Q stringlengths 4 3.96k | A stringlengths 1 3k | Result stringclasses 4
values |
|---|---|---|
Theorem 2.112 (Lagrange multiplier rule). Let \( X, Y \) be Banach spaces, let \( W \) be an open subset of \( X \), let \( f : W \rightarrow \mathbb{R} \) be differentiable at \( a \), and let \( g : W \rightarrow Y \) be circa-differentiable at a with \( {g}^{\prime }\left( a\right) \left( X\right) = Y \) . Let \( b ... | Example. Let us find the shape of a box having a given volume \( v > 0 \) and minimal area. Denoting by \( x, y, z \) the lengths of the sides of the box, we are led to minimize\n\n\[ \nf\left( {x, y, z}\right) \mathrel{\text{:=}} 2\left( {{xy} + {yz} + {zx}}\right) \;\text{ subject to }g\left( {x, y, z}\right) \mathre... | Yes |
Lemma 2.113. Given \( U, L \), and \( j \) as above, the set \( W \mathrel{\text{:=}} \left\{ {x \in X : \operatorname{cl}\left( {{J}^{1}x\left( T\right) }\right) \subset U}\right\} \) , where \( {J}^{1}x\left( T\right) \mathrel{\text{:=}} \left\{ {\left( {x\left( t\right) ,{x}^{\prime }\left( t\right), t}\right) : t \... | Proof. By Proposition 2.16, for all \( x \in W \), the set \( \operatorname{cl}\left( {{J}^{1}x\left( T\right) }\right) \) is a compact subset of \( E \times E \times T \) . Thus, there exists some \( r > 0 \) such that \( B\left( {{J}^{1}x\left( T\right), r}\right) \subset U \) . Then for all \( w \in X \) satisfying ... | Yes |
Proposition 2.114. Suppose \( L \) is continuous on \( U \) and has partial derivatives with respect to its first and second variables that are continuous on \( U \) . Then \( j \) is Hadamard differentiable on \( W \) and for \( \bar{x} \in W, x \in X \) one has\n\n\[ \n{j}^{\prime }\left( \bar{x}\right) x = {\int }_{... | Proof. Let us set \( {L}_{t}\left( {e, v}\right) = L\left( {e, v, t}\right) \) for \( \left( {e, v, t}\right) \in U \) and\n\n\[ \nY \mathrel{\text{:=}} \left\{ {\left( {{e}_{1},{e}_{2},{v}_{1},{v}_{2}, t}\right) : \forall s \in \left\lbrack {0,1}\right\rbrack ,\left( {\left( {1 - s}\right) {e}_{1} + s{e}_{2},\left( {1... | Yes |
Proposition 2.115. Suppose \( L \) satisfies the assumptions of the preceding proposition and \( \bar{x} \) is a local minimizer of \( j \) on \( W\left( {{e}_{0},{e}_{1}}\right) \) . Then \( \bar{x} \) is a critical point of \( j \) on \( W\left( {{e}_{0},{e}_{1}}\right) \) in the following sense:\n\n\[ \n{j}^{\prime ... | Proof. Let \( N \) be a neighborhood of \( \bar{x} \) in \( X \) such that \( j\left( w\right) \geq j\left( \bar{x}\right) \) for every \( w \in N \cap \) \( W\left( {{e}_{0},{e}_{1}}\right) \) . Given \( v \in {X}_{0} \), for \( r \in \mathbb{R} \) with \( \left| r\right| \) small enough, we have \( w \mathrel{\text{:... | Yes |
Theorem 2.116 (Euler-Lagrange condition). Suppose \( L \) satisfies the assumptions of Proposition 2.114 and \( \bar{x} \in W \) is a critical point of \( j \) on \( W\left( {{e}_{0},{e}_{1}}\right) \) . Then the function \( {D}_{1}L\left( {\bar{x}\left( \cdot \right) ,{\bar{x}}^{\prime }\left( \cdot \right) , \cdot }\... | \[ \frac{\mathrm{d}}{\mathrm{d}t}\left( {{D}_{2}L\left( {\bar{x}\left( t\right) ,{\bar{x}}^{\prime }\left( t\right) }\right), t}\right) = {D}_{1}L\left( {\bar{x}\left( t\right) ,{\bar{x}}^{\prime }\left( t\right), t}\right) . \] | Yes |
Lemma 2.117. Let \( f \) be a nonnegative element of the space \( {R}_{n}\left( {T,\mathbb{R}}\right) \) of normalized regulated functions on \( T \) such that \( {\int }_{0}^{1}f\left( t\right) \mathrm{d}t = 0 \) . Then \( f = 0 \) . | Proof. Suppose, to the contrary, that there exists some \( r \in T \) such that \( f\left( r\right) > 0 \) . When \( r < 1 \), using the right continuity of \( f \) at \( r \) we can find some \( \alpha ,\delta > 0 \) such that \( r + \delta < 1 \) and \( f\left( s\right) \geq \alpha \) for \( s \in \left\lbrack {r, r ... | Yes |
Lemma 2.118. Let \( F \in {R}_{n}\left( {T,{E}^{ * }}\right) \) be such that for all \( x \in {X}_{0} \mathrel{\text{:=}} \{ x \in X : x\left( 0\right) = 0 = \) \( x\left( 1\right) \} \) one has \( {\int }_{0}^{1}F\left( t\right) \cdot {x}^{\prime }\left( t\right) \mathrm{d}t = 0 \) . Then \( F\left( \cdot \right) \) i... | Proof. Since \( {\int }_{0}^{1}{e}^{ * } \cdot {x}^{\prime }\left( t\right) \mathrm{d}t = 0 \) for all \( x \in {X}_{0} \), subtracting from \( F \) its means \( {e}^{ * } \), we are reduced to showing that \( F\left( \cdot \right) = 0 \) when \( {\int }_{0}^{1}F\left( t\right) \cdot {x}^{\prime }\left( t\right) \mathr... | Yes |
Lemma 2.119 (Dubois-Reymond lemma). Let \( A, B \in {R}_{n}\left( {T,{E}^{ * }}\right) \) be such that\n\n\[ \forall x \in {X}_{0},\;{\int }_{0}^{1}\left\lbrack {A\left( t\right) x\left( t\right) + B\left( t\right) {x}^{\prime }\left( t\right) }\right\rbrack \mathrm{d}t = 0. \]\n\nThen \( B \) is a primitive of \( A \)... | Proof. Let us set \( C\left( t\right) \mathrel{\text{:=}} B\left( 0\right) + {\int }_{0}^{t}A\left( s\right) \mathrm{d}s \) . Then for each \( x \in {X}_{0} \) the function \( t \mapsto \) \( C\left( t\right) x\left( t\right) \) has a right derivative \( t \mapsto A\left( t\right) x\left( t\right) + C\left( t\right) {x... | Yes |
Corollary 2.120. Suppose the Lagrangian \( L \) is independent of \( e : L\left( {e, v, t}\right) = {\widehat{L}}_{t}\left( v\right) \) . Then for every extremal \( \bar{x}\left( \cdot \right) \), the function \( t \mapsto D{\widehat{L}}_{t}\left( {{\bar{x}}^{\prime }\left( t\right) }\right) \) is a constant. | Proof. Since \( {D}_{1}L = 0 \) ,(2.39) is reduced to \( \frac{\mathrm{d}}{\mathrm{d}t}{D}_{2}L\left( {t,\bar{x}\left( t\right) ,{\bar{x}}^{\prime }\left( t\right) }\right) = 0 \), and hence \( \widehat{L}\left( {\cdot ,{\bar{x}}^{\prime }\left( \cdot \right) }\right) \) is constant. | Yes |
Theorem 2.121 (Hamilton). Suppose that for all \( \left( {e, t}\right) \in T \times E \), the map \( {D}_{2}L\left( {e,\cdot, t}\right) \) is a diffeomorphism from \( {U}_{e, t} \) onto its image \( {V}_{e, t} \) . Let \( \bar{x} \) be an extremal and let \( \bar{y}\left( t\right) \mathrel{\text{:=}} {D}_{2}L\left( {\b... | Proof. Plugging \( e = \bar{x}\left( t\right), v = {\bar{x}}^{\prime }\left( t\right), p \mathrel{\text{:=}} \bar{y}\left( t\right) \) into the relation \( v = {D}_{2}H\left( {e, p, t}\right) \), we get the first equation. By the Euler-Lagrange equation (2.39) and relation (2.40), we have\n\n\[ \n{\bar{y}}^{\prime }\le... | Yes |
Proposition 3.1. Every local minimizer of a convex function \( f : X \rightarrow {\mathbb{R}}_{\infty } \mathrel{\text{:=}} \) \( \mathbb{R} \cup \{ + \infty \} \) on a normed space (or topological vector space) \( X \) is a global minimizer. | Proof. Let \( \bar{x} \in X \) and let \( V \) be a neighborhood of \( \bar{x} \) such that \( f\left( \bar{x}\right) \leq f\left( v\right) \) for all \( v \in V \) . Given \( x \in X \), one can find \( t \in \left( {0,1}\right) \) such that \( v \mathrel{\text{:=}} \bar{x} + t\left( {x - \bar{x}}\right) \in V \) . Th... | Yes |
Proposition 3.2. Let \( f : X \rightarrow {\mathbb{R}}_{\infty } \) be a convex function on a normed space (or topological vector space) \( X \) . If \( f \) is finite at some \( \bar{x} \in X \), the following assertions are equivalent:\n\n(a) \( f \) is bounded above on some neighborhood \( V \) of \( \bar{x} \) ;\n\... | Proof. The implications \( \left( c\right) \Rightarrow \left( b\right) \Rightarrow \left( a\right) \) are obvious. Let us prove \( \left( a\right) \Rightarrow \left( b\right) \) and \( \left( b\right) \Rightarrow \left( c\right) \) . We may suppose that \( \bar{x} = 0, f\left( \bar{x}\right) = 0 \) by performing a tran... | Yes |
Proposition 3.3. Suppose \( f : X \rightarrow {\mathbb{R}}_{\infty } \) is a convex function on a finite-dimensional space \( X \) . Then \( f \) is continuous on the interior of its domain \( {D}_{f} \mathrel{\text{:=}} \operatorname{dom}f \mathrel{\text{:=}} {f}^{-1}\left( \mathbb{R}\right) \) . | Proof. Given \( \bar{x} \in \operatorname{int}{D}_{f} \), let \( {x}_{1},\ldots ,{x}_{n} \in {D}_{f} \) be such that \( \bar{x} \) belongs to the interior of the convex hull \( C \) of \( \left\{ {{x}_{1},\ldots ,{x}_{n}}\right\} \) (for instance, one can take for \( C \) a ball with center \( \bar{x} \) for some polyh... | Yes |
Proposition 3.4. Let \( f : X \rightarrow {\mathbb{R}}_{\infty } \) be a lower semicontinuous convex function on a Banach space \( X \) . Then \( f \) is continuous on the core of its domain \( {D}_{f} \) (which coincides with the interior of \( {D}_{f} \) ). | Proof. Given \( \bar{x} \in \operatorname{core}{D}_{f} \), let \( m > f\left( \bar{x}\right) \) and let \( C \mathrel{\text{:=}} \{ x \in X : f\left( x\right) \leq m\} \) . Again we may suppose \( \bar{x} = 0 \) . Then \( C \) is a closed convex subset of \( X \) that is absorbing: for all \( x \in X \) we can find \( ... | Yes |
Proposition 3.5. Let \( f : X \rightarrow {\mathbb{R}}_{\infty } \) be a convex function on a normed space \( X \) . If \( f \) is continuous at some \( \bar{x} \in {D}_{f} \mathrel{\text{:=}} \operatorname{dom}f \) then \( f \) is continuous on the interior of \( {D}_{f} \) . | Proof. Given \( {x}_{0} \in \operatorname{int}{D}_{f} \), let us prove that \( f \) is continuous at \( {x}_{0} \) . Using a translation, we may suppose \( {x}_{0} = 0 \) . Then since \( {D}_{f} \) is a neighborhood of 0, there exists some \( r > 0 \) such that \( \bar{y} \mathrel{\text{:=}} - r\bar{x} \in {D}_{f} \) .... | Yes |
Theorem 3.6. If \( f \) is a convex function that is lower semicontinuous on a normed space \( X \), then \( f \) is lower semicontinuous on \( X \) endowed with the weak topology. | Proof. This is an immediate consequence of Mazur's theorem: for every real number \( r \) the sublevel set \( \left\lbrack {f \leq r}\right\rbrack \mathrel{\text{:=}} \{ x \in X : f\left( x\right) \leq r\} \) of \( f \) is closed and convex, hence weakly closed. | Yes |
Corollary 3.7. A coercive lower semicontinuous convex function \( f \) on a reflexive Banach space \( X \) attains its infimum. | Proof. The result is obvious if \( f \) takes only the value \( + \infty \) (i.e., \( f = + {\infty }^{X} \) ). For \( f \neq \) \( + {\infty }^{\widetilde{X}} \) we pick \( {x}_{0} \in \operatorname{dom}f \) and use coercivity to get some \( r > 0 \) such that \( f\left( x\right) > f\left( {x}_{0}\right) \) for \( x \... | Yes |
Proposition 3.8. Let \( f \) be a convex function on a convex subset \( C \) of a normed space \( X \) and let \( \alpha ,\beta \in \mathbb{R},\rho > 0 \) . Suppose \( f \) is bounded below by \( \beta \) on a subset \( B \) of \( C \) and is bounded above by \( \alpha \) on a subset \( A \) of \( C \) such that \( B +... | Proof. Given \( x, y \in B \) and \( \delta > \parallel x - y\parallel \), let \( z \mathrel{\text{:=}} y + \rho {\delta }^{-1}\left( {y - x}\right) \in A \), since \( B + \rho {U}_{X} \subset A \) . Then \( y = x + t\left( {z - x}\right) \), where \( t \mathrel{\text{:=}} \delta {\left( \delta + \rho \right) }^{-1} \i... | Yes |
Corollary 3.9. Suppose the convex function \( f \) on the normed space \( X \) is bounded above by \( \alpha \) on some ball \( B\left( {\bar{x}, r}\right) \) . Then for every \( s \in \left( {0, r}\right) \) the function \( f \) is Lipschitzian on the ball \( B\left( {\bar{x}, s}\right) \) with rate \( 2{\left( r - s\... | Proof. Taking \( A \mathrel{\text{:=}} B\left( {\bar{x}, r}\right), B \mathrel{\text{:=}} B\left( {\bar{x}, s}\right) ,\rho \mathrel{\text{:=}} r - s,\beta \mathrel{\text{:=}} {2f}\left( \bar{x}\right) - \alpha \), it suffices to observe that for all \( x \in B \) one has \( f\left( x\right) \geq \beta \) by convexity. | No |
Lemma 3.12. For every lower semicontinuous convex function \( f : X \rightarrow {\mathbb{R}}_{\infty } \) there exists a continuous affine function \( g \) such that \( g \leq f \) . Moreover, if \( w \in \operatorname{dom}f \) and \( r < f\left( w\right) \), we may require that \( g\left( w\right) > r \) . | Proof. The case \( f = + {\infty }^{X} \) is obvious. Let us suppose \( f \neq + {\infty }^{X} \), so that the epigraph \( {E}_{f} \) of \( f \) is nonempty. Let \( w \in \operatorname{dom}f \) and \( r < f\left( w\right) \) . The Hahn-Banach theorem allows us to separate the compact set \( \{ \left( {w, r}\right) \} \... | Yes |
Theorem 3.13. Let \( W, X \) be Banach spaces, and let \( F : W \rightrightarrows X \) be a multimap with closed convex graph. If \( W \) is reflexive, then for every \( \left( {\bar{w},\bar{x}}\right) \) in (the graph of) \( F \) such that \( X = {\mathbb{R}}_{ + }\left( {F\left( W\right) - \bar{x}}\right) \), i.e., \... | Proof. Let us define a function \( f : X \rightarrow {\mathbb{R}}_{\infty } \) by\n\n\[ f\left( x\right) \mathrel{\text{:=}} d\left( {\bar{w},{F}^{-1}\left( x\right) }\right) \mathrel{\text{:=}} \inf \{ \parallel w - \bar{w}\parallel : w \in W, x \in F\left( w\right) \} ,\]\n\nwith the convention that \( \inf \varnothi... | Yes |
Corollary 3.14. Let \( F : W \rightrightarrows X \) be a multimap with convex graph between two normed spaces. Suppose that for some \( \rho, r > 0 \) and some \( \left( {\bar{w},\bar{x}}\right) \in F \) one has \( B\left( {\bar{x}, r}\right) \subset F\left( {B\left( {\bar{w},\rho }\right) }\right) \) . Then for every ... | Proof. Given \( s \in (0, r/3\rbrack \), let \( c \mathrel{\text{:=}} 4{\left( r - s\right) }^{-1}\rho \) and let \( \left( {w, z}\right) \in B\left( {\bar{w},\rho }\right) \times B\left( {\bar{x}, s}\right) \) with \( z \in F\left( w\right) \) . Then for every \( x \in B\left( {\bar{x}, s}\right), y \in F\left( w\righ... | Yes |
Lemma 3.15. If \( f : I \rightarrow \mathbb{R} \) is a finite convex function on some interval \( I \) of \( \mathbb{R} \), then for every \( s \in I \smallsetminus \{ \sup I\} \) the right derivative \( {D}_{r}f\left( s\right) \mathrel{\text{:=}} {f}_{ + }^{\prime }\left( s\right) \) of \( f \) at \( s \) exists in \(... | Proof. The first assertion is a direct consequence of the existence of a limit for the nondecreasing function \( t \mapsto {\left( t - s\right) }^{-1}\left( {f\left( t\right) - f\left( s\right) }\right) \) on \( \left( {s,\sup I}\right) \) . The second assertion stems from the fact that when \( s \in \operatorname{int}... | Yes |
Let \( f : I \rightarrow \mathbb{R} \) be a differentiable function on an open interval of \( \mathbb{R} \). Then \( f \) is convex if and only if its derivative is nondecreasing. If \( f \) is twice differentiable, then \( f \) is convex if and only if for all \( r \in I \) one has \( {f}^{\prime \prime }\left( r\righ... | The necessary condition is a consequence of Lemma 3.15. Let us prove the sufficient condition. Let \( f \) be differentiable with nondecreasing derivative. Given \( r, t \in I \) and \( s \in \left( {r, t}\right) \), we have \( s = {ar} + {bt} \) with \( a = {\left( t - r\right) }^{-1}\left( {t - s}\right) \geq 0 \), \... | Yes |
Proposition 3.18. If \( f : X \rightarrow {\mathbb{R}}_{\infty } \mathrel{\text{:=}} \mathbb{R} \cup \{ + \infty \} \) is a convex function on a vector space \( X \), then for all \( x \in \operatorname{dom}f \) and for all \( v \in X \) the radial derivative\n\n\[ \n{d}_{r}f\left( {x, v}\right) \mathrel{\text{:=}} \ma... | Proof. Let \( g \) be given by \( g\left( t\right) = f\left( {x + {tv}}\right) \) . Then \( g \) is convex and its right derivative at 0 is \( {d}_{r}f\left( {x, v}\right) \) . It exists in \( \lbrack - \infty , + \infty ) \) if \( \left( {x + \left( {0,\infty }\right) v}\right) \cap \operatorname{dom}f \) is nonempty,... | Yes |
Proposition 3.19. If \( f : X \rightarrow {\mathbb{R}}_{\infty } \) is a convex function on a vector space \( X \), then for all \( x \in \operatorname{dom}f \), the radial derivative \( {d}_{r}f\left( {x, \cdot }\right) \) is a sublinear function. | Proof. Clearly \( {d}_{r}f\left( {x, \cdot }\right) \) is positively homogeneous. Let us prove that it is subadditive: for every \( v, w \in X \) we have \( f\left( {x + \frac{1}{2}t\left( {v + w}\right) }\right) \leq \frac{1}{2}f\left( {x + {tv}}\right) + \frac{1}{2}f\left( {x + {tw}}\right) \) ; hence\n\n\[ {d}_{r}f\... | Yes |
Proposition 3.21. A function \( f \) on a normed space \( X \) attains its minimum at \( x \in \) \( \operatorname{dom}f \) if and only if \( 0 \in \partial f\left( x\right) \) . | The result is an immediate consequence of the definition. Calculus rules will make it efficient. In particular, they enable us to give optimality conditions for problems with constraints. | No |
Theorem 3.22. If \( f \) is a convex function on a normed space \( X \) and \( x \in \operatorname{dom}f \), then\n\n\[ \n{x}^{ * } \in \partial f\left( x\right) \Leftrightarrow \forall v \in X\left\langle {{x}^{ * }, v}\right\rangle \leq {df}\left( {x, v}\right) \n\]\n\n\[ \n\Leftrightarrow \forall v \in X\left\langle... | Proof. Given \( {x}^{ * } \in \partial f\left( x\right) \), for every \( t > 0, u \in X \) we have\n\n\[ \n\left\langle {{x}^{ * },{tu}}\right\rangle \leq f\left( {x + {tu}}\right) - f\left( x\right) \n\]\n\nDividing by \( t \) and taking the liminf as \( \left( {t, u}\right) \rightarrow \left( {{0}_{ + }, v}\right) \)... | Yes |
Proposition 3.23. For a convex function \( f \) on a normed space \( X \) and \( x \in \operatorname{dom}f \), one has the following equivalence in which \( {E}_{f} \) is the epigraph of \( f \) and \( {x}_{f} \mathrel{\text{:=}} \left( {x, f\left( x\right) }\right) \) :\n\n\[ \n{x}^{ * } \in \partial f\left( x\right) ... | The proof is immediate from the definition of \( \partial f\left( x\right) \) :\n\n\[ \n\left( {{x}^{ * }, - 1}\right) \in N\left( {{E}_{f},{x}_{f}}\right) \Leftrightarrow \forall \left( {w, r}\right) \in {E}_{f}\left| {\;\left\langle {{x}^{ * }, w - x}\right\rangle - \left( {r - f\left( x\right) }\right) \leq 0 \Leftr... | Yes |
For a convex subset \( C \) of a normed space \( X \), the normal cone to \( C \) at \( x \in C \) is the subdifferential of the indicator function \( {\iota }_{C} \) to \( C \) at \( x \) . It is also the cone \( {\mathbb{R}}_{ + }\partial {d}_{C}\left( x\right) \) generated by the subdifferential of the distance func... | Proof. By definition, \( {x}^{ * } \in N\left( {C, x}\right) \) iff \( \left\langle {{x}^{ * }, w - x}\right\rangle \leq 0 \) for all \( x \in C \) . Since \( {\iota }_{C}\left( w\right) = 0 \) for \( w \in C \) and \( {\iota }_{C}\left( w\right) = \infty \) for \( w \in X \smallsetminus C \), this property is equivale... | Yes |
Theorem 3.25 (Moreau). If a convex function \( f \) on a normed space \( X \) is finite and continuous at \( x \), then \( \partial f\left( x\right) \) is nonempty and weak* compact. Moreover, for all \( u \in X \)\n\n\[{f}^{\prime }\left( {x, u}\right) = \max \left\{ {\left\langle {{x}^{ * }, u}\right\rangle : {x}^{ *... | Proof. For every \( r > f\left( x\right) \) there exists a neighborhood \( V \) of \( x \) such that \( V \times \left( {r,\infty }\right) \) is contained in the epigraph \( {E}_{f} \) of \( f \) . Thus the interior of \( {E}_{f} \) is convex and nonempty. It does not contain \( {x}_{f} \mathrel{\text{:=}} \left( {x, f... | Yes |
Corollary 3.26. Let \( f : X \rightarrow {\mathbb{R}}_{\infty } \) be a convex function finite and continuous at \( x \in X \) . If \( \partial f\left( x\right) \) is a singleton \( \left\{ {x}^{ * }\right\} \), then \( f \) is Gâteaux and Hadamard differentiable at \( x \) and \( {Df}\left( x\right) = {x}^{ * }. | Proof. The preceding theorem ensures that \( {f}^{\prime }\left( {x, \cdot }\right) = {x}^{ * } \) . Thus \( f \) is Gâteaux differentiable. Since \( f \) is Lipschitzian around \( x \), it is Hadamard differentiable. | Yes |
Corollary 3.27. Let \( f \) be a convex function on a normed space \( X \) . Suppose the restriction of \( f \) to the affine subspace \( A \) generated by \( \operatorname{dom}f \) is continuous at \( x \in \) \( \operatorname{dom}f \) . Then \( \partial f\left( x\right) \) is nonempty. | Proof. Without loss of generality, we may suppose \( x = 0 \), so that \( A \) is the vector subspace generated by \( \operatorname{dom}f \) . The preceding theorem ensures that the restriction \( f \mid A \) of \( f \) to \( A \) is subdifferentiable at 0 . Then every continuous linear extension of every element of \(... | Yes |
Corollary 3.28. Let \( f \) be a convex function on a finite-dimensional normed space \( X \) and let \( x \in \operatorname{ri}\operatorname{dom}f \) (i.e., be such that \( {\mathbb{R}}_{ + }\left( {\operatorname{dom}f - x}\right) \) is a linear subspace). Then \( \partial f\left( x\right) \) is nonempty. | Proof. Recall that for a subset \( D \) of \( X,\operatorname{ri}D \) is the set of points that belong to the interior of \( D \) in the affine subspace \( Y \) generated by \( D \) . Taking \( D = \operatorname{dom}f \), we have that the restriction \( g \) of \( f \) to \( Y \) is continuous at \( x \) . The precedin... | No |
Proposition 3.29. A convex function \( f : X \rightarrow {\mathbb{R}}_{\infty } \) finite at some \( \bar{x} \in X \) is subdiffer-entiable at \( \bar{x} \) iff it is globally calm at \( \bar{x} \), if and only if it is calm at \( \bar{x} \). Moreover, the calmness rate of \( f \) at \( \bar{x} \) is equal to the remot... | Proof. If \( \partial f\left( \bar{x}\right) \) is nonempty, for every element \( {\bar{x}}^{ * } \in \partial f\left( \bar{x}\right) \) one can take \( c = \begin{Vmatrix}{\bar{x}}^{ * }\end{Vmatrix} \) to get global calmness, so that \( {\gamma }_{f}\left( \bar{x}\right) \leq \rho \left( {\partial f\left( \bar{x}\rig... | Yes |
Proposition 3.30. A convex function \( f : X \rightarrow {\mathbb{R}}_{\infty } \) finite and continuous at some point \( x \in X \) is Fréchet (resp. Hadamard) differentiable at \( x \) if and only if \( {r}_{x} \) is a remainder (resp. if for all \( u \in {S}_{X} \) one has \( {\sigma }_{x}\left( {t, u}\right) \right... | Proof. Necessity is obtained by addition directly from the definitions. Let us prove sufficiency in the Fréchet case. Since \( f \) is finite and continuous at \( x,\partial f\left( x\right) \) is nonempty. Let \( {x}^{ * } \in \partial f\left( x\right) \) . Then the definition of \( \partial f\left( x\right) \) and (3... | Yes |
Proposition 3.31. If \( f : W \rightarrow \mathbb{R} \) is continuous and convex on an open convex subset \( W \) of a normed space \( X \) and if \( f \) is Gâteaux differentiable at \( x \in W \), then \( f \) is Hadamard differentiable at \( x \) and \( {df} \) is continuous at \( \left( {x, v}\right) \) for all \( ... | Proof. For every \( r > {df}\left( {x, v}\right) \) one can find \( s > 0 \) such that \( r > {s}^{-1}\lbrack f\left( {x + {sv}}\right) - \) \( f\left( x\right) \rbrack \) . Thus for \( \left( {{x}^{\prime },{v}^{\prime }}\right) \) close enough to \( \left( {x, v}\right) \) one has \( r > {s}^{-1}\left\lbrack {f\left(... | Yes |
Proposition 3.32. Let \( f : W \rightarrow \mathbb{R} \) be a convex function on some open convex subset \( W \) of a normed space \( X \) . If \( f \) is Fréchet differentiable at some \( x \in W \) and Gâteaux differentiable on \( W \), then its derivative is continuous at \( x \) . | Proof. It suffices to prove the second assertion. The differentiability of \( f \) at \( \bar{x} \) entails continuity of \( f \) on \( W \), hence that \( \partial f\left( w\right) \neq \varnothing \) for all \( w \in W \) . Let \( {x}^{ * } \mathrel{\text{:=}} {Df}\left( x\right) \) . Given \( \varepsilon \in \left( ... | Yes |
Theorem 3.34. (a) Let \( f : W \rightarrow \mathbb{R} \) be a continuous convex function on some open convex subset \( W \) of a normed space \( X \) . Then the set \( F \) of points in \( W \) of Fréchet differentiability of \( f \) is a (possibly empty) \( {\mathcal{G}}_{\delta } \) subset of \( W \) . | Proof. (a) For \( u \in {S}_{X} \), let \( {\sigma }_{x}\left( {\cdot, u}\right) \) be the function of relation (3.5), and let\n\n\[ \n{G}_{n} \mathrel{\text{:=}} \left\{ {x \in W : \exists t > 0 : \mathop{\sup }\limits_{{u \in {S}_{X}}}{\sigma }_{x}\left( {t, u}\right) < \frac{1}{n}}\right\} .\n\]\n\nSince for all \( ... | Yes |
Lemma 3.36. Let \( {\left( {f}_{i}\right) }_{i \in I} \) be a finite family of functions and let \( \bar{x} \in \mathop{\bigcap }\limits_{{i \in I}}\operatorname{dom}{f}_{i} \) . If \( f \mathrel{\text{:=}} \mathop{\inf }\limits_{{i \in I}}{f}_{i} \) and if \( {f}_{i}\left( \bar{x}\right) = f\left( \bar{x}\right) \) fo... | Proof. The inclusion \( \partial f\left( \bar{x}\right) \subset \mathop{\bigcap }\limits_{{i \in I}}\partial {f}_{i}\left( \bar{x}\right) \) stems from the preceding lemma. For the opposite inclusion, we note that for all \( {\bar{x}}^{ * } \in \mathop{\bigcap }\limits_{{i \in I}}\partial {f}_{i}\left( \bar{x}\right) \... | Yes |
Proposition 3.37. Let \( f : W \times X \rightarrow {\mathbb{R}}_{\infty } \), where \( W \) and \( X \) are normed spaces. Let \( p \) be the performance function given by \( p\left( w\right) \mathrel{\text{:=}} \inf \{ f\left( {w, x}\right) : x \in X\} \) and let \( S : W \rightrightarrows X \) be the solution multim... | Proof. For all \( \bar{x} \in S\left( \bar{w}\right) ,\left( {w, x}\right) \in W \times X \), one has \( f\left( {\bar{w},\bar{x}}\right) = p\left( \bar{w}\right), f\left( {w, x}\right) \geq p\left( w\right) \) , whence\n\n\[{\bar{w}}^{ * } \in \partial p\left( \bar{w}\right) \Leftrightarrow \forall w \in W,\;p\left( w... | Yes |
Theorem 3.39. Let \( f \) and \( g \) be convex functions on a normed space \( X \) . If \( f \) and \( g \) are finite at \( \bar{x} \) and if \( f \) is continuous at some point of \( \operatorname{dom}f \cap \operatorname{dom}g \), then\n\n\[ \partial \left( {f + g}\right) \left( \bar{x}\right) = \partial f\left( \b... | Proof. The inclusion \( \partial f\left( \bar{x}\right) + \partial g\left( \bar{x}\right) \subset \partial \left( {f + g}\right) \left( \bar{x}\right) \) is an immediate consequence of the definition of the subdifferential. Let us prove the reverse inclusion under the assumptions of the theorem. Let \( {\bar{x}}^{ * } ... | Yes |
Theorem 3.40 (Chain rule). Let \( X \) and \( Y \) be normed spaces, let \( A : X \rightarrow Y \) be a linear continuous map, and let \( g : Y \rightarrow {\mathbb{R}}_{\infty } \) be finite at \( \bar{y} \mathrel{\text{:=}} A\left( \bar{x}\right) \) and continuous at some point of \( A\left( X\right) \) . Then for \(... | Proof. The inclusion \( \partial g\left( \bar{y}\right) \circ A \subset \partial f\left( \bar{x}\right) \) is immediate, without any assumption on \( g \) . Let us first observe that the reverse inclusion is valid without any assumption in the case \( X \mathrel{\text{:=}} W \times Y,\bar{x} \mathrel{\text{:=}} \left( ... | Yes |
Proposition 3.41 (Valadier). Let \( f \mathrel{\text{:=}} \mathop{\sup }\limits_{{s \in S}}{f}_{s} \), as above. For \( \bar{x} \in \operatorname{dom}f \) and \( u \in X \) such that \( {f}^{\prime }\left( {\bar{x}, u}\right) < + \infty \) one has\n\n\[ \n{f}^{\prime }\left( {\bar{x}, u}\right) \leq \mathop{\limsup }\l... | Proof. Since for all \( V \in \mathcal{N}\left( \bar{x}\right) \) we have \( \bar{x} + {tu} \in V \) for \( t > 0 \) small enough, to prove (3.9) it suffices to show that\n\n\[ \n{f}^{\prime }\left( {\bar{x}, u}\right) \leq \mathop{\inf }\limits_{{t > 0}}\mathop{\inf }\limits_{{\varepsilon > 0}}\mathop{\sup }\limits_{{... | Yes |
Proposition 3.42 (Rockafellar). Let \( S \) be a compact topological space, let \( {\left( {f}_{s}\right) }_{s \in S} \) be a family of convex functions on some convex open subset \( U \) of a normed space \( X \) , let \( f \mathrel{\text{:=}} \mathop{\sup }\limits_{{s \in S}}{f}_{s} \), and let \( \bar{x} \in \operat... | Proof. Assumption (a) ensures that \( S\left( \bar{x}\right) = { \cap }_{\varepsilon > 0}S\left( \varepsilon \right) \) is nonempty and compact. Moreover, given \( u \in X \) and \( s \in S\left( \bar{x}\right) \), since \( {f}_{s}^{\prime }\left( {\bar{x}, u}\right) = \mathop{\inf }\limits_{{t > 0}}\left( {1/t}\right)... | Yes |
Theorem 3.44. For every function \( f : X \rightarrow \overline{\mathbb{R}} \) one has \( {f}^{* * } \leq f \) . If \( f \) is closed proper convex (or if \( f = + {\infty }^{X} \) or \( f = - {\infty }^{X} \), the constant functions with values \( + \infty \) and \( - \infty \) respectively) then \( {f}^{* * } = f \) ... | Proof. Given \( x \in X \), for every function \( f : X \rightarrow \overline{\mathbb{R}} \) and every \( {x}^{ * } \in {X}^{ * } \) we have \( - {f}^{ * }\left( {x}^{ * }\right) \leq f\left( x\right) - \left\langle {{x}^{ * }, x}\right\rangle \) hence \( {f}^{* * }\left( x\right) = \sup \left\{ {\left\langle {{x}^{ * ... | Yes |
Corollary 3.45. For every function \( f : X \rightarrow \overline{\mathbb{R}} \) bounded below by a continuous affine function and with nonempty domain, the greatest closed proper convex function on \( X \) bounded above by \( f \) is \( {f}^{* * } \mid X \) . If \( f \) is not bounded below by a continuous affine func... | Proof. The last assertion is obvious, since \( {f}^{ * } = + {\infty }^{{X}^{ * }} \) when \( f \) is not bounded below by a continuous affine function (since \( {f}^{ * }\left( {w}^{ * }\right) < c \) for some \( {w}^{ * } \in {X}^{ * }, c \in \mathbb{R} \) implies that \( f\left( x\right) \geq \left\langle {{w}^{ * }... | Yes |
Corollary 3.46. For every function \( f : X \rightarrow \overline{\mathbb{R}} \) one has \( {f}^{* * * } = {f}^{ * } \) . | Proof. The result is obvious if \( {f}^{ * } = + {\infty }^{{X}^{ * }} \) or if \( {f}^{ * } = - {\infty }^{{X}^{ * }} \) ; otherwise, \( {f}^{ * } \) is closed proper convex. | No |
Theorem 3.47 (Young-Fenchel). For every function \( f : X \rightarrow \overline{\mathbb{R}} \) and for every \( x \in \) \( X,{x}^{ * } \in {X}^{ * } \) one has \( f\left( x\right) + {f}^{ * }\left( {x}^{ * }\right) \geq \left\langle {{x}^{ * }, x}\right\rangle \) . | Proof. The first assertion is a direct consequence of the definition. When \( f\left( x\right) \in \mathbb{R} \) , the equality \( f\left( x\right) + {f}^{ * }\left( {x}^{ * }\right) = \left\langle {{x}^{ * }, x}\right\rangle \) is equivalent to each of the following assertions:\n\n\[ f\left( x\right) + {f}^{ * }\left(... | Yes |
Theorem 3.48. For every function \( f : X \rightarrow \overline{\mathbb{R}} \) one has \( {f}^{* * }\left( x\right) = f\left( x\right) \) whenever \( \partial f\left( x\right) \neq \varnothing \) . | Proof. Given \( {x}^{ * } \in \partial f\left( x\right) \), let \( g : w \mapsto \left\langle {{x}^{ * }, w - x}\right\rangle + f\left( x\right) \) . Then \( g \) is a continuous affine function satisfying \( g \leq f \), so that \( g \leq {f}^{* * } \) and \( g\left( x\right) = f\left( x\right) \geq {f}^{* * }\left( x... | Yes |
Corollary 3.49. When \( f = {f}^{* * } \) the multimap \( \partial {f}^{ * } \) is the inverse of the multimap \( \partial f \) : | \[ {x}^{ * } \in \partial f\left( x\right) \Leftrightarrow x \in \partial {f}^{ * }\left( {x}^{ * }\right) . \] | Yes |
When \( {f}^{* * }\left( 0\right) = f\left( 0\right) \in \mathbb{R} \) the set of minimizers of \( {f}^{ * } \) is \( \partial f\left( 0\right) \) . For every function \( g \) with finite infimum, the set \( \partial {g}^{ * }\left( 0\right) \) is the set of minimizers of \( {g}^{* * } \) . When \( {f}^{* * } = f \) an... | The first assertion follows from the equivalences \( {x}^{ * } \in \partial f\left( 0\right) \Leftrightarrow 0 \in \partial {f}^{ * }\left( {x}^{ * }\right) \) \( \Leftrightarrow {x}^{ * } \) is a minimizer of \( {f}^{ * } \) . The second one ensues because \( {g}^{ * }\left( 0\right) = - \inf g\left( X\right) \) and \... | Yes |
Lemma 3.51. Let \( f : X \rightarrow {\mathbb{R}}_{\infty } \) be proper and let \( r, c \in {\mathbb{R}}_{ + }, a, b \in \mathbb{R} \). (a) If \( f \) is such that \( f \geq a \) on \( r{B}_{X} \) and \( f\left( \cdot \right) \geq c\parallel \cdot \parallel - b \) on \( X \smallsetminus r{B}_{X} \), then for \( y \in ... | Proof. (a) For \( y \in c{B}_{Y} \), setting \( s \mathrel{\text{:=}} \parallel x\parallel \), one has \[ {f}^{ * }\left( y\right) \leq \max \left( {\mathop{\sup }\limits_{{x \in r{B}_{X}}}\left( {\langle y, x\rangle - a}\right) ,\mathop{\sup }\limits_{{x \in X \smallsetminus r{B}_{X}}}\left( {\langle y, x\rangle - c\p... | Yes |
Proposition 3.52. Let \( X \) be a normed space, let \( f : X \rightarrow {\mathbb{R}}_{\infty } \) be closed convex proper; and let \( c \in {\mathbb{R}}_{ + }, a, b \in \mathbb{R} \) . Then the following assertions are equivalent:\n\n(a) \( f \) is supercoercive: \( {\alpha }_{f} \mathrel{\text{:=}} \mathop{\liminf }... | Proof. (a) \( \Rightarrow \) (b) Since \( f \) is bounded below by a continuous affine function, it is bounded below on balls. Given \( c \in \left( {0,{\alpha }_{f}}\right) \), we can find \( r > 0 \) such that \( f\left( \cdot \right) \geq c\parallel \cdot \parallel \) on \( X \smallsetminus r{B}_{X} \) and \( a \in ... | Yes |
Theorem 3.53. Let \( f \) be a closed proper convex function finite and continuous at \( \bar{x} \in X \) . If \( {f}^{ * } \) is strictly convex (resp. uniformly convex), then \( f \) is Hadamard (resp. Fréchet) differentiable at \( \bar{x} \) . | Proof. For \( {\bar{x}}^{ * } \in \partial f\left( \bar{x}\right) \) one has \( \bar{x} \in \partial {f}^{ * }\left( {\bar{x}}^{ * }\right) \) by Theorem 3.48, whence \( 0 \in \partial \left( {{f}^{ * } - }\right. \) \( \bar{x})\left( {\bar{x}}^{ * }\right) \) and \( {\bar{x}}^{ * } \) is a minimizer of \( {f}^{ * } - ... | No |
Proposition 3.54. Suppose the performance function \( p \) is convex and finite at 0 . Then there is no duality gap if and only if \( p \) is lower semicontinuous at 0 . | Proof. Since \( {p}^{* * } \leq p \) and \( {p}^{* * } \) is lower semicontinuous, the equality \( {p}^{* * }\left( 0\right) = p\left( 0\right) \) entails that \( p \) is lower semicontinuous at 0 . Conversely, when \( p \) is convex, finite, and lower semicontinuous at 0, its lower semicontinuous hull \( \bar{p} \) sa... | Yes |
Proposition 3.55. If the Moreau-Rockafellar subdifferential \( \partial p\left( 0\right) \) of \( p \) at 0 is nonempty, then strong duality holds: one has \( \inf \left( \mathcal{P}\right) = \max \left( \mathcal{D}\right) \), and \( \left( \mathcal{D}\right) \) has optimal solutions. More precisely, the set \( {S}^{ *... | Proof. Let \( {\bar{w}}^{ * } \in \partial p\left( 0\right) \) : for all \( w \in W \) one has \( p\left( w\right) \geq p\left( 0\right) + \left\langle {{\bar{w}}^{ * }, w}\right\rangle \) . Thus \( - p\left( 0\right) \geq \) \( {p}^{ * }\left( {\bar{w}}^{ * }\right) \) ; hence \( p\left( 0\right) \leq - {p}^{ * }\left... | Yes |
Corollary 3.56. Suppose \( p \) is convex and \( \inf \left( \mathcal{P}\right) \) is finite. Suppose there exists some \( \bar{x} \in X \) such that \( P\left( {\cdot ,\bar{x}}\right) \) is finite and continuous at 0 . More generally, denoting by \( V \) the vector space generated by \( \operatorname{dom}p \), suppose... | Proof. Under the general assumption, \( p \) is majorized on \( B\left( {0, r}\right) \cap V \), since for \( w \in B\left( {0, r}\right) \cap V \) one has \( p\left( w\right) \leq P\left( {w, x\left( w\right) }\right) \leq m \) . Thus \( p \mid V \) is continuous, and by Corollary 3.27, \( p \) is subdifferentiable at... | Yes |
Corollary 3.57. Let \( W, X \) be Banach spaces and let \( p \) be the performance function associated to a perturbation \( P : W \times X \rightarrow {\mathbb{R}}_{\infty } \) that is convex, lower semicontinuous, and such that \[ Z \mathrel{\text{:=}} \mathop{\bigcup }\limits_{{x \in X}}{\mathbb{R}}_{ + }\operatornam... | Proof. By Corollary 3.27, we may suppose \( Z = W \) . The set \[ F \mathrel{\text{:=}} \{ \left( {x, r, w}\right) \in X \times \mathbb{R} \times W : P\left( {w, x}\right) \leq r\} , \] being the image of the epigraph of \( P \) under an isomorphism (an interchange of components), is closed and convex. Relation (3.23) ... | Yes |
Lemma 3.58 (Bourass-Giner [163]). If \( \left( {f, g}\right) \) is convexlike and \( h \) is convex, then \( p \) is convex. In fact, \( p \) is the infimal convolution \( h▱\widetilde{q} \) of \( h \) and \( \widetilde{q} \), where \( \widetilde{q}\left( w\right) \mathrel{\text{:=}} q\left( {-w}\right) \) for \( w \in... | Proof. For all \( w \in W \) we have\n\n\[ \left( {h▱\widetilde{q}}\right) \left( w\right) = \mathop{\inf }\limits_{{v \in W}}\left( {h\left( {w + v}\right) + q\left( v\right) }\right) = \mathop{\inf }\limits_{{w \in W}}\inf \{ h\left( {w + g\left( x\right) }\right) + f\left( x\right) : x \in D, g\left( x\right) = v\} ... | Yes |
Theorem 3.60 (Attouch-Brézis). Let \( X, Y \) be Banach spaces, let \( A : X \rightarrow Y \) be a continuous linear map, and let \( f : X \rightarrow {\mathbb{R}}_{\infty }, g : Y \rightarrow {\mathbb{R}}_{\infty } \) be closed proper convex functions such that the following cone is closed and symmetric (i.e., \( Z = ... | Proof. Taking \( W \mathrel{\text{:=}} Y \), we define the perturbation function \( P \) as in the preceding proof. Then for \( x \in X \), we have \( w \in \operatorname{dom}P\left( {\cdot, x}\right) \) if and only if \( x \in \operatorname{dom}f \) and \( w \in \operatorname{dom}g - {Ax} \), so that the cone generate... | Yes |
Proposition 3.61. Let \( f \) be a convex function on a normed space \( X \), let \( x \in \operatorname{dom}f \) , \( {\left( {x}_{i}\right) }_{i \in I}{ \rightarrow }_{f}x \), and let \( {x}_{i}^{ * } \in \partial f\left( {x}_{i}\right) \) be such that \( {\left( {x}_{i}^{ * }\right) }_{i \in I}\overset{ * }{ \righta... | Proof. It suffices to observe that for all \( w \in X \) one has\n\n\[ \left\langle {{x}^{ * }, w - x}\right\rangle = \mathop{\lim }\limits_{i}\left\langle {{x}_{i}^{ * }, w - x}\right\rangle = \mathop{\lim }\limits_{i}\left\langle {{x}_{i}^{ * }, w - {x}_{i}}\right\rangle \leq \mathop{\lim }\limits_{i}\left( {f\left( ... | Yes |
Theorem 3.63. Let \( X \) and \( Y \) be reflexive Banach spaces, let \( A : X \rightarrow Y \) be linear and continuous, and let \( C \mathrel{\text{:=}} {A}^{-1}\left( D\right) \), where \( D \) is a closed convex subset of \( Y \) . Let \( \bar{x} \in C \) , \( \bar{y} \mathrel{\text{:=}} A\left( \bar{x}\right) \) .... | Proof. Sufficiency: given \( \left( {x}_{n}\right) ,\left( {y}_{n}\right) ,\left( {y}_{n}^{ * }\right) \) as in the statement, for all \( x \in C \) we have\n\n\[ \n\left\langle {{\bar{x}}^{ * }, x - \bar{x}}\right\rangle - \left\langle {{y}_{n}^{ * },{Ax} - A{x}_{n}}\right\rangle = \left\langle {{\bar{x}}^{ * } - {A}^... | Yes |
Theorem 3.64. Let \( X \) and \( Y \) be Banach spaces, \( X \) being reflexive, let \( A \in L\left( {X, Y}\right) \) , and let \( f \mathrel{\text{:=}} g \circ A \), where \( g : Y \rightarrow {\mathbb{R}}_{\infty } \) is lower semicontinuous and convex. Let \( \bar{x} \in \) \( \operatorname{dom}f,{\bar{x}}^{ * } \i... | Proof. Let us first observe that when \( {\bar{x}}^{ * } \) satisfies the above conditions, it belongs to \( \partial f\left( \bar{x}\right) \), since for all \( x \in X \), by (3.32) we have\n\n\[ f\left( x\right) - f\left( \bar{x}\right) = g\left( {A\left( x\right) }\right) - g\left( \bar{y}\right) = \mathop{\lim }\l... | Yes |
Theorem 3.65. Let \( X, Y \) be Banach spaces, \( A : X \rightarrow Y \) a continuous linear map, and let \( f \mathrel{\text{:=}} g \circ A \), where \( g : Y \rightarrow {\mathbb{R}}_{\infty } \) is lower semicontinuous and convex. Let \( \bar{x} \in \operatorname{dom}f,{\bar{x}}^{ * } \in {X}^{ * } \). Then \( {\bar... | Proof. The proof of the sufficient condition is similar to the one given above. The second assertion is a simple application of Theorem 3.64, denoting by \( B \) : \( W \rightarrow X \) the canonical inclusion and observing that \( f \circ B = g \circ \left( {A \circ B}\right) \), that \( {\bar{x}}^{ * }{ \mid }_{W} \i... | No |
Theorem 3.66. Let \( {f}_{1},{f}_{2} \) be lower semicontinuous proper convex functions on a Banach space \( X \) and let \( f \mathrel{\text{:=}} {f}_{1} + {f}_{2} \) be finite at \( \bar{x} \in X \) . Let \( {\bar{x}}^{ * } \in {X}^{ * } \) . Then \( {\bar{x}}^{ * } \in \partial f\left( \bar{x}\right) \) if and only ... | Proof. The sufficient condition is a simple verification. We first note that\n\n\[ \left\langle {{x}_{1, i}^{ * },{x}_{1, i} - \bar{x}}\right\rangle + \left\langle {{x}_{2, i}^{ * },{x}_{2, i} - \bar{x}}\right\rangle = \left\langle {{x}_{1, i}^{ * } + {x}_{2, i}^{ * },{x}_{1, i} - \bar{x}}\right\rangle + \left\langle {... | Yes |
Proposition 3.67. Let \( h \mathrel{\text{:=}} f + g \), where \( f, g \) are convex, lower semicontinuous, finite at \( \bar{x} \in X \) . Suppose there exists some \( \gamma > 0 \) such that \( K \mathrel{\text{:=}} \left\{ {x \in \bar{x} + \gamma {B}_{X} : g\left( x\right) \leq }\right. \) \( g\left( \bar{x}\right) ... | Proof. Changing \( f \) into \( f - {\bar{x}}^{ * } \) and performing a translation, we may suppose \( {\bar{x}}^{ * } = 0 \) and \( \bar{x} = 0 \) . Let \( \mu \) be a modulus of lower semicontinuity of \( \left( {w, z}\right) \mapsto f\left( w\right) + g\left( z\right) \) at \( \left( {0,0}\right) \), i.e., a modulus... | Yes |
For a closed proper convex function \( f \) on a Banach space \( X \), the set of points \( x \in X \) such that \( \partial f\left( x\right) \) is nonempty is dense in \( \operatorname{dom}f \). More precisely, for every \( \bar{x} \in \operatorname{dom}f \) there exists a sequence \( \left( {x}_{n}\right) { \rightarr... | Given \( \bar{x} \in \operatorname{dom}f \), let \( S \mathrel{\text{:=}} \{ \bar{x}\} \) and \( g \mathrel{\text{:=}} {\iota }_{S} \) ; then \( g \) satisfies the compactness assumption of Proposition 3.67. Since \( {f}_{S} \mathrel{\text{:=}} f + {\iota }_{S} \) attains its minimum at \( \bar{x} \), one has \( 0 \in ... | Yes |
Proposition 3.69. Let \( f : X \rightarrow {\mathbb{R}}_{\infty } \) be a convex function on a normed space \( X \) and let \( x \in \operatorname{dom}f \) .\n\n(a) \( {u}^{ * } \in {\partial }_{\infty }f\left( x\right) \) whenever there exist nets \( {\left( {t}_{i}\right) }_{i \in I} \rightarrow {0}_{ + },{\left( {x}... | Proof. (a) Given \( {\left( {t}_{i}\right) }_{i \in I},{\left( {x}_{i}\right) }_{i \in I},{\left( {x}_{i}^{ * }\right) }_{i \in I} \) as in the statement, setting \( {z}_{i} \mathrel{\text{:=}} \left( {{x}_{i}, f\left( {x}_{i}\right) }\right) \) , \( z \mathrel{\text{:=}} {x}_{f} \mathrel{\text{:=}} \left( {x, f\left( ... | Yes |
Theorem 3.75 (Fuzzy mean value theorem). Let \( X \) be a Banach space and let \( f : X \rightarrow {\mathbb{R}}_{\infty } \) be lower semicontinuous, convex, and 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 \( f\left( \bar... | Proof. Let \( {e}^{ * } \in {X}^{ * } \) be such that \( \left\langle {{e}^{ * },\bar{y} - \bar{x}}\right\rangle = f\left( \bar{x}\right) - r \) and let \( g : X \rightarrow {\mathbb{R}}_{\infty } \) be defined by \( g\left( x\right) \mathrel{\text{:=}} f\left( x\right) + \left\langle {{e}^{ * }, x}\right\rangle + {\io... | Yes |
Corollary 3.76 (Usual mean value theorem). Let \( W \) be an open convex subset of a Banach space and let \( f : W \rightarrow \mathbb{R} \) be convex and continuous. Then for every \( x, y \in W \) , there exist \( u \in \lbrack \bar{x},\bar{y}) \) and \( {u}^{ * } \in \partial f\left( u\right) \), such that\n\n\[ \le... | Proof. We extend \( f \) by \( + \infty \) on \( X \smallsetminus V \), where \( V \) is a closed convex neighborhood of \( \left\lbrack {\bar{x},\bar{y}}\right\rbrack \) contained in \( W \), and we pass to the limit in (3.55), using the fact that the multimap \( \partial f \) is locally bounded and closed by Proposit... | No |
Theorem 3.77 (Multidirectional Rolle’s theorem, compact case). Let \( C \) be a weakly compact convex subset of a Banach space \( X \) and let \( \bar{x} \in X \smallsetminus C, D \mathrel{\text{:=}} \left\lbrack {\bar{x}, C}\right\rbrack \) . Suppose \( f : X \rightarrow {\mathbb{R}}_{\infty } \) is convex, lower semi... | Proof. In fact, the result is valid for every \( u \in D \smallsetminus C \) such that \( f\left( u\right) = \inf f\left( D\right) \) , as we shall see. Without loss of generality, using the translation by \( - \bar{x} \), we may assume \( \bar{x} = 0 \) . Since \( D \) is weakly compact and \( f \) is lower semicontin... | Yes |
Theorem 3.78 (Multidirectional mean value theorem, compact case). Let \( C \) be a weakly compact convex subset of a Banach space \( X \) and let \( \bar{x} \in X \smallsetminus C, D \mathrel{\text{:=}} \left\lbrack {\bar{x}, C}\right\rbrack \) . Suppose \( f : X \rightarrow {\mathbb{R}}_{\infty } \) is convex, lower s... | Proof. Let \( {C}^{\prime } \mathrel{\text{:=}} C \times \{ 1\} \subset {X}^{\prime } \mathrel{\text{:=}} X \times \mathbb{R},{\overrightarrow{x}}^{\prime } \mathrel{\text{:=}} \left( {\bar{x},0}\right) \), and let \( {f}^{\prime } : {X}^{\prime } \rightarrow {\mathbb{R}}_{\infty } \) be given by \( {f}^{\prime }\left(... | Yes |
A sufficient condition for \( \bar{x} \in C \) to be a solution to ( \( \mathcal{C} \) ) is\n\n\[ 0 \in \partial f\left( \bar{x}\right) + N\left( {C,\bar{x}}\right) \] | Proof. Suppose \( \bar{x} \in C \) is such that \( 0 \in \partial f\left( \bar{x}\right) + N\left( {C,\bar{x}}\right) \) . Let \( {\bar{x}}^{ * } \in \partial f\left( \bar{x}\right) \) be such that \( - {\bar{x}}^{ * } \in N\left( {C,\bar{x}}\right) \) . Then \( f\left( \bar{x}\right) \) is finite and for all \( x \in ... | Yes |
A necessary and sufficient condition for \( \bar{x} \in C \) to be a solution to \( \left( \mathcal{C}\right) \) is that there exist sequences \( \left( {x}_{n}\right) ,\left( {w}_{n}\right) \rightarrow \bar{x},\left( {w}_{n}^{ * }\right) ,\left( {x}_{n}^{ * }\right) \) such that \( \left( {f\left( {x}_{n}\right) }\rig... | The condition stems from the fuzzy sum rule used in transcribing the inclusion \( 0 \in \partial \left( {f + {\iota }_{C}}\right) \left( \bar{x}\right) \) that characterizes \( \bar{x} \) as a minimizer of \( g + {\iota }_{C} \) . Let us give a direct proof of sufficiency. Given sequences as in the statement, noting th... | Yes |
Lemma 3.81. Let \( g : X \rightarrow {\mathbb{R}}_{\infty } \) be a convex function and let \( C \mathrel{\text{:=}} \{ x \in X : g\left( x\right) \leq 0\} \) , \( \bar{x} \in {g}^{-1}\left( 0\right) \) . Suppose \( {C}^{\prime } \mathrel{\text{:=}} \{ x \in X : g\left( x\right) < 0\} \) is nonempty and \( g \) is cont... | Proof. For all \( {x}^{\prime } \in {C}^{\prime } \) the set \( C \) is a neighborhood of \( {x}^{\prime } \), so that \( N\left( {C,{x}^{\prime }}\right) = \{ 0\} \) . The inclusion \( N\left( {C,\bar{x}}\right) \supset {\mathbb{R}}_{ + }\partial g\left( \bar{x}\right) \) is obvious: given \( r \in {\mathbb{R}}_{ + } ... | Yes |
Lemma 3.82. Let \( {C}_{1},\ldots ,{C}_{k} \) be convex subsets of \( X \) and let \( \bar{x} \in C \mathrel{\text{:=}} {C}_{1} \cap \cdots \cap {C}_{k} \) . Then\n\n\[ N\left( {C,\bar{x}}\right) = N\left( {{C}_{1},\bar{x}}\right) + \cdots + N\left( {{C}_{k},\bar{x}}\right) \]\n\nwhenever one of the following assumptio... | Proof. Assumption (a) ensures that \( \partial \left( {{\iota }_{{C}_{1}} + \cdots + {\iota }_{{C}_{k}}}\right) \left( \bar{x}\right) = \partial {\iota }_{{C}_{1}}\left( \bar{x}\right) + \cdots + \partial {\iota }_{{C}_{k}}\left( \bar{x}\right) \) , since for \( i \neq j \) the function \( {\iota }_{{C}_{i}} \) is fini... | Yes |
Lemma 3.83. Let \( {g}_{i} : X \rightarrow {\mathbb{R}}_{\infty } \) be convex, let \( {C}_{i} \mathrel{\text{:=}} \left\{ {x \in X : {g}_{i}\left( x\right) \leq 0}\right\} \) for \( i \in I \mathrel{\text{:=}} \) \( \{ 1,\ldots, k\} \), let \( \bar{x} \in C \mathrel{\text{:=}} {C}_{1} \cap \cdots \cap {C}_{k} \), and ... | Proof. The sufficient condition is immediate: if \( {\bar{x}}^{ * } = {y}_{1}{\bar{x}}_{1}^{ * } + \cdots + {y}_{k}{\bar{x}}_{k}^{ * } \) with \( {\bar{x}}_{i}^{ * } \in \) \( \partial {g}_{i}\left( \bar{x}\right) \) and \( {y}_{i} \in {\mathbb{R}}_{ + } \) with \( {y}_{i}{g}_{i}\left( \bar{x}\right) = 0 \), for all \(... | Yes |
Theorem 3.84 (Karush-Kuhn-Tucker theorem). Let \( f : X \rightarrow {\mathbb{R}}_{\infty },{g}_{1},\ldots ,{g}_{k} \) be as in the preceding lemma and let \( \bar{x} \in C \) . Suppose \( f \) is continuous at \( \bar{x} \) and the Slater condition holds: there exists some \( {x}_{0} \in \operatorname{dom}f \) such tha... | \[ 0 \in \partial f\left( \bar{x}\right) + {\bar{y}}_{1}\partial {g}_{1}\left( \bar{x}\right) + \cdots + {\bar{y}}_{k}\partial {g}_{k}\left( \bar{x}\right) ,\;{\bar{y}}_{1}{g}_{1}\left( \bar{x}\right) = 0,\ldots ,{\bar{y}}_{k}{g}_{k}\left( \bar{x}\right) = 0. \] | Yes |
Corollary 3.86. Let \( w,{w}^{\prime } \in {\mathbb{R}}^{k} \) and let \( y,{y}^{\prime } \) be multipliers for the problems of minimizing \( f\left( x\right) \) under the constraints \( g\left( x\right) + w \leq 0 \) and \( g\left( x\right) + {w}^{\prime } \leq 0 \) respectively. Then the values \( p\left( w\right) \)... | Proof. Given \( w \in {\mathbb{R}}^{k} \), let \( h : x \mapsto g\left( x\right) + w \) and let \( q : z \mapsto \inf \{ f\left( x\right) : h\left( x\right) + z \leq 0\} \) . If \( y \) is a multiplier for this problem, one has \( y \in \partial q\left( 0\right) \), so that for \( {w}^{\prime } \in {\mathbb{R}}^{k}, z ... | Yes |
Lemma 3.89. For a normed space \( \left( {X,\parallel \cdot \parallel }\right) \) the following assertions are equivalent:\n\n(a) \( \parallel \cdot \parallel \) is rotund;\n\n(b) If \( x, y \in {S}_{X} \) satisfy \( \parallel x + y\parallel = 2 \), then \( x = y \) ;\n\n(c) If \( x, y \in X \) satisfy \( \parallel x +... | Proof. (a) \( \Leftrightarrow \) (b) is a reformulation, since \( \parallel x + y\parallel = 2 \) means that \( \frac{1}{2}\left( {x + y}\right) \in {S}_{X} \) .\n\n(c) \( \Rightarrow \) (b) is immediate. (b) \( \Rightarrow \) (c) For \( x, y \in X \), since\n\n\[ 2\parallel x{\parallel }^{2} + 2\parallel y{\parallel }... | Yes |
Lemma 3.90. For a normed space \( \\left( {X,\\parallel \\cdot \\parallel }\\right) \) the following assertions are equivalent:\n\n(a) \( \\parallel \\cdot \\parallel \) is locally uniformly rotund;\n\n(b) If \( x,{x}_{n} \\in {S}_{X} \) for \( n \\in \\mathbb{N} \) satisfy \( \\left( \\begin{Vmatrix}{x + {x}_{n}}\\end... | Proof. (a) \\Rightarrow (b) is obvious. The converse is obtained by considering (in the nontrivial case \( x \\neq 0 \) ) \( u \\mathrel{\\text{:=}} x/\\parallel x\\parallel ,{u}_{n} \\mathrel{\\text{:=}} {x}_{n}/\\begin{Vmatrix}{x}_{n}\\end{Vmatrix} \) (for \( n \) large enough).\n\n(c) \\Rightarrow (b) is immediate. ... | Yes |
Proposition 3.91. If \( \parallel \cdot \parallel \) is a LUR norm, then \( X \) has the (sequential) Kadec-Klee property: a sequence \( \left( {x}_{n}\right) \) of \( X \) converges to \( x \in X \) whenever it weakly converges to \( x \) and \( \left( \begin{Vmatrix}{x}_{n}\end{Vmatrix}\right) \rightarrow \parallel x... | Proof. Let \( x \in X \) and let \( {\left( {x}_{n}\right) }_{n \in I} \) be a weakly convergent sequence whose limit \( x \) is such that \( \left( \begin{Vmatrix}{x}_{n}\end{Vmatrix}\right) \rightarrow \parallel x\parallel \) . Then \( \mathop{\limsup }\limits_{n}\begin{Vmatrix}{x + {x}_{n}}\end{Vmatrix} \leq \mathop... | Yes |
Proposition 3.92. Let \( \parallel \cdot \parallel \) be a norm on \( X \) and let \( \parallel \cdot {\parallel }_{ * } \) be its dual norm.\n\n(a) If \( \parallel \cdot {\parallel }_{ * } \) is a rotund norm, then \( \parallel \cdot \parallel \) is Hadamard differentiable on \( X \smallsetminus \{ 0\} \) . | Proof. (a) By Corollary 3.26, it suffices to show that for every \( x \in X \smallsetminus \{ 0\} \) ,\n\n\[ S\left( x\right) \mathrel{\text{:=}} \partial \parallel \cdot \parallel \left( x\right) = \left\{ {{x}^{ * } \in {X}^{ * } : {\begin{Vmatrix}{x}^{ * }\end{Vmatrix}}_{ * } = 1,\left\langle {{x}^{ * }, x}\right\ra... | Yes |
Proposition 3.93. Let \( \left( {X,\parallel \cdot \parallel }\right) \) be a normed space. If the dual norm \( \parallel \cdot {\parallel }_{ * } \) is LUR, then \( \parallel \cdot \parallel \) is Fréchet differentiable on \( X \smallsetminus \{ 0\} \) . | Proof. We use Šmulian test (c). Let \( x \in {S}_{X} \) . Using a corollary of the Hahn-Banach theorem, we pick \( f \in {S}_{{X}^{ * }} \) such that \( f\left( x\right) = 1 \) . Let \( \left( {f}_{n}\right) \) be a sequence of \( {S}_{{X}^{ * }} \) such that \( \left( {{f}_{n}\left( x\right) }\right) \rightarrow 1 \) ... | Yes |
Lemma 3.94. An equivalent norm \( \parallel \cdot \parallel \) on the dual \( {X}^{ * } \) of a Banach space \( X \) is the dual norm of an equivalent norm \( \parallel \cdot {\parallel }_{X} \) on \( X \) if and only if it is weak* lower semicontinuous. | Proof. If \( \parallel \cdot \parallel \) is the dual norm of an equivalent norm \( \parallel \cdot {\parallel }_{X} \), then \( \parallel \cdot \parallel = \sup \{ \langle x, \cdot \rangle : x \in \) \( \left. {X,\parallel x{\parallel }_{X} = 1}\right\} \) is weak* lower semicontinuous as a supremum of weak* continuou... | Yes |
Theorem 3.95. (a) Every separable Banach space \( X \) has an equivalent norm that is Hadamard differentiable on \( X \smallsetminus \{ 0\} \) . | Proof. (a) Let \( {\left( {e}_{n}\right) }_{n \in \mathbb{N}} \) be a countable dense subset of \( {B}_{X} \) . Define a norm by\n\n\[ \parallel f\parallel = {\left\lbrack \parallel f{\parallel }_{0}^{2} + \mathop{\sum }\limits_{{n = 0}}^{\infty }{2}^{-n}{f}^{2}\left( {e}_{n}\right) \right\rbrack }^{1/2},\;f \in {X}^{ ... | Yes |
Corollary 3.101. If \( X \) is an Asplund space, then for all \( n \in \mathbb{N} \smallsetminus \{ 0\} ,{X}^{n} \) is an Asplund space. | Proof. Let us prove the result by induction. Assume that \( {X}^{n} \) is Asplund. The graph \( Y \) of the map \( s : \left( {{x}_{1},\ldots ,{x}_{n}}\right) \mapsto {x}_{1} + \cdots + {x}_{n} \) is isomorphic to \( {X}^{n} \), hence is Asplund by assumption. The quotient of \( {X}^{n + 1} \) by \( Y \) is isomorphic ... | Yes |
Proposition 3.103. For every separable Asplund space \( X \) there exists a norm inducing the topology of \( X \) that is Fréchet differentiable on \( X \smallsetminus \{ 0\} \) . | ## Proof. This is a consequence of Proposition 3.99 and Theorem 3.95. | No |
Proposition 3.105. Let \( \left( {X,\parallel \cdot \parallel }\right) \) be a Banach space whose dual space does not have the dual Radon-Nikodým Property. Then there exist \( c > 0 \) and an equivalent norm \( \parallel \cdot {\parallel }^{\prime } \) on \( X \) such that for all \( x \in X \), \[ \mathop{\limsup }\li... | Proof. Since \( {X}^{ * } \) does not have the dual Radon-Nikodým property, there exist \( c > 0 \) and a nonempty bounded subset \( A \) of \( {X}^{ * } \) whose weak* slices have diameter greater than \( {3c} \). In particular, for all \( x \in X \smallsetminus \{ 0\} \), the weak* slice \( S\left( {x, A, c}\right) \... | Yes |
Corollary 3.111. The closed unit ball of the dual \( {X}^{ * } \) of a WCG space is sequentially compact for the weak* topology in the sense that every sequence of \( {B}_{{X}^{ * }} \) has a weak* convergent subsequence. | Proof. Given a bounded sequence \( \left( {x}_{n}^{ * }\right) \) of \( {X}^{ * } \), let \( F\left( n\right) \mathrel{\text{:=}} \left\{ {{x}_{p} : p \geq n}\right\} \) for \( n \in \mathbb{N} \) and let \( {x}^{ * } \) be a weak* cluster point of \( \left( {x}_{n}^{ * }\right) \), i.e., a point in \( {\operatorname{c... | Yes |
Lemma 3.112. Let \( T \) be a compact topological space such that every nonempty closed subset \( S \) of \( T \) has a \( {\mathcal{G}}_{\delta } \) -point \( s \), i.e., a point \( s \in S \) such that \( \{ s\} = { \cap }_{n}{S}_{n} \), where \( {S}_{n} \) is an open subset of \( S \) . Then \( T \) is sequentially ... | Proof. Let \( \left( {t}_{n}\right) \) be a sequence of \( T \) . For \( m \in \mathbb{N} \), let \( {T}_{m} \mathrel{\text{:=}} \operatorname{cl}\left( \left\{ {{t}_{n} : n \geq m}\right\} \right) \) . Then \( S \mathrel{\text{:=}} { \cap }_{m}{T}_{m} \) is the set of cluster points of \( \left( {t}_{n}\right) \), hen... | Yes |
Theorem 3.113 (Hagler, Johnson). Let \( X \) be a Banach space such that every continuous sublinear function on \( X \) has a point of Gâteaux differentiability. Then the closed unit ball \( {B}_{{X}^{ * }} \) of \( {X}^{ * } \) is sequentially compact for the weak* topology. | Proof. In view of the lemma, it suffices to show that every closed nonempty subset \( S \) of the closed unit ball \( T \) of \( {X}^{ * } \) endowed with the weak* topology has a \( {\mathcal{G}}_{\delta } \) -point. Let \( h : X \rightarrow \mathbb{R} \) be the support function of \( S : h\left( x\right) \mathrel{\te... | Yes |
Proposition 4.3. A continuous linear form \( {\bar{x}}^{ * } \) belongs to \( {\partial }_{D}f\left( \bar{x}\right) \) iff it is bounded above by the lower directional (or contingent or Hadamard) (sub)derivate \( {f}^{D}\left( {\bar{x}, \cdot }\right) \) defined by\n\n\[ \n{f}^{D}\left( {\bar{x}, u}\right) \mathrel{\te... | Proof. This follows from (4.6), since \( \mathop{\lim }\limits_{{\left( {t, v}\right) \rightarrow \left( {{0}_{ + }, u}\right) }}{t}^{-1}\left( \left\langle {{\bar{x}}^{ * },{tv}}\right\rangle \right) = \left\langle {{\bar{x}}^{ * }, u}\right\rangle \) . | No |
Proposition 4.4. For \( f : X \rightarrow \overline{\mathbb{R}} \) finite at \( x \in X \), the following assertions are equivalent and are satisfied when \( {\partial }_{D}f\left( x\right) \) is nonempty:\n\n(a) \( f \) is calm at \( x \) ;\n\n(b) There exists some \( c \in {\mathbb{R}}_{ + } \) such that \( {f}^{D}\l... | Proof. The implications (a) \( \Rightarrow \) (b) \( \Rightarrow \) (c) \( \Rightarrow \) (d) are immediate, taking into account the fact that either \( {f}^{D}\left( {x,0}\right) = 0 \) or \( - \infty \) . Let us prove that (d) \( \Rightarrow \) (a). If (a) does not hold, there exists a sequence \( \left( {u}_{n}\righ... | Yes |
Corollary 4.5. The function \( f \) is Hadamard differentiable at \( \bar{x} \) iff both \( {\partial }_{D}f\left( \bar{x}\right) \) and \( {\widetilde{\partial }}_{D}f\left( \bar{x}\right) \mathrel{\text{:=}} - {\partial }_{D}\left( {-f}\right) \left( \bar{x}\right) \), the directional superdifferential of \( f \) at ... | Proof. If \( {\bar{x}}^{ * } \in {\partial }_{D}f\left( \bar{x}\right) \) and \( {\widetilde{x}}^{ * } \in - {\partial }_{D}\left( {-f}\right) \left( \bar{x}\right) \), then for all \( u \in X \), one has\n\n\[ \left\langle {{\bar{x}}^{ * }, u}\right\rangle \leq \mathop{\liminf }\limits_{{\left( {t, v}\right) \rightarr... | Yes |
Proposition 4.6. If \( X \) is finite-dimensional, then \( {\partial }_{F}f\left( \bar{x}\right) = {\partial }_{D}f\left( \bar{x}\right) \) . | Proof. Suppose, to the contrary, that there exists \( {\bar{x}}^{ * } \in {\partial }_{D}f\left( \bar{x}\right) \smallsetminus {\partial }_{F}f\left( \bar{x}\right) \) . By definition of \( {\partial }_{F}f\left( \bar{x}\right) \), there exists \( \varepsilon > 0 \) such that for all \( n \in \mathbb{N} \smallsetminus ... | Yes |
Proposition 4.7. For every normed space \( X \) and every function \( f \) finite at \( \bar{x} \), the firm (resp. directional) subdifferential of \( f \) at \( \bar{x} \) coincides with the set of derivatives at \( \bar{x} \) of functions \( \varphi \) that are Fréchet (resp. Hadamard) differentiable at \( \bar{x} \)... | Proof. Clearly, if \( \varphi \) is Fréchet (resp. Hadamard) differentiable and such that \( \varphi \leq f \) , \( \varphi \left( \bar{x}\right) = f\left( \bar{x}\right) \), one has \( {\varphi }^{\prime }\left( \bar{x}\right) \in {\partial }_{F}f\left( \bar{x}\right) \) (resp. \( \left. {{\varphi }^{\prime }\left( \b... | Yes |
Proposition 4.9. If \( f \) is convex, then \( {\partial }_{F}f\left( \bar{x}\right) \) and \( {\partial }_{D}f\left( \bar{x}\right) \) coincide with the Moreau-Rockafellar subdifferential \( {\partial }_{MR}f\left( \bar{x}\right) \) : | Proof. It is clear that \( {\partial }_{F}f\left( \bar{x}\right) \) and \( {\partial }_{D}f\left( \bar{x}\right) \) contain \( {\partial }_{MR}f\left( \bar{x}\right) \) . Let \( f \) be convex and let \( {\bar{x}}^{ * } \in {\partial }_{D}f\left( \bar{x}\right) \) . Then \( {\bar{x}}^{ * } \) is bounded above by \( {f}... | Yes |
Proposition 4.10. For every function \( f \) finite at \( \bar{x},{\partial }_{F}f\left( \bar{x}\right) \) (resp. \( {\partial }_{D}f\left( \bar{x}\right) \) ) is a closed (resp. weak* closed) convex subset of \( {X}^{ * } \) . | Proof. The weak* closedness and convexity of \( {\partial }_{D}f\left( \bar{x}\right) \) stem from Proposition 4.3. Let \( {\bar{x}}^{ * } \in {X}^{ * } \) be in the closure of \( {\partial }_{F}f\left( \bar{x}\right) \) . For every \( \varepsilon > 0 \) there exists \( {x}^{ * } \in {\partial }_{F}f\left( \bar{x}\righ... | Yes |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.