}\right) = d\left( {\left( {x, y}\right), C \times Y}\right) \), one sees that (6.26) means that \( F \) and \( C \times Y \) are (linearly) coherent around \( \left( {\bar{x},\bar{y}}\right) \) in the sense of Theorem 6.41, so t... | Yes |
Proposition 6.56. Let \( F : X \rightrightarrows Y \) be a multimap with closed graph between two Asplund spaces and let \( Q \) be a closed subset of \( Y, P \mathrel{\text{:=}} {F}^{-1}\left( Q\right) ,\bar{x} \in P, B \subset F\left( \bar{x}\right) \cap Q \) . Suppose that the multimap \( x \mapsto F\left( x\right) ... | Proof. (a) Let \( {\bar{x}}^{ * } \in {N}_{L}\left( {P,\bar{x}}\right) \), so that by Proposition 6.8, \( {\bar{x}}^{ * } = r{\bar{u}}^{ * } \) for some \( r \in {\mathbb{R}}_{ + } \) , \( {\bar{u}}^{ * } \in {\partial }_{L}{d}_{P}\left( \bar{x}\right) \) . Let \( j : X \times Y \rightarrow \mathbb{R} \) be given by \(... | Yes |
Corollary 6.57. Let \( X \) and \( Y \) be Asplund spaces, let \( Q \) be a closed subset of \( Y \), let \( F : X \rightrightarrows Y \) be a multimap with closed graph, and let \( \bar{x} \in P \mathrel{\text{:=}} {F}^{-1}\left( Q\right) \) be a minimizer of a Lipschitzian function \( f : X \rightarrow \mathbb{R} \) ... | \[ 0 \in {\partial }_{L}f\left( \bar{x}\right) + {D}_{L}^{ * }F\left( {\bar{x},\bar{y}}\right) \left( {y}^{ * }\right) . \] | Yes |
Lemma 6.58. Suppose \( C \) is lower semicontinuous at \( \left( {\left( {\bar{x},\bar{z}}\right), B}\right) \) on \( E \) for some subset \( B \) of \( C\left( {\bar{x},\bar{z}}\right) \) and that\n\n\[ \n{D}_{M}^{ * }C\left( {\left( {\bar{x},\bar{z}}\right) ,\bar{y}}\right) \left( 0\right) \subset \mathop{\bigcup }\l... | Proof. Let \( {z}^{ * } \in {Z}^{ * } \) and let \( {x}^{ * } \in {D}_{L}^{ * }E\left( {\bar{x},\bar{z}}\right) \left( {z}^{ * }\right) \) . Then \( \left( {{x}^{ * }, - {z}^{ * }}\right) \in {N}_{L}\left( {E,\left( {\bar{x},\bar{z}}\right) }\right) \), and since \( C \) is lower semicontinuous at \( \left( {\bar{x},\b... | Yes |
Proposition 6.59. Suppose \( X, Y, Z \) are Asplund spaces, \( F \) and \( G \) have closed graphs, \( C \) is lower semicontinuous at \( \left( {\left( {\bar{x},\bar{z}}\right), B}\right) \) on \( E \), and for every \( \bar{y} \in B \) there are \( c > 0 \) and a neighborhood \( U \) of \( \left( {\bar{x},\bar{z},\ba... | Proof. Let us set\n\n\[ j\left( {x, z, y}\right) \mathrel{\text{:=}} {cd}\left( {\left( {x, y}\right), F}\right) + {cd}\left( {\left( {y, z}\right), G}\right) = {cd}\left( {\left( {x, z, y}\right) ,{F}_{Z}}\right) + {cd}\left( {\left( {x, z, y}\right) ,{G}_{X}^{-1}}\right) ,\]\n\nso that (6.36) can be rewritten \( {d}_... | Yes |
Theorem 6.60. Suppose \( X, Y, Z \) are Asplund spaces, \( F \) and \( G \) have closed graphs. If for some subset \( B \) of \( C\left( {\bar{x},\bar{z}}\right), C \) is lower semicontinuous at \( \left( {\left( {\bar{x},\bar{z}}\right), B}\right) \) on \( E \) and if \( {F}^{-1} \) and \( G \) are cooperative at \( \... | In particular, if \( C \) is lower semicontinuous at \( \left( {\bar{x},\bar{z},\bar{y}}\right) \) and if \( {F}^{-1} \) and \( G \) are cooperative at \( \left( {\bar{y},\bar{x},\bar{z}}\right) \), then (6.32) and (6.34) with \( B \mathrel{\text{:=}} \{ \bar{y}\} \) hold. | No |
Corollary 6.61. Suppose \( X, Y, Z \) are Asplund spaces, \( {F}^{-1} \) and \( G \) have closed graphs and are coordinated at \( \left( {\bar{y},\bar{x},\bar{z}}\right) \) for all \( \bar{y} \in B \subset C\left( {\bar{x},\bar{z}}\right) \) . Suppose \( C \) is lower semicontinuous at \( \left( {\left( {\bar{x},\bar{z... | Proof. We apply Corollary 6.53 with \( {X}_{1} \mathrel{\text{:=}} X,{X}_{2} \mathrel{\text{:=}} Z,{G}_{1} \mathrel{\text{:=}} F,{G}_{2} \mathrel{\text{:=}} {G}^{-1} \), since \( C\left( {x, z}\right) = F\left( x\right) \cap {G}^{-1}\left( z\right) \) for all \( \left( {x, z}\right) \in X \times Z \) and since (6.37) c... | Yes |
Corollary 6.63. If \( G \) is a single-valued map that is circa-differentiable at \( \bar{y} \), then\n\n\[ \n{D}_{L}^{ * }\left( {G \circ F}\right) \left( {\bar{x},\bar{z}}\right) \subset {D}_{L}^{ * }F\left( {\bar{x},\bar{y}}\right) \circ {\left( {G}^{\prime }\left( \bar{y}\right) \right) }^{\top }.\n\] | Proof. It is easy to see that \( C \) is lower semicontinuous at \( \left( {\left( {\bar{x},\bar{z}}\right) ,\bar{y}}\right) \) on \( E \mathrel{\text{:=}} G \circ F \) for \( z \mathrel{\text{:=}} G\left( \bar{y}\right) \) . Since \( G \) is circa-differentiable at \( \bar{y} \), it is Lipschitzian around \( \bar{y} \... | Yes |
Theorem 6.64. Suppose that \( {F}_{1} \) and \( {F}_{2} \) have closed graphs, \( C \) is lower semicontinuous at \( \left( {\left( {\bar{x},\bar{z}}\right), B}\right) \) on \( S \) for some subset \( B \) of \( C\left( {\bar{x},\bar{z}}\right) \), and for every \( \bar{y} \mathrel{\text{:=}} \left( {{\bar{y}}_{1},{\ba... | Proof. This is a consequence of Corollary 6.63, \( G \) being linear continuous, hence circa-differentiable, with \( {\left( {G}^{\prime }\left( \bar{y}\right) \right) }^{\top }\left( {z}^{ * }\right) = \left( {{z}^{ * },{z}^{ * }}\right) \), so that Corollary 6.63 yields\n\n\[ \n\forall {z}^{ * } \in {Y}^{ * }\;{D}_{L... | Yes |
Corollary 6.65. Let \( X,{Y}_{1},{Y}_{2} \) be Asplund spaces and let the multimaps \( {F}_{1} : X \rightrightarrows {Y}_{1} \) , \( {F}_{2} : X \rightrightarrows {Y}_{2} \) have closed graphs. If \( C \) is lower semicontinuous at \( \left( {\left( {\bar{x},\bar{z}}\right), B}\right) \) on \( S \) for some subset \( B... | \[ \left( {-{D}_{M}^{ * }{F}_{1}\left( {\bar{x},{\bar{y}}_{1}}\right) }\right) \left( 0\right) \cap {D}_{M}^{ * }{F}_{2}\left( {\bar{x},{\bar{y}}_{2}}\right) \left( 0\right) = \{ 0\} . \] | Yes |
Corollary 6.66. Suppose \( {F}_{2} \) is a single-valued map that is circa-differentiable at \( \bar{x} \) . Let \( {y}_{2} \in Y \) be such that \( {F}_{2}\left( \bar{x}\right) = \left\{ {y}_{2}\right\} \) . Then for all \( {\bar{y}}_{1} \in {F}_{1}\left( \bar{x}\right) \) and all \( {y}^{ * } \in {Y}^{ * } \) , \n\n\... | Proof. It is easy to see that the multimaps \( {F}_{1},{F}_{2} \) are coordinated at \( \left( {\bar{x},{\bar{y}}_{1},{\bar{y}}_{2}}\right) \) and (6.50) holds. Moreover, \( C \) is lower semicontinuous at \( \left( {\left( {\bar{x},\bar{z}}\right) ,\left( {{\bar{y}}_{1},{\bar{y}}_{2}}\right) }\right) \) on \( S \) . T... | Yes |
Theorem 6.67 (Chain rule for limiting subdifferentials). Let \( X \) and \( Y \) be Asplund spaces, let \( g : X \rightarrow Y \) be continuous around \( \bar{x} \in X \), let \( h \in \mathcal{F}\left( Y\right) \), let \( f \mathrel{\text{:=}} h \circ g \), and let \( \bar{y} \mathrel{\text{:=}} g\left( \bar{x}\right)... | Proof. Let \( S \mathrel{\text{:=}} \{ \left( {x, y, r}\right) \in X \times Y \times \mathbb{R} : \left( {x, r}\right) \in \) epi \( f\} \) and let \( j : X \times Y \times \mathbb{R} \rightarrow \mathbb{R} \) be given by \( j\left( {x, y, r}\right) \mathrel{\text{:=}} {cd}\left( {\left( {y, r}\right) \text{, epi }h}\r... | Yes |
Theorem 6.68 (Chain rule for limiting subdifferentials). Let \( X \) and \( Y \) be As-plund spaces, let \( g : X \rightarrow Y \) be continuous around \( \bar{x} \in X \), and let \( h \in \mathcal{F}\left( Y\right) \) be such that \( g \) and the epigraph multimap \( {E}_{h} \) associated with \( h \) are cooperative... | Proof. Let us apply Theorem 6.60 and Corollary 6.61 with \( Z \mathrel{\text{:=}} \mathbb{R}, F \mathrel{\text{:=}} g, G \mathrel{\text{:=}} {E}_{h} \) , \( B \mathrel{\text{:=}} \{ \bar{y}\} \) . Since \( g \) is continuous, the resultant multimap \( C \) is lower semicontinuous at \( \left( {\left( {\bar{x},\bar{z}}\... | Yes |
Theorem 6.70. Let \( X \) be an Asplund space and let \( {f}_{1},{f}_{2} \in \mathcal{F}\left( X\right) \) be finite at \( \bar{x} \) and such that\n\n\[ \n{\partial }_{L}^{\infty }{f}_{1}\left( \bar{x}\right) \cap \left( {-{\partial }_{L}^{\infty }{f}_{2}\left( \bar{x}\right) }\right) = \{ 0\} .\n\]\n\nThen if \( {f}_... | Proof. It suffices to observe that condition (6.59) amounts to (6.50) and to apply (6.49) to \( {z}^{ * } = 1 \) and \( {z}^{ * } = 0 \), with \( B \mathrel{\text{:=}} \left\{ \left( {{f}_{1}\left( \bar{x}\right) ,{f}_{2}\left( \bar{x}\right) }\right) \right\} \) . | Yes |
Corollary 6.71. Let \( X \) be an Asplund space and let \( f = {f}_{1} + \cdots + {f}_{k} \), where \( {f}_{i} \in \) \( \mathcal{F}\left( X\right) \) is finite at \( \bar{x} \) for \( i \in {\mathbb{N}}_{k} \) and such that\n\n\[ \n{x}_{1}^{ * } + \cdots + {x}_{k}^{ * } = 0,{x}_{i}^{ * } \in {\partial }_{L}^{\infty }{... | Proof. The epigraph multimap \( {E}_{f} \) of \( f \) is the sum of the epigraph multimaps \( {F}_{i} \mathrel{\text{:=}} \) \( {E}_{{f}_{i}} \) for \( i \in {\mathbb{N}}_{k} \) . The case \( k = 2 \) stems from the theorem, since (6.62) reduces to (6.59). Then assuming that \( {f}_{2},\ldots ,{f}_{k} \) are subdiffere... | Yes |
Lemma 6.73. Let \( S \) be a closed subset of an Asplund space \( X \) and let \( {\bar{x}}^{ * } \in {\partial }_{L}^{ > }{d}_{S}\left( \bar{x}\right) \) with \( \bar{x} \in S \) . Then there exist sequences \( \left( {x}_{n}\right) \rightarrow \bar{x} \) in \( S,\left( {u}_{n}\right) \) in \( {S}_{X},\left( {t}_{n}\r... | Proof. Given \( {\bar{x}}^{ * } \in {\partial }_{L}^{ > }{d}_{S}\left( \bar{x}\right) \), let \( \left( {w}_{n}\right) { \rightarrow }_{X \smallsetminus S}\bar{x},\left( {w}_{n}^{ * }\right) \overset{ * }{ \rightarrow }{\bar{x}}^{ * } \) with \( {w}_{n}^{ * } \in {\partial }_{F}{d}_{S}\left( {w}_{n}\right) \) for all \... | Yes |
Theorem 6.74. Let \( X \) be an Asplund space and let \( f \in \mathcal{F}\left( X\right) \) be nonnegative. Then the conditioning rate \( {\gamma }_{f}\left( \bar{x}\right) \mathrel{\text{:=}} \mathop{\liminf }\limits_{{x \rightarrow \bar{x}, x \in X \smallsetminus S}}f\left( x\right) /{d}_{S}\left( x\right) \) of \( ... | Proof. By Theorems 1.114 and 4.80, setting \( \bar{c} \mathrel{\text{:=}} d\left( {0,{\partial }_{L}^{ > }f\left( \bar{x}\right) }\right) \), it suffices to prove that for all \( c \in \left( {0,\bar{c}}\right) \) there exists \( r > 0 \) such that \( \begin{Vmatrix}{x}^{ * }\end{Vmatrix} > c \) whenever \( {x}^{ * } \... | Yes |
Proposition 6.75. Let \( X \) be an Asplund space and let \( f \in \mathcal{F}\left( X\right) \) be nonnegative and such that for some \( \bar{x} \in S \mathrel{\text{:=}} {f}^{-1}\left( 0\right) \), one has \( 0 \notin {\partial }_{L}^{ > }f\left( \bar{x}\right) \). Then \( f \) is linearly conditioned around \( \bar{... | Proof. Again it suffices to prove that there exist \( r, c > 0 \) such that \( \begin{Vmatrix}{x}^{ * }\end{Vmatrix} > c \) whenever \( {x}^{ * } \in {\partial }_{F}f\left( x\right) \) for some \( x \in B\left( {\bar{x},{2r}}\right) \smallsetminus S \) satisfying \( f\left( x\right) < {cr} \). If it is not the case, on... | Yes |
Corollary 6.76. Let \( \bar{x} \in S \mathrel{\text{:=}} {g}^{-1}\left( C\right) ,\bar{y} \mathrel{\text{:=}} g\left( \bar{x}\right) \), where \( g : X \rightarrow Y \) is a continuous map between two Asplund spaces and \( C \) is a closed subset of \( Y \) . Suppose there exists \( \bar{c} > 0 \) such that \( \begin{V... | Proof. Let \( f \mathrel{\text{:=}} d\left( {g\left( \cdot \right), C}\right) \) . By Theorem 6.74, to prove inequality (6.66), it suffices to show that \( \inf \left\{ {\begin{Vmatrix}{x}^{ * }\end{Vmatrix} : {x}^{ * } \in {\partial }_{L}^{ > }f\left( \bar{x}\right) }\right\} \geq \bar{c} \), or, by our assumption, th... | Yes |
Corollary 6.77. Let \( F \) and \( G \) be two closed subsets of an Asplund space and let \( \bar{x} \in E \mathrel{\text{:=}} F \cap G \) . Suppose there exists \( \bar{c} > 0 \) such that \( \begin{Vmatrix}{{\bar{y}}^{ * } + {\bar{z}}^{ * }}\end{Vmatrix} \geq \bar{c} \) for all \( \left( {{\bar{y}}^{ * },{\bar{z}}^{ ... | Proof. Let \( g : x \mapsto \left( {x, x}\right) \) be the diagonal map, so that \( E = {g}^{-1}\left( {F \times G}\right) \) . It is easy to see that \( {D}_{L}^{ * }g\left( \bar{x}\right) \left( {{y}^{ * },{z}^{ * }}\right) = {y}^{ * } + {z}^{ * } \) for all \( {y}^{ * },{z}^{ * } \in {X}^{ * } \) and \( {\partial }_... | Yes |
Theorem 6.78. Let \( Y \) and \( Z \) be Asplund spaces and let \( M : Y \rightrightarrows Z \) be a multimap with closed graph. Then the following assertions are equivalent:\n\n(a) \( M \) is pseudo-Lipschitzian around \( \left( {\bar{y},\bar{z}}\right) \in M \) ;\n\n(b) \( M \) is coderivatively compact at \( \left( ... | Proof. (a) \( \Rightarrow \) (b) and the estimates of the exact pseudo-Lipschitz rate of \( M \) are given in Proposition 6.34. (b) \( \Rightarrow \) (c) is obvious from the relation \( \begin{Vmatrix}{y}^{ * }\end{Vmatrix} \leq c\begin{Vmatrix}{z}^{ * }\end{Vmatrix} \) for \( c \mathrel{\text{:=}} \) \( \parallel {D}_... | Yes |
Theorem 6.82. (a) Let \( X \) be a WCG space and let \( f \) be an element of the set \( \mathcal{L}\left( X\right) \) of locally Lipschitzian functions on \( X \) . Then for all \( x \in X \), one has\n\n\[{\partial }_{\ell }f\left( x\right) = {\mathrm{w}}^{ * } - \mathop{\limsup }\limits_{{w \rightarrow x}}{\partial ... | Proof. (a) Let \( {x}^{ * } \in {\partial }_{\ell }f\left( x\right) \) : there exist sequences \( \left( {x}_{n}\right) \rightarrow x,\left( {x}_{n}^{ * }\right) { \rightarrow }^{ * }{x}^{ * } \) such that \( {x}_{n}^{ * } \in {\partial }_{D}f\left( {x}_{n}\right) \) for all \( n \in \mathbb{N} \) . Since \( X \) is a ... | Yes |
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