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Proposition 43.2. If \( A, D \) and both Schur complements \( A - B{D}^{-1}C \) and \( D - C{A}^{-1}B \) are all invertible, then\n\n\[ \n{\left( \begin{array}{ll} A & B \\ C & D \end{array}\right) }^{-1} = \left( \begin{matrix} {\left( A - B{D}^{-1}C\right) }^{-1} & - {A}^{-1}B{\left( D - C{A}^{-1}B\right) }^{-1} \\ -... | If we set \( D = I \) and change \( B \) to \( - B \), we get\n\n\[ \n{\left( A + BC\right) }^{-1} = {A}^{-1} - {A}^{-1}B{\left( I - C{A}^{-1}B\right) }^{-1}C{A}^{-1},\n\]\n\na formula known as the matrix inversion lemma (see Boyd and Vandenberghe [29], Appendix C.4, especially C.4.3). | No |
Proposition 43.3. For any symmetric matrix \( M \) of the form\n\n\[ M = \left( \begin{matrix} A & B \\ {B}^{\top } & C \end{matrix}\right) \]\n\nif \( C \) is invertible, then the following properties hold:\n\n(1) \( M \succ 0 \) iff \( C \succ 0 \) and \( A - B{C}^{-1}{B}^{\top } \succ 0 \) .\n\n(2) If \( C \succ 0 \... | Proof. (1) Observe that\n\n\[ {\left( \begin{matrix} I & B{C}^{-1} \\ 0 & I \end{matrix}\right) }^{-1} = \left( \begin{matrix} I & - B{C}^{-1} \\ 0 & I \end{matrix}\right) \]\n\nand we know that for any symmetric matrix \( T \) and any invertible matrix \( N \), the matrix \( T \) is positive definite \( \left( {T \suc... | Yes |
Theorem 43.5. Given any symmetric matrix\n\n\[ \nM = \left( \begin{matrix} A & B \\ {B}^{\top } & C \end{matrix}\right) \n\]\n\nthe following conditions are equivalent:\n\n(1) \( M \succcurlyeq 0 \) ( \( M \) is positive semidefinite).\n\n(2) \( A \succcurlyeq 0,\;\left( {I - A{A}^{ + }}\right) B = 0,\;C - {B}^{\top }{... | If \( M \succcurlyeq 0 \) as in Theorem 43.5, then it is easy to check that we have the following factorizations (using the fact that \( {A}^{ + }A{A}^{ + } = {A}^{ + } \) and \( {C}^{ + }C{C}^{ + } = {C}^{ + } \) ):\n\n\[ \n\left( \begin{matrix} A & B \\ {B}^{\top } & C \end{matrix}\right) = \left( \begin{matrix} I & ... | Yes |
Proposition 44.2. Every polyhedral cone \( C \) is closed. | Proof. This is proved by showing that\n\n1. Every primitive cone is closed.\n\nAssume that \( \\left( {{a}_{1},\\ldots ,{a}_{m}}\\right) \) are linearly independent vectors in \( {\\mathbb{R}}^{n} \), and consider any sequence \( {\\left( {x}^{\\left( k\\right) }\\right) }_{k \\geq 0} \n\n\\[ \n{x}^{\\left( k\\right) }... | Yes |
Example 45.1.\n\n\\[ \n\\text{maximize}{x}_{1} + {x}_{2} \n\\]\n\nsubject to\n\n\\[ \n{x}_{2} - {x}_{1} \\leq 1 \n\\]\n\n\\[ \n{x}_{1} + 6{x}_{2} \\leq {15} \n\\]\n\n\\[ \n4{x}_{1} - {x}_{2} \\leq {10} \n\\]\n\n\\[ \n{x}_{1} \\geq 0,{x}_{2} \\geq 0 \n\\]\n\nand in matrix form\n\n\\[ \n\\text{maximize}\\left( \\begin{ar... | It turns out that \\( {x}_{1} = 3,{x}_{2} = 2 \\) yields the maximum of the objective function \\( {x}_{1} + {x}_{2} \\) , which is 5 . This is illustrated in Figure 45.1. Observe that the set of points that satisfy the above constraints is a convex region cut out by half planes determined by the lines of equations\n\n... | Yes |
\[ \text{maximize}{x}_{1} + {x}_{2} \] \[ \text{subject to} \] \[ {x}_{2} - {x}_{1} \leq 1 \] \[ {x}_{1} \geq 0,{x}_{2} \geq 0 \] | Otherwise, we will prove shortly that if \( \mu \) is the least upper bound of the set \( \{ {cx} \in \mathbb{R} \mid \) \( x \in \mathcal{P}\left( {A, b}\right) \} \), then there is some \( p \in \mathcal{P}\left( {A, b}\right) \) such that \[ {cp} = \mu \] that is, the objective function \( x \mapsto {cx} \) has a ma... | No |
\[ \mathop{\operatorname{maximize}}\limits_{0}\;\frac{1}{6}{x}_{1} + {x}_{2} \] subject to \[ {x}_{2} - {x}_{1} \leq 1 \] \[ {x}_{1} + 6{x}_{2} \leq {15} \] \[ 4{x}_{1} - {x}_{2} \leq {10} \] \[ {x}_{1} \geq 0,{x}_{2} \geq 0 \] | The proof that if the set \( \{ {cx} \in \mathbb{R} \mid x \in \mathcal{P}\left( {A, b}\right) \} \) is nonempty and bounded above, then there is an optimal solution \( p \in \mathcal{P}\left( {A, b}\right) \), is not as trivial as it might seem. It relies on the fact that a polyhedral cone is closed, a fact that was s... | No |
Proposition 45.1. Let \( \left( {P}_{2}\right) \) be a linear program in standard form, with equality constraint \( {Ax} = b \) . If \( \mathcal{P}\left( {A, b}\right) \) is nonempty and bounded above, and if \( \mu \) is the least upper bound of the set \( \{ {cx} \in \mathbb{R} \mid x \in \mathcal{P}\left( {A, b}\rig... | Proof. Since \( \mu = \sup \{ {cx} \in \mathbb{R} \mid x \in \mathcal{P}\left( {A, b}\right) \} \), there is a sequence \( {\left( {x}^{\left( k\right) }\right) }_{k \geq 0} \) of vectors \( {x}^{\left( k\right) } \in \mathcal{P}\left( {A, b}\right) \) such that \( \mathop{\lim }\limits_{{k \mapsto \infty }}c{x}^{\left... | Yes |
Proposition 45.2. Given any Standard Linear Program \( \left( {P}_{2}\right) \) where \( {Ax} = b \) and \( A \) is an \( m \times n \) matrix of rank \( m \), for any feasible solution \( x \), if \( {J}_{ > } = \left\{ {j \in \{ 1,\ldots, n\} \mid {x}_{j} > 0}\right\} \), then \( x \) is a basic feasible solution iff... | Proof. If \( x \) is a basic feasible solution, then there is some subset \( K \subseteq \{ 1,\ldots, n\} \) of size \( m \) such that the columns of \( {A}_{K} \) are linearly independent and \( {x}_{j} = 0 \) for all \( j \notin K \), so by definition, \( {J}_{ > } \subseteq K \), which implies that the columns of th... | Yes |
Theorem 45.4. Let \( \left( {P}_{2}\right) \) be any standard linear program with objective function \( {cx} \), where \( {Ax} = b \) and \( A \) is an \( m \times n \) matrix of rank \( m \) . If \( \left( {P}_{2}\right) \) has some feasible solution and if it is bounded above, then some basic feasible solution \( \wi... | Proof. By Proposition 45.3, for any feasible solution \( x \) there is some basic feasible solution \( \widetilde{x} \) such that \( {cx} \leq c\widetilde{x} \) . But there are only finitely many basic feasible solutions, so one of them has to yield the maximum of the objective function. | Yes |
Proposition 45.5. Let \( {Ax} = b \) be a linear system where \( A \) is an \( m \times n \) matrix of rank \( m \) . For any subset \( K \subseteq \{ 1,\ldots, n\} \) of size \( m \), if \( {A}_{K} \) is invertible, then there is at most one basic feasible solution \( x \in {\mathbb{R}}^{n} \) with \( {x}_{j} = 0 \) f... | Proof. In order for \( x \) to be feasible we must have \( {Ax} = b \) . Write \( N = \{ 1,\ldots, n\} - K,{x}_{K} \) for the vector consisting of the coordinates of \( x \) with indices in \( K \), and \( {x}_{N} \) for the vector consisting of the coordinates of \( x \) with indices in \( N \) . Then\n\n\[ \n{Ax} = {... | Yes |
Theorem 45.6. Let \( \left( P\right) \) be a linear program in standard form, where \( {Ax} = b \) and \( A \) is an \( m \times n \) matrix of rank \( m \) . For every \( v \in \mathcal{P}\left( {A, b}\right) \), the following conditions are equivalent:\n\n(1) \( v \) is a vertex of the Polyhedron \( \mathcal{P}\left(... | Proof. First, assume that \( v \) is a vertex of \( \mathcal{P}\left( {A, b}\right) \), and let \( \varphi \left( x\right) = {cx} - \mu \) be a linear form such that \( {cy} < \mu \) for all \( y \in \mathcal{P}\left( {A, b}\right) \) and \( {cv} = \mu \) . This means that \( v \) is the unique point of \( \mathcal{P}\... | Yes |
Theorem 45.7. Let \( \left( P\right) \) be a linear program in standard form, where \( {Ax} = b \) and \( A \) is an \( m \times n \) matrix of rank \( m \) . If \( \mathcal{P}\left( {A, b}\right) \) is nonempty (there is a feasible solution), then \( \mathcal{P}\left( {A, b}\right) \) has some vertex; equivalently, \(... | Proof. The proof relies on a trick, which is to add slack variables \( {x}_{n + 1},\ldots ,{x}_{n + m} \) and use the new objective function \( - \left( {{x}_{n + 1} + \cdots + {x}_{n + m}}\right) \) .\n\nIf we let \( \widehat{A} \) be the \( m \times \left( {m + n}\right) \) -matrix, and \( x,\bar{x} \), and \( \wideh... | Yes |
maximize \( {x}_{2} \) subject to \[ - {x}_{1} + {x}_{2} + {x}_{3} = 0 \] \[ {x}_{1} + {x}_{4} = 2 \] \[ {x}_{1} \geq 0,{x}_{2} \geq 0,{x}_{3} \geq 0,{x}_{4} \geq 0. \] | The matrix \( A \) and the vector \( b \) are given by \[ A = \left( \begin{matrix} - 1 & 1 & 1 & 0 \\ 1 & 0 & 0 & 1 \end{matrix}\right) ,\;b = \left( \begin{array}{l} 0 \\ 2 \end{array}\right) \] and if \( x = \left( {0,0,0,2}\right) \), then \( {J}_{ > }\left( x\right) = \{ 4\} \) . There are two ways of forming a se... | No |
Let \( \left( P\right) \) be the following linear program in standard form.\n\n\[ \n\text{maximize} {x}_{1} + {x}_{2} \n\]\n\nsubject to\n\n\[ \n- {x}_{1} + {x}_{2} + {x}_{3} = 1 \n\]\n\n\[ \n{x}_{1} + {x}_{4} = 3 \n\]\n\n\[ \n{x}_{2} + {x}_{5} = 2 \n\]\n\n\[ \n{x}_{1} \geq 0, {x}_{2} \geq 0, {x}_{3} \geq 0, {x}_{4} \g... | The vector \( {u}_{0} = \left( {0,0,1,3,2}\right) \) corresponding to the basis \( K = \{ 3,4,5\} \) is a basic feasible solution, and the corresponding value of the objective function is \( 0 + 0 = 0 \). Since the columns \( \left( {{A}^{3},{A}^{4},{A}^{5}}\right) \) corresponding to \( K = \{ 3,4,5\} \) are linearly ... | Yes |
Example 46.3. Let \( \left( P\right) \) be the following linear program in standard form.\n\nmaximize \( {x}_{1} \)\n\nsubject to\n\n\[ \n{x}_{1} - {x}_{2} + {x}_{3} = 1 \n\]\n\n\[ \n- {x}_{1} + {x}_{2} + {x}_{4} = 2 \n\]\n\n\[ \n{x}_{1} \geq 0,{x}_{2} \geq 0,{x}_{3} \geq 0,{x}_{4} \geq 0. \n\] | The matrix \( A \) and the vector \( b \) are given by\n\n\[ \nA = \left( \begin{matrix} 1 & - 1 & 1 & 0 \\ - 1 & 1 & 0 & 1 \end{matrix}\right) ,\;b = \left( \begin{array}{l} 1 \\ 2 \end{array}\right) .\n\]\n\nThe vector \( {u}_{0} = \left( {0,0,1,2}\right) \) corresponding to the basis \( K = \{ 3,4\} \) is a basic fe... | Yes |
Proposition 46.2. Given any Linear Program (P2) in standard form maximize \( \;{cx} \) subject to \( {Ax} = b \) and \( x \geq 0 \), where \( A \) is an \( m \times n \) matrix of rank \( m \), if \( \left( {u, K}\right) \) is a basic (not feasible) solution of \( \left( {P2}\right) \) and if \( {K}^{ + } = \left( {K -... | Proof. Without any loss of generality and to simplify notation assume that \( K = \left( {1,\ldots, m}\right) \) and write \( j \) for \( {j}^{ + } \) and \( \ell \) for \( {k}_{m} \) . Since \( {\gamma }_{K}^{i} = {A}_{K}^{-1}{A}^{i},{\gamma }_{{K}^{ + }}^{i} = {A}_{{K}^{ + }}^{-1}{A}^{i} \), and \( {A}_{{K}^{ + }} = ... | Yes |
Proposition 47.1. Let \( C \subseteq {\mathbb{R}}^{n} \) be a closed nonempty cone. For any point \( a \in {\mathbb{R}}^{n} \), if \( a \notin C \) , then there is a linear hyperplane \( H \) (through 0) such that\n\n1. \( C \) lies in one of the two half-spaces determined by \( H \) .\n\n2. \( a \notin H \)\n\n3. a li... | Proposition 47.1 is an easy consequence of another separation theorem that asserts that given any two nonempty closed convex sets \( A \) and \( B \) with \( A \) compact, there is a hyperplane \( H \) strictly separating \( A \) and \( B \) (which means that \( A \cap H = \varnothing, B \cap H = \varnothing \), that \... | No |
Proposition 47.2. (Farkas-Minkowski) Let \( C \subseteq {\mathbb{R}}^{n} \) be a nonempty polyhedral cone \( C = \) \( \operatorname{cone}\left( \left\{ {{a}_{1},\ldots ,{a}_{n}}\right\} \right) \) . For any point \( b \in {\mathbb{R}}^{n} \), if \( b \notin C \), then there is a linear hyperplane \( H \) (through 0) s... | A direct proof of the Farkas-Minkowski proposition not involving Proposition 47.1 is given at the end of this section. | Yes |
Proposition 47.3. (Farkas Lemma, Version I) Let \( A \) be an \( m \times n \) matrix and let \( b \in {\mathbb{R}}^{m} \) be any vector. The linear system \( {Ax} = b \) has no solution \( x \geq 0 \) iff there is some nonzero linear form \( y \in {\left( {\mathbb{R}}^{m}\right) }^{ * } \) such that \( {yA} \geq {0}_{... | Proof. First, assume that there is some nonzero linear form \( y \in {\left( {\mathbb{R}}^{m}\right) }^{ * } \) such that \( {yA} \geq 0 \) and \( {yb} < 0 \) . If \( x \geq 0 \) is a solution of \( {Ax} = b \), then we get\n\n\[ \n{yAx} = {yb} \n\]\n\nbut if \( {yA} \geq 0 \) and \( x \geq 0 \), then \( {yAx} \geq 0 \... | Yes |
Proposition 47.4. (Farkas Lemma, Version II) Let \( A \) be an \( m \times n \) matrix and let \( b \in {\mathbb{R}}^{m} \) be any vector. The system of inequalities \( {Ax} \leq b \) has no solution \( x \geq 0 \) iff there is some nonzero linear form \( y \in {\left( {\mathbb{R}}^{m}\right) }^{ * } \) such that \( y ... | Proof. We use the trick of linear programming which consists of adding \ | No |
Proposition 47.5. Let \( X \subseteq {\mathbb{R}}^{n} \) be any nonempty set and let \( a \in {\mathbb{R}}^{n} \) be any point. If \( X \) is closed, then there is some \( z \in X \) such that \( \parallel a - z\parallel = d\left( {a, X}\right) \) . | Proof. Since \( X \) is nonempty, pick any \( {x}_{0} \in X \), and let \( r = \begin{Vmatrix}{a - {x}_{0}}\end{Vmatrix} \) . If \( {B}_{r}\left( a\right) \) is the closed ball \( {B}_{r}\left( a\right) = \left\{ {x \in {\mathbb{R}}^{n} \mid \parallel x - a\parallel \leq r}\right\} \), then clearly\n\n\[ d\left( {a, X}... | Yes |
Consider the linear program illustrated by Figure 47.2\n\n\\[ \n\\text{maximize}\\;2{x}_{1} + 3{x}_{2}\n\\]\n\nsubject to\n\n\\[ \n4{x}_{1} + 8{x}_{2} \\leq {12}\n\\]\n\n\\[ \n2{x}_{1} + {x}_{2} \\leq 3\n\\]\n\n\\[ \n3{x}_{1} + 2{x}_{2} \\leq 4\n\\]\n\n\\[ \n{x}_{1} \\geq 0,{x}_{2} \\geq 0\n\\] | It can be checked that \\( \\left( {{x}_{1},{x}_{2}}\\right) = \\left( {1/2,5/4}\\right) \\) is an optimal solution of the primal linear\n\nprogram, with the maximum value of the objective function \\( 2{x}_{1} + 3{x}_{2} \\) equal to \\( {19}/4 \\), and that \\( \\left( {{y}_{1},{y}_{2},{y}_{3}}\\right) = \\left( {5/{... | Yes |
Theorem 47.8. Consider the Linear Program (P), maximize \( cx \) subject to \( Ax \leq b \) and \( x \geq 0 \), its equivalent version \( (P2) \) in standard form, maximize \( \widehat{c}\widehat{x} \) subject to \( \widehat{A}\widehat{x} = b \) and \( \widehat{x} \geq 0 \), where \( \widehat{A} \) is an \( m \times (n... | Proof. We know that \( K^* \) is a subset of \( \{ 1, \ldots, n + m \} \) consisting of \( m \) indices such that the corresponding columns of \( \widehat{A} \) are linearly independent. Let \( N^* = \{ 1, \ldots, n + m \} - K^* \). The simplex method terminates with an optimal solution in Case (A), namely when \( \wid... | Yes |
Theorem 47.10. (Equilibrium Theorem) For any linear program (P) and its dual linear program (D) (with set of inequalities \( {Ax} \leq b \) where \( A \) is an \( m \times n \) matrix, and objective function \( x \mapsto {cx} \) ), for any feasible solution \( x \) of \( \left( P\right) \) and any feasible solution \( ... | Proof. First, assume that \( \left( { * }_{D}\right) \) and \( \left( { * }_{P}\right) \) hold. The equations in \( \left( { * }_{D}\right) \) say that \( {y}_{i} = 0 \) unless \( \mathop{\sum }\limits_{{j = 1}}^{n}{a}_{ij}{x}_{j} = {b}_{i} \), hence \[ {yb} = \mathop{\sum }\limits_{{i = 1}}^{m}{y}_{i}{b}_{i} = \mathop... | Yes |
Theorem 47.11. Consider the linear program (P2) in standard form\n\nmaximize \( \;{cx} \)\n\n\[ \text{subject to}{Ax} = b\text{and}x \geq 0\text{,} \]\nand its dual \( \left( D\right) \) given by\n\n\[ \text{minimize}{yb} \]\n\n\[ \text{subject to}{yA} \geq c\text{,} \]\n\nwhere \( y \in {\left( {\mathbb{R}}^{m}\right)... | Proof. The proof of Theorem 47.8 applies with \( A \) instead of \( \widehat{A} \) and we can show that\n\n\[ {c}_{{K}^{ * }}{A}_{{K}^{ * }}^{-1}{A}_{{N}^{ * }} \geq {c}_{{N}^{ * }} \]\n\nand that \( {y}^{ * } = {c}_{{K}^{ * }}{A}_{{K}^{ * }}^{-1} \) satisfies, \( c{u}^{ * } = {y}^{ * }b \), and\n\n\[ {y}^{ * }{A}_{{K}... | Yes |
Proposition 47.13. Every \( j \in J \) such that \( {A}^{j} \) is in the basis of the optimal solution \( {\xi }^{ * } \) belongs to the next index set \( {J}^{ + } \) . | Proof. Such an index \( j \in J \) correspond to a variable \( {\xi }_{j} \) such that \( {\xi }_{j} > 0 \), so by complementary slackness, the constraint \( {z}^{ * }{A}^{j} \geq 0 \) of the dual program \( \left( {DRP}\right) \) must be an equality, that is, \( {z}^{ * }{A}^{j} = 0 \) . But then, we have\n\n\[ \n{y}^... | Yes |
Consider the following linear program in standard form:\n\nMaximize \( \; - {x}_{1} - 3{x}_{2} - 3{x}_{3} - {x}_{4} \)\n\nsubject to \( \left( \begin{matrix} 3 & 4 & - 3 & 1 \\ 3 & - 2 & 6 & - 1 \\ 6 & 4 & 0 & 1 \end{matrix}\right) \left( \begin{array}{l} {x}_{1} \\ {x}_{2} \\ {x}_{3} \\ {x}_{4} \end{array}\right) = \l... | The associated dual program \( \left( D\right) \) is\n\n\[ \text{Minimize}\;2{y}_{1} + {y}_{2} + 4{y}_{3} \]\n\n\[ \text{subject to}\;\left( \begin{array}{lll} {y}_{1} & {y}_{2} & {y}_{3} \end{array}\right) \left( \begin{matrix} 3 & 4 & - 3 & 1 \\ 3 & - 2 & 6 & - 1 \\ 6 & 4 & 0 & 1 \end{matrix}\right) \geq \left( \begi... | Yes |
The space \( {l}^{2} \) of all countably infinite sequences \( x = {\left( {x}_{i}\right) }_{i \in \mathbb{N}} \) of complex numbers such that \( \mathop{\sum }\limits_{{i = 0}}^{\infty }{\left| {x}_{i}\right| }^{2} < \infty \) is a Hilbert space. | It will be shown later that the map \( \varphi : {l}^{2} \times {l}^{2} \rightarrow \mathbb{C} \) defined such that\n\n\[ \varphi \left( {{\left( {x}_{i}\right) }_{i \in \mathbb{N}},{\left( {y}_{i}\right) }_{i \in \mathbb{N}}}\right) = \mathop{\sum }\limits_{{i = 0}}^{\infty }{x}_{i}\overline{{y}_{i}} \]\n\nis well def... | No |
Theorem 48.1. Given a Hermitian space \( \left( {E,\langle -, - \rangle }\right) \) (resp. Euclidean space), there is a Hilbert space \( \left( {{E}_{h},\langle -, - {\rangle }_{h}}\right) \) and a linear map \( \varphi : E \rightarrow {E}_{h} \), such that\n\n\[ \langle u, v\rangle = \langle \varphi \left( u\right) ,\... | Proof. Let \( \left( {\widehat{E},\parallel {\parallel }_{\widehat{E}}}\right) \) be the Banach space, and let \( \varphi : E \rightarrow \widehat{E} \) be the linear isometry, given by Theorem 37.63. Let \( \parallel u\parallel = \sqrt{\langle u, u\rangle } \) and \( {E}_{h} = \widehat{E} \) . If \( E \) is a real vec... | Yes |
Proposition 48.3. If \( E \) is a Hermitian space, given any \( d,\delta \in \mathbb{R} \) such that \( 0 \leq \delta < d \), let\n\n\[ B = \{ u \in E \mid \parallel u\parallel < d\} \;\text{ and }\;C = \{ u \in E \mid \parallel u\parallel \leq d + \delta \} .\n\]\n\nFor any convex set such \( A \) that \( A \subseteq ... | Proof. Since \( A \) is convex, \( \frac{1}{2}\left( {u + v}\right) \in A \) if \( u, v \in A \), and thus, \( \begin{Vmatrix}{\frac{1}{2}\left( {u + v}\right) }\end{Vmatrix} \geq d \) . From the parallelogram inequality written in the form\n\n\[ {\begin{Vmatrix}\frac{1}{2}\left( u + v\right) \end{Vmatrix}}^{2} + {\beg... | Yes |
Proposition 48.5. (Projection lemma) Let \( E \) be a Hilbert space.\n\n(1) For any nonempty convex and closed subset \( X \subseteq E \), for any \( u \in E \), there is a unique vector \( {p}_{X}\left( u\right) \in X \) such that\n\n\[ \begin{Vmatrix}{u - {p}_{X}\left( u\right) }\end{Vmatrix} = \mathop{\inf }\limits_... | Proof. (1) Let \( d = \mathop{\inf }\limits_{{v \in X}}\parallel u - v\parallel = d\left( {u, X}\right) \) . We define a sequence \( {X}_{n} \) of subsets of \( X \) as\n\nfollows: for every \( n \geq 1 \) ,\n\n\[ {X}_{n} = \left\{ {v \in X \mid \parallel u - v\parallel \leq d + \frac{1}{n}}\right\} . \]\n\nIt is immed... | Yes |
Proposition 48.6. Let \( E \) be a Hilbert space. For any nonempty convex and closed subset \( X \subseteq E \), the map \( {p}_{X} : E \rightarrow X \) is continuous. In fact, \( {p}_{X} \) satisfies the Lipschitz condition \[ \begin{Vmatrix}{{p}_{X}\left( v\right) - {p}_{X}\left( u\right) }\end{Vmatrix} \leq \paralle... | Proof. For any two vectors \( u, v \in E \), let \( x = {p}_{X}\left( u\right) - u, y = {p}_{X}\left( v\right) - {p}_{X}\left( u\right) \), and \( z = v - {p}_{X}\left( v\right) \). Clearly, (as illustrated in Figure 48.6), \[ v - u = x + y + z, \] and from Proposition 48.5 (2), we also have \[ \Re \langle x, y\rangle ... | Yes |
Proposition 48.7. Let \( E \) be a Hilbert space.\n\n(1) For any closed subspace \( V \subseteq E \), we have \( E = V \oplus {V}^{ \bot } \), and the map \( {p}_{V} : E \rightarrow V \) is linear and continuous.\n\n(2) For any \( u \in E \), the projection \( {p}_{V}\left( u\right) \) is the unique vector \( w \in E \... | Proof. (1) First, we prove that \( u - {p}_{V}\left( u\right) \in {V}^{ \bot } \) for all \( u \in E \) . For any \( v \in V \), since \( V \) is a subspace, \( z = {p}_{V}\left( u\right) + {\lambda v} \in V \) for all \( \lambda \in \mathbb{C} \), and since \( V \) is convex and nonempty (since it is a subspace), and ... | Yes |
Proposition 48.10. Given a Hilbert space \( E \), for every continuous linear map \( f : E \rightarrow E \) , there is a unique continuous linear map \( {f}^{ * } : E \rightarrow E \), such that\n\n\[ \langle f\left( u\right), v\rangle = \left\langle {u,{f}^{ * }\left( v\right) }\right\rangle \;\text{ for all }u, v \in... | Proof. The proof is adapted from Rudin [139] (Section 12.9). By the Cauchy-Schwarz inequality\n\n\[ \left| {\langle x, y\rangle }\right| \leq \parallel x\parallel \parallel y\parallel \]\n\nwe see that the sesquilinear map \( \left( {x, y}\right) \mapsto \langle x, y\rangle \) on \( E \times E \) is continuous. Let \( ... | Yes |
Theorem 48.11. (Farkas-Minkowski Lemma in Hilbert Spaces) Let \( \\left( {V,\\langle -, - \\rangle }\\right) \) be a real Hilbert space. For any finite sequence of vectors \( \\left( {{a}_{1},\\ldots ,{a}_{m}}\\right) \) with \( {a}_{i} \\in V \), if \( C \) is the polyhedral cone \( C = \\operatorname{cone}\\left( {{a... | Proof. We follow Ciarlet [41] (Chapter 9, Theorem 9.1.1). We already established in Proposition 44.2 that the polyhedral cone \( C = \\operatorname{cone}\\left( {{a}_{1},\\ldots ,{a}_{m}}\\right) \) is closed. Next we claim the following:\n\nClaim: If \( C \) is a nonempty, closed, convex subset of a Hilbert space \( V... | Yes |
Proposition 49.1. Let \( U \) be a nonempty, closed subset of \( {\mathbb{R}}^{n} \), and let \( J : {\mathbb{R}}^{n} \rightarrow \mathbb{R} \) be a continuous function which is coercive if \( U \) is unbounded. Then there is a least one element \( u \in {\mathbb{R}}^{n} \) such that\n\n\[ u \in U\;\text{ and }\;J\left... | Proof. Since \( U \neq \varnothing \), pick any \( {u}_{0} \in U \) . Since \( J \) is coercive, there is some \( r > 0 \) such that for all \( v \in {\mathbb{R}}^{n} \), if \( \parallel v\parallel > r \) then \( J\left( {u}_{0}\right) < J\left( v\right) \) . It follows that \( J \) is minimized over the set\n\n\[ {U}_... | Yes |
Proposition 49.3. If \( J \) is a quadratic functional, then\n\n\[ J\left( {u + {\rho v}}\right) = \frac{{\rho }^{2}}{2}a\left( {v, v}\right) + \rho \left( {a\left( {u, v}\right) - h\left( v\right) }\right) + J\left( u\right) . \] | Proof. Since \( a \) is symmetric bilinear and \( h \) is linear, we have\n\n\[ J\left( {u + {\rho v}}\right) = \frac{1}{2}a\left( {u + {\rho v}, u + {\rho v}}\right) - h\left( {u + {\rho v}}\right) \]\n\n\[ \frac{{\rho }^{2}}{2}a\left( {v, v}\right) + {\rho a}\left( {u, v}\right) + \frac{1}{2}a\left( {u, u}\right) - h... | Yes |
Theorem 49.4. Given any Hilbert space \( V \), let \( J : V \rightarrow \mathbb{R} \) be a quadratic functional of the form\n\n\[ J\left( v\right) = \frac{1}{2}a\left( {v, v}\right) - h\left( v\right) \]\n\nAssume that there is some real number \( \alpha > 0 \) such that\n\n\[ a\left( {v, v}\right) \geq \alpha \paralle... | Proof. The key point is that the bilinear form \( a \) is actually an inner product in \( V \) . This is because it is positive definite, since \( \left( { * }_{\alpha }\right) \) implies that\n\n\[ \sqrt{\alpha }\parallel v\parallel \leq {\left( a\left( v, v\right) \right) }^{1/2} \]\n\nand on the other hand the conti... | Yes |
Let \( V \) be a Hilbert space. (1) An elliptic functional \( J : V \rightarrow \mathbb{R} \) is strictly convex and coercice. Furthermore, it satisfies the identity \[ J\left( v\right) - J\left( u\right) \geq \left\langle {\nabla {J}_{u}, v - u}\right\rangle + \frac{\alpha }{2}\parallel v - u{\parallel }^{2}\;\text{ f... | Since \( J \) is a \( {C}^{1} \) -function, by Taylor’s formula with integral remainder in the case \( m = 0 \) (Theorem 39.25), we get \[ J\left( v\right) - J\left( u\right) = {\int }_{0}^{1}d{J}_{u + t\left( {v - u}\right) }\left( {v - u}\right) {dt} \] \[ = {\int }_{0}^{1}\left\langle {\nabla {J}_{u + t\left( {v - u... | Yes |
Proposition 49.11. If \( J \) is a quadratic elliptic functional of the form\n\n\[ J\left( v\right) = \frac{1}{2}a\left( {v, v}\right) - h\left( v\right) \]\n\nthen given \( {d}_{k} \), there is a unique \( {\rho }_{k} \) solving the line search in Step (2). | Proof. This is because, by Proposition 49.3, we have\n\n\[ J\left( {{u}_{k} + \rho {d}_{k}}\right) = \frac{{\rho }^{2}}{2}a\left( {{d}_{k},{d}_{k}}\right) + \rho \left\langle {\nabla {J}_{{u}_{k}},{d}_{k}}\right\rangle + J\left( {u}_{k}\right) ,\]\n\nand since \( a\left( {{d}_{k},{d}_{k}}\right) > 0 \) (because \( J \)... | Yes |
Proposition 49.14. Let \( J : V \rightarrow \mathbb{R} \) be a continuously differentiable functional defined on a Hilbert space \( V \) . Suppose there exists two constants \( \alpha > 0 \) and \( M > 0 \) such that\n\n\[ \left\langle {\nabla {J}_{v} - \nabla {J}_{u}, v - u}\right\rangle \geq \alpha \parallel v - u{\p... | Proof. By hypothesis the functional \( J \) is elliptic, so by Theorem 49.8(2) it has a unique minimum \( u \) characterized by the fact that \( \nabla {J}_{u} = 0 \) . Then since \( {u}_{k + 1} = {u}_{k} - {\rho }_{k}\nabla {J}_{{u}_{k}} \), we can write\n\n\[ {u}_{k + 1} - u = \left( {{u}_{k} - u}\right) - {\rho }_{k... | Yes |
Proposition 49.15. Assume that \( \nabla {J}_{{u}_{i}} \neq 0 \) for \( i = 0,\ldots, k \) . Then the minimization problem, find \( {u}_{k + 1} \) such that\n\n\[ \n{u}_{k + 1} \in {u}_{k} + {\mathcal{G}}_{k}\;\text{ and }\;J\left( {u}_{k + 1}\right) = \mathop{\inf }\limits_{{v \in {u}_{k} + {\mathcal{G}}_{k}}}J\left( ... | Proof. The affine space \( {u}_{\ell } + {\mathcal{G}}_{\ell } \) is closed and convex, and since \( J \) is a quadratic elliptic functional it is coercise and strictly convex, so by Theorem 49.8(2) it has a unique minimum in \( {u}_{\ell } + {\mathcal{G}}_{\ell } \) . This minimum \( {u}_{\ell + 1} \) is also the mini... | Yes |
Let \( J : {\mathbb{R}}^{2} \rightarrow \mathbb{R} \) be the function given by\n\n\[ J\left( {{v}_{1},{v}_{2}}\right) = \frac{1}{2}\left( {{\alpha }_{1}{v}_{1}^{2} + {\alpha }_{2}{v}_{2}^{2}}\right) \]\n\nwhere \( 0 < {\alpha }_{1} < {\alpha }_{2} \). The minimum of \( J \) is attained at \( \left( {0,0}\right) \). Unl... | Observe that\n\n\[ \nabla {J}_{\left( {v}_{1},{v}_{2}\right) } = \left( \begin{array}{l} {\alpha }_{1}{v}_{1} \\ {\alpha }_{2}{v}_{2} \end{array}\right) . \]\n\nAs a consequence, given \( {u}_{k} \), the line search for finding \( {\rho }_{k} \) and \( {u}_{k + 1} \) yields \( {u}_{k + 1} = \left( {0,0}\right) \) iff t... | Yes |
Proposition 49.16. Assume that \( \nabla {J}_{{u}_{i}} \neq 0 \) for \( i = 0,\ldots, k \), and let \( {\Delta }_{\ell } = {u}_{\ell + 1} - {u}_{\ell } \), for \( \ell = 0,\ldots, k \). Then \( {\Delta }_{\ell } \neq 0 \) for \( \ell = 0,\ldots, k \), and\n\n\[ \left\langle {A{\Delta }_{\ell },{\Delta }_{i}}\right\rang... | Proof. Since \( J \) is a quadratic functional we have\n\n\[ \nabla {J}_{v + w} = A\left( {v + w}\right) - b = {Av} - b + {Aw} = \nabla {J}_{v} + {Aw}. \]\n\nIt follows that\n\n\[ \nabla {J}_{{u}_{\ell + 1}} = \nabla {J}_{{u}_{\ell } + {\Delta }_{\ell }} = \nabla {J}_{{u}_{\ell }} + A{\Delta }_{\ell },\;0 \leq \ell \le... | Yes |
Proposition 49.17. Assume that \( \nabla {J}_{{u}_{i}} \neq 0 \) for \( i = 0,\ldots, k \) . If we write\n\n\[ \n{d}_{\ell } = \mathop{\sum }\limits_{{i = 0}}^{{\ell - 1}}{\lambda }_{i}^{\ell }\nabla {J}_{{u}_{i}} + \nabla {J}_{{u}_{\ell }},\;0 \leq \ell \leq k,\n\]\n\nthen we have\n\n\[ \n\text{(} \dagger \text{)}\;\l... | Proof. Since by \( \left( { * }_{4}\right) \) we have \( {\Delta }_{k} = {\delta }_{k}^{k}{d}_{k},{\delta }_{k}^{k} \neq 0 \) ,(by Proposition 49.16) we have\n\n\[ \n\left\langle {A{\Delta }_{\ell },{\Delta }_{i}}\right\rangle = 0,\;0 \leq i < \ell \leq k.\n\]\n\nBy \( \left( { * }_{1}\right) \) we have \( \nabla {J}_{... | Yes |
Let us take the example of Section 49.6 and apply the conjugate gradient procedure. Recall that\n\n\\[ \nJ\\left( {x, y}\\right) = \\frac{1}{2}\\left( \\begin{array}{ll} x & y \\end{array}\\right) \\left( \\begin{array}{ll} 3 & 2 \\\\ 2 & 6 \\end{array}\\right) \\left( \\begin{array}{l} x \\\\ y \\end{array}\\right) - ... | Step 1 involves calculating\n\n\\[ \n{\\rho }_{0} = \\frac{\\left\\langle \\nabla {J}_{{u}_{0}},{d}_{0}\\right\\rangle }{\\left\\langle A{d}_{0},{d}_{0}\\right\\rangle } = \\frac{13}{75} \n\\]\n\n\\[ \n{u}_{1} = {u}_{0} - {\\rho }_{0}{d}_{0} = \\left( {-2, - 2}\\right) - \\frac{13}{75}\\left( {-{12}, - 8}\\right) = \\l... | Yes |
Proposition 49.18. Let \( J : V \rightarrow \mathbb{R} \) be a continuously differentiable functional defined on a Hilbert space \( V \), and let \( U \) be nonempty, convex, closed subset of \( V \). Suppose there exists two constants \( \alpha > 0 \) and \( M > 0 \) such that\n\n\[ \left\langle {\nabla {J}_{v} - \nab... | Proof. For every \( {\rho }_{k} \geq 0 \), define the function \( {g}_{k} : V \rightarrow U \) by\n\n\[ {g}_{k}\left( v\right) = {p}_{U}\left( {v - {\rho }_{k}\nabla {J}_{v}}\right) \]\n\nBy Proposition 48.6, the projection map \( {p}_{U} \) has Lipschitz constant 1, so using the inequalities assumed to hold in the pro... | Yes |
Example 50.1. In \( V = {\mathbb{R}}^{2} \), let \( {\varphi }_{1} \) and \( {\varphi }_{2} \) be given by\n\n\[ \n{\varphi }_{1}\left( {{u}_{1},{u}_{2}}\right) = - {u}_{1} - {u}_{2} \n\] \n\n\[ \n{\varphi }_{2}\left( {{u}_{1},{u}_{2}}\right) = {u}_{1}\left( {{u}_{1}^{2} + {u}_{2}^{2}}\right) - \left( {{u}_{1}^{2} - {u... | The region \( U \) is shown in Figure 50.4 and is bounded by the curve given by the equation \( {\varphi }_{1}\left( {{u}_{1},{u}_{2}}\right) = 0 \), that is, \( - {u}_{1} - {u}_{2} = 0 \), the line of slope -1 through the origin, and the curve given by the equation \( {u}_{1}\left( {{u}_{1}^{2} + {u}_{2}^{2}}\right) -... | Yes |
Proposition 50.1. Let \( U \) be any nonempty subset of a normed vector space \( V \). (1) For any \( u \in U \), the cone \( C\left( u\right) \) of feasible directions at \( u \) is closed. | Proof. (1) Let \( {\left( {w}_{n}\right) }_{n \geq 0} \) be a sequence of vectors \( {w}_{n} \in C\left( u\right) \) converging to a limit \( w \in V \). We may assume that \( w \neq 0 \), since \( 0 \in C\left( u\right) \) by definition, and thus we may also assume that \( {w}_{n} \neq 0 \) for all \( n \geq 0 \). By ... | Yes |
Consider the region \( U \subseteq {\mathbb{R}}^{2} \) determined by the two curves given by \[ {\varphi }_{1}\left( {{u}_{1},{u}_{2}}\right) = {u}_{2} - \max \left( {0,{u}_{1}^{3}}\right) \] \[ {\varphi }_{2}\left( {{u}_{1},{u}_{2}}\right) = {u}_{1}^{4} - {u}_{2} \] | We have \( I\left( {0,0}\right) = \{ 1,2\} \), and since \( {\left( {\varphi }_{1}\right) }_{\left( 0,0\right) }^{\prime }\left( {{w}_{1},{w}_{2}}\right) = \left( \begin{array}{ll} 0 & 1 \end{array}\right) \left( \begin{array}{l} {w}_{1} \\ {w}_{2} \end{array}\right) = {w}_{2} \) and \( {\left( {\varphi }_{2}^{\prime }... | Yes |
Proposition 50.2. Let \( u \) be any point of the set\n\n\[ U = \\left\\{ {x \\in \\Omega \\mid {\\varphi }_{i}\\left( x\\right) \\leq 0,1 \\leq i \\leq m}\\right\\} \]\n\nwhere \( \\Omega \) is an open subset of the normed vector space \( V \), and assume that the functions \( {\\varphi }_{i} \) are differentiable at ... | Proof. (1) For every \( i \\in I\\left( u\\right) \), since \( {\\varphi }_{i}\\left( v\\right) \\leq 0 \) for all \( v \\in U \) and \( {\\varphi }_{i}\\left( u\\right) = 0 \), the function \( - {\\varphi }_{i} \) has a local minimum at \( u \) with respect to \( U \), so by Proposition 50.1(2), we have\n\n\[ {\\left(... | Yes |
Proposition 50.4. (Farkas-Minkowski) Let \( V \) be a Euclidean space of finite dimension with inner product \( \langle - , - \rangle \) (more generally, a Hilbert space). For any finite family \( \left( {{a}_{1},\ldots ,{a}_{m}}\right) \) of \( m \) vectors \( {a}_{i} \in V \) and any vector \( b \in V \), for any \( ... | Proposition 50.4 is the special case of Theorem 48.11 which holds for real Hilbert spaces. | Yes |
Theorem 50.5. Let \( {\varphi }_{i} : \Omega \rightarrow \mathbb{R} \) be \( m \) constraints defined on some open subset \( \Omega \) of a finite-dimensional Euclidean vector space \( V \) (more generally, a real Hilbert space \( V \) ), let \( J : \Omega \rightarrow \mathbb{R} \) be some function, and let \( U \) be ... | Proof. By Proposition 50.1(2), we have\n\n\[ {J}_{u}^{\prime }\left( w\right) \geq 0\;\text{ for all }w \in C\left( u\right) ,\]\n\n\( \left( { * }_{1}\right) \)\n\nand by Proposition \( {50.2}\left( 2\right) \), we have \( C\left( u\right) = {C}^{ * }\left( u\right) \), where\n\n\[ {C}^{ * }\left( u\right) = \left\{ {... | Yes |
Theorem 50.6. Let \( {\varphi }_{i} : \Omega \rightarrow \mathbb{R} \) be \( m \) convex constraints defined on some open convex subset \( \Omega \) of a finite-dimensional Euclidean vector space \( V \) (more generally, a real Hilbert space \( V \) ), let \( J : \Omega \rightarrow \mathbb{R} \) be some function, let \... | Proof. (1) It suffices to prove that if the convex constraints are qualified according to Definition 50.6 , then they are qualified according to Definition 50.5 , since in this case we can apply Theorem 50.5.\n\nIf \( v \in \Omega \) is a vector such that Condition (b) of Definition 50.6 holds and if \( v \neq u \), fo... | Yes |
Proposition 50.7. If \( U \) is given by\n\n\[ U = \{ x \in \Omega \mid {Ax} \leq b\} \]\n\nwhere \( \Omega \) is an open convex subset of \( {\mathbb{R}}^{n} \) and \( A \) is an \( m \times n \) matrix, and if \( J \) is differentiable at \( u \) and \( J \) has a local minimum at \( u \), then there exist some vecto... | If the function \( J \) is convex, then the above conditions are also sufficient for \( J \) to have a minimum at \( u \in U \) . | No |
We would like to find necessary conditions for \( {f}_{\mu } \) to have a maximum on\n\n\[ U = \\left\\{ {x \\in {\\mathbb{R}}_{+ + }^{n} \\mid {Ax} = b}\\right\\} \]\n\nor equivalently to solve the following problem:\n\n\[ \\text{maximize}\\{f}_{\\mu }\\left( x\\right) \]\n\n\[ \\text{subject to} \]\n\n\[ {Ax} = b \]\... | Since maximizing \( {f}_{\mu } \) is equivalent to minimizing \( - {f}_{\mu } \), by Proposition 50.9, if \( x \) is an optimal of the above problem then there is some \( y \\in {\\mathbb{R}}^{m} \) such that\n\n\[ - \\nabla {f}_{\\mu }\\left( x\\right) + {A}^{\\top }y = 0. \]\n\nSince\n\n\[ \\nabla {f}_{\\mu }\\left( ... | Yes |
Proposition 50.11. If \( \left( {w, b,\delta }\right) \) is an optimal solution of Problem \( \left( {\mathrm{{SVM}}}_{h1}\right) \), so in particular \( \delta > 0 \), then we must have \( \parallel w\parallel = 1 \) . | Proof. First, if \( w = 0 \), then we get the two inequalities\n\n\[ \n- b \geq \delta ,\;b \geq \delta , \n\]\n\nwhich imply that \( b \leq - \delta \) and \( b \geq \delta \) for some positive \( \delta \), which is impossible. But then, if \( w \neq 0 \) and \( \parallel w\parallel < 1 \), by dividing both sides of ... | Yes |
Theorem 50.12. If two disjoint subsets of \( p \) blue points \( {\left\{ {u}_{i}\right\} }_{i = 1}^{p} \) and \( q \) red points \( {\left\{ {v}_{j}\right\} }_{j = 1}^{q} \) are linearly separable, then Problem \( \left( {\mathrm{{SVM}}}_{h1}\right) \) has a unique optimal solution consisting of a hyperplane of equati... | Proof. Our proof is adapted from Vapnik [180] (Chapter 10, Theorem 10.1). For any separating hyperplane \( H \), since\n\n\[ \nd\left( {{u}_{i}, H}\right) = {w}^{\top }{u}_{i} - b \n\]\n\n\[ \ni = 1,\ldots, p \n\]\n\n\[ \nd\left( {{v}_{j}, H}\right) = - {w}^{\top }{v}_{j} + b\;j = 1,\ldots, q, \n\]\nand since the small... | Yes |
Proposition 50.13. If \( \left( {u,\lambda }\right) \) is a saddle point of a function \( L : \Omega \times M \rightarrow \mathbb{R} \), then\n\n\[ \mathop{\sup }\limits_{{\mu \in M}}\mathop{\inf }\limits_{{v \in \Omega }}L\left( {v,\mu }\right) = L\left( {u,\lambda }\right) = \mathop{\inf }\limits_{{v \in \Omega }}\ma... | Proof. First we prove that the following inequality always holds:\n\n\[ \mathop{\sup }\limits_{{\mu \in M}}\mathop{\inf }\limits_{{v \in \Omega }}L\left( {v,\mu }\right) \leq \mathop{\inf }\limits_{{v \in \Omega }}\mathop{\sup }\limits_{{\mu \in M}}L\left( {v,\mu }\right) \]\n\n\( \left( { * }_{1}\right) \)\n\nPick any... | Yes |
(1) If \( \left( {u,\lambda }\right) \in \Omega \times {\mathbb{R}}_{ + }^{m} \) is a saddle point of the Lagrangian \( L \) associated with Problem \( \left( P\right) \) , then \( u \in U, u \) is a solution of Problem \( \left( P\right) \), and \( J\left( u\right) = L\left( {u,\lambda }\right) \) . | Proof. (1) Since \( \left( {u,\lambda }\right) \) is a saddle point of \( L \) we have \( \mathop{\sup }\limits_{{\mu \in {\mathbb{R}}_{ + }^{m}}}L\left( {u,\mu }\right) = L\left( {u,\lambda }\right) \) which implies that \( L\left( {u,\mu }\right) \leq L\left( {u,\lambda }\right) \) for all \( \mu \in {\mathbb{R}}_{ +... | Yes |
Consider the Linear Program \( \\left( P\\right) \)\n\n\\[ \n\\text{minimize}\\,{c}^{\\top }v\n\\]\n\n\\[ \n\\text{subject to}\\,{Av} \\leq b, v \\geq 0\\text{,}\n\\] | where \( A \) is an \( m \\times n \) matrix. The constraints \( v \\geq 0 \) are rewritten as \( - {v}_{i} \\leq 0 \), so we introduce Lagrange multipliers \( \\mu \\in {\\mathbb{R}}_{ + }^{m} \) and \( \\nu \\in {\\mathbb{R}}_{ + }^{n} \), and we have the Lagrangian\n\n\\[ \nL\\left( {v,\\mu ,\\nu }\\right) = {c}^{\\... | Yes |
Proposition 50.15. (Complementary Slackness) Given the Minimization Problem (P)\n\n\\[ \n\\text{minimize}\\;J\\left( v\\right) \n\\]\n\n\\[ \n\\text{subject to}{\\varphi }_{i}\\left( v\\right) \\leq 0,\\;i = 1,\\ldots, m\\text{,}\n\\]\n\nand its Dual Problem (D)\n\n\\[ \n\\text{maximize}\\;G\\left( \\mu \\right) \n\\]\... | Proof. Since \\( J\\left( u\\right) = G\\left( \\lambda \\right) \\) we have\n\n\\[ \nJ\\left( u\\right) = G\\left( \\lambda \\right) \n\\]\n\n\\[ \n= \\mathop{\\inf }\\limits_{{v \\in \\Omega }}\\left( {J\\left( v\\right) + \\mathop{\\sum }\\limits_{{i = 1}}^{m}{\\lambda }_{i}{\\varphi }_{i}\\left( v\\right) }\\right)... | Yes |
Theorem 50.16. Consider the Minimization Problem (P):\n\n\\[ \n\\text{minimize}\\;J\\left( v\\right) \n\\]\n\n\\[ \n\\text{subject to}{\\varphi }_{i}\\left( v\\right) \\leq 0,\\;i = 1,\\ldots, m\\text{,}\n\\]\n\nwhere the functions \\( J \\) and \\( {\\varphi }_{i} \\) are defined on some open subset \\( \\Omega \\) of... | Proof. (1) Our goal is to prove that for any solution \\( \\lambda \\) of Problem \\( \\left( D\\right) \\), the pair \\( \\left( {{u}_{\\lambda },\\lambda }\\right) \\) is a saddle point of \\( L \\) . By Theorem 50.14(1), the point \\( {u}_{\\lambda } \\in U \\) is a solution of Problem \\( \\left( P\\right) \\) .\n\... | Yes |
Consider the quadratic objective function\n\n\[ J\left( v\right) = \frac{1}{2}{v}^{\top }{Av} - {v}^{\top }b \]\n\nwhere \( A \) is an \( n \times n \) matrix which is symmetric positive definite, \( b \in {\mathbb{R}}^{n} \), and the constraints are affine inequality constraints of the form\n\n\[ {Cv} \leq d \]\n\nwhe... | Since \( A \) is symmetric positive definite, \( J \) is strictly convex, as implied by Proposition 40.9 (see Example 40.1). The Lagrangian of this quadratic optimization problem is given by\n\n\[ L\left( {v,\mu }\right) = \frac{1}{2}{v}^{\top }{Av} - {v}^{\top }b + {\left( Cv - d\right) }^{\top }\mu \]\n\n\[ = \frac{1... | Yes |
Theorem 50.17. Let \( {\varphi }_{i} : \Omega \rightarrow \mathbb{R} \) be \( m \) convex inequality constraints and \( {\psi }_{j} : \Omega \rightarrow \mathbb{R} \) be \( p \) affine equality constraints defined on some open convex subset \( \Omega \) of a finite-dimensional Euclidean vector space \( V \) (more gener... | Equivalently, in terms of gradients, the above conditions are expressed as\n\n\[ \\nabla {J}_{u} + \\mathop{\\sum }\\limits_{{i = 1}}^{m}{\\lambda }_{i}\\nabla {\\left( {\\varphi }_{i}\\right) }_{u} + \\mathop{\\sum }\\limits_{{j = 1}}^{p}{\\nu }_{j}\\nabla {\\left( {\\psi }_{j}\\right) }_{u} = 0 \]\n\nand\n\n\[ \\math... | Yes |
Proposition 50.19. Consider Problem (P), \n\n\[ \n\\text{minimize}\;J\\left( v\\right) \n\] \n\n\[ \n\\text{subject to}{Av} \\leq b \n\] \n\n\[ \n{Cv} = d, \n\] \n\nwith affine inequality and equality constraints (with \( A \) an \( m \\times n \) matrix, \( C \) an \( p \\times n \) matrix, \( \\left. {b \\in {\\mathb... | Proof. The Lagrangian associated with the above program is \n\n\[ \nL\\left( {v,\\lambda ,\\nu }\\right) = J\\left( v\\right) + {\\left( Av - b\\right) }^{\\top }\\lambda + {\\left( Cv - d\\right) }^{\\top }\\nu \n\] \n\n\[ \n= - {b}^{\\top }\\lambda - {d}^{\\top }\\nu + J\\left( v\\right) + {\\left( {A}^{\\top }\\lamb... | Yes |
Consider the following problem:\n\n\[ \text{minimize}\parallel v\parallel \]\n\n\[ \text{subject to}{Av} = b\text{,} \] | Using the result of Example 50.8(6), we obtain\n\n\[ G\left( \nu \right) = - {b}^{\top }\nu - {\begin{Vmatrix}-{A}^{\top }\nu \end{Vmatrix}}^{ * }, \]\n\nthat is,\n\n\[ G\left( \nu \right) = \left\{ \begin{array}{ll} - {b}^{\top }\nu & \text{ if }{\begin{Vmatrix}{A}^{\top }\nu \end{Vmatrix}}^{D} \leq 1 \\ - \infty & \t... | Yes |
As a concrete example, consider the following unconstrained program:\n\n\[ \text{ minimize }f\left( x\right) = \log \left( {\mathop{\sum }\limits_{{i = 1}}^{n}{e}^{{\left( {A}^{i}\right) }^{\top }x + {b}_{i}}}\right) \] \n\nwhere \( {A}^{i} \) is a column vector in \( {\mathbb{R}}^{n} \). | We reformulate the problem by introducing new variables and equality constraints as follows:\n\n\[ \text{ minimize }\;f\left( y\right) = \log \left( {\mathop{\sum }\limits_{{i = 1}}^{n}{e}^{{y}_{i}}}\right) \] \n\n\[ \text{subject to} \] \n\n\[ {Ax} + b = y, \] \n\nwhere \( A \) is the \( n \times n \) matrix whose col... | Yes |
Similarly the unconstrained norm minimization problem\n\n\[ \n\\text{minimize}\\parallel {Ax} - b\\parallel \\text{,}\n\]\n\nwhere \( \\parallel \\parallel \) is any norm on \( {\\mathbb{R}}^{m} \), has a dual function which is a constant, and is not useful. This problem can be reformulated as\n\nminimize \( \\parallel... | By Example 50.8(6), the conjugate of the norm is given by\n\n\[ \n\\parallel y{\\parallel }^{ * } = \\left\\{ \\begin{array}{ll} 0 & \\text{ if }\\parallel y{\\parallel }^{D} \\leq 1 \\\\ + \\infty & \\text{ otherwise,} \\end{array}\\right.\n\]\n\nso the dual of the reformulated program is:\n\n\[ \n\\text{maximize}\;{b... | Yes |
The norm minimization of Example 50.13 can be reformulated as\n\n\\[ \n\\text{minimize}\\;\\frac{1}{2}\\parallel y{\\parallel }^{2} \n\\]\n\n\\[ \n\\text{subject to} \n\\]\n\n\\[ \n{Ax} - b = y. \n\\] | This program is obviously equivalent to the original one. By Example 50.8(7), the conjugate of the square norm is given by\n\n\\[ \n\\frac{1}{2}{\\left( \\parallel y{\\parallel }^{D}\\right) }^{2} \n\\]\n\nso the dual of the reformulated program is\n\n\\[ \n\\text{maximize}\\; - \\frac{1}{2}{\\left( \\parallel \\mu {\\... | Yes |
Theorem 50.20. Suppose \( J : {\mathbb{R}}^{n} \rightarrow \mathbb{R} \) is an elliptic functional, which means that \( J \) is continuously differentiable on \( {\mathbb{R}}^{n} \), and there is some constant \( \alpha > 0 \) such that\n\n\[ \left\langle {\nabla {J}_{v} - \nabla {J}_{u}, v - u}\right\rangle \geq \alph... | Proof.\n\nStep 1. We establish algebraic conditions relating the unique minimizer \( u \in U \) of \( J \) over \( U \) and some \( \lambda \in {\mathbb{R}}_{ + }^{m} \) such that \( \left( {u,\lambda }\right) \) is a saddle point.\n\nSince \( J \) is elliptic and \( U \) is nonempty closed and convex, by Theorem 49.8,... | No |
Example 51.1. The above fact is illustrated by the function \( f : \mathbb{R} \rightarrow \mathbb{R} \cup \{ - \infty , + \infty \} \) where\n\n\[ f\left( x\right) = \left\{ \begin{array}{ll} - {x}^{2} & \text{ if }x \geq 0 \\ + \infty & \text{ if }x < 0 \end{array}\right. \]\n\nThe epigraph of this function is illustr... | If \( f \) is a convex function, since \( \operatorname{dom}\left( f\right) \) is the image of \( \operatorname{epi}\left( f\right) \) by a linear map (a projection), it is convex.\n\nBy definition, \( \operatorname{\mathbf{e} \mathbf{p} \mathbf{i} }\left( {f \mid S}\right) \) is convex iff for any \( \left( {{x}_{1},{... | Yes |
Here is an example of an improper convex function \( f : \mathbb{R} \rightarrow \mathbb{R} \cup \{ - \infty , + \infty \} \) :\n\n\[ f\left( x\right) = \left\{ \begin{array}{ll} - \infty & \text{ if }\left| x\right| < 1 \\ 0 & \text{ if }\left| x\right| = 1 \\ + \infty & \text{ if }\left| x\right| > 1 \end{array}\right... | Observe that \( \operatorname{dom}\left( f\right) = \left\lbrack {-1,1}\right\rbrack \), and that \( \operatorname{epi}\left( f\right) \) is not closed. See Figure 51.4. | Yes |
For an example of Propositions 51.6 and 51.5, let \( f : \mathbb{R} \rightarrow \mathbb{R} \cup \{ + \infty \} \) be the proper convex function\n\n\[ f\left( x\right) = \left\{ \begin{array}{ll} {x}^{2} & \text{ if }x < 1 \\ + \infty & \text{ if }\left| x\right| \geq 1 \end{array}\right. \]\n\nThen \( \operatorname{cl}... | and \( \operatorname{cl}f\left( x\right) = f\left( x\right) \) whenever \( x \in \left( {-\infty ,1}\right) = \operatorname{relint}\left( {\operatorname{dom}\left( f\right) }\right) = \operatorname{dom}\left( f\right) \) . Furthermore, since \( \operatorname{relint}\left( {\operatorname{dom}\left( f\right) }\right) = \... | Yes |
Consider the proper convex function (on \( {\mathbb{R}}^{2} \) ) given by\n\n\[ f\left( {x, y}\right) = \left\{ \begin{array}{ll} {y}^{2}/\left( {2x}\right) & \text{ if }x > 0 \\ 0 & \text{ if }x = 0, y = 0 \\ + \infty & \text{ otherwise. } \end{array}\right. \] | The function \( f \) is continuous on the open right half-plane \( \left\{ {\left( {x, y}\right) \in {\mathbb{R}}^{2} \mid x > 0}\right\} \), but not at \( \left( {0,0}\right) \) . The limit of \( f\left( {x, y}\right) \) when \( \left( {x, y}\right) \) approaches \( \left( {0,0}\right) \) on the parabola of equation \... | Yes |
Proposition 51.9. Let \( \varphi : {\mathbb{R}}^{n} \rightarrow \mathbb{R} \) be a nonconstant affine form. Then the map \( \omega : {\mathbb{R}}^{n + 1} \rightarrow \) \( \mathbb{R} \) given by\n\n\[ \omega \left( {x,\alpha }\right) = \varphi \left( x\right) - \alpha ,\;x \in {\mathbb{R}}^{n},\alpha \in \mathbb{R}, \]... | Proof. Indeed, \( \varphi \) is of the form \( \varphi \left( x\right) = h\left( x\right) + c \) for some nonzero linear form \( h \), so\n\n\[ \omega \left( {x,\alpha }\right) = h\left( x\right) - \alpha + c. \]\n\nSince \( h \) is linear, the map \( \left( {x,\alpha }\right) = h\left( x\right) - \alpha \) is obviousl... | Yes |
The \( {\ell }^{2} \) norm \( f\left( x\right) = \parallel x{\parallel }_{2} \) is subdifferentiable for all \( x \in {\mathbb{R}}^{n} \), in fact differentiable for all \( x \neq 0 \) . For \( x = 0 \), the set \( \partial f\left( 0\right) \) consists of all \( u \in {\mathbb{R}}^{n} \) such that | \[ \parallel z{\parallel }_{2} \geq \langle z, u\rangle \;\text{ for all }z \in {\mathbb{R}}^{n}, \] namely (by Cauchy-Schwarz), the Euclidean unit ball \( \left\{ {u \in {\mathbb{R}}^{n} \mid \parallel u{\parallel }_{2} \leq 1}\right\} \) . See Figure 51.14. | Yes |
For the \( {\ell }^{\infty } \) norm if \( f\left( x\right) = \parallel x{\parallel }_{\infty } \), we leave it as an exercise to show that \( \partial f\left( 0\right) \) is the polyhedron | \[ \partial f\left( 0\right) = \operatorname{conv}\left\{ {\pm {e}_{1},\ldots , \pm {e}_{n}}\right\} . \] | No |
The following function is an example of a proper convex function which is not subdifferentiable everywhere: | \[ f\left( x\right) = \left\{ \begin{array}{ll} - {\left( 1 - {\left| x\right| }^{2}\right) }^{1/2} & \text{ if }\left| x\right| \leq 1 \\ + \infty & \text{ otherwise. } \end{array}\right. \] | Yes |
The subdifferential of an indicator function is interesting. Let \( C \) be a nonempty convex set. By definition, \( u \in \partial {I}_{C}\left( x\right) \) iff\n\n\[ \n{I}_{C}\left( z\right) \geq {I}_{C}\left( x\right) + \langle z - x, u\rangle \;\text{ for all }z \in {\mathbb{R}}^{n}.\n\] | Since \( C \) is nonempty, there is some \( z \in C \) such that \( {I}_{C}\left( z\right) = 0 \), so the above condition implies that \( x \in C \) (otherwise \( {I}_{C}\left( x\right) = + \infty \) but \( 0 \geq + \infty + \langle z - u, u\rangle \) is impossible), so \( 0 \geq \langle z - x, u\rangle \) for all \( z... | Yes |
The subdifferentials of the indicator function \( f \) of the nonnegative orthant of \( {\mathbb{R}}^{n} \) reveal a connection to complementary slackness conditions. Recall that this indicator function is given by\n\n\[ f\left( {{x}_{1},\ldots ,{x}_{n}}\right) = \left\{ \begin{array}{ll} 0 & \text{ if }{x}_{i} \geq 0,... | By Example 51.8, the subgradients \( y \) of \( f \) at \( x \geq 0 \) form the normal cone to the nonnegative orthant at \( x \) . This means that \( y \in {N}_{C}\left( x\right) \) iff\n\n\[ \langle z - x, y\rangle \leq 0\;\text{ for all }z \geq 0 \]\n\niff\n\n\[ \langle z, y\rangle \leq \langle x, y\rangle \;\text{ ... | Yes |
Theorem 51.10. (Minkowski) Let \( C \) be a nonempty convex set in \( {\mathbb{R}}^{n} \) . For any point \( a \in \) \( C - \operatorname{relint}\left( C\right) \), there is a supporting hyperplane \( H \) to \( C \) at a. | Theorem 51.10 is proven in Rockafellar [136] (Theorem 11.6). See also Berger [11] (Proposition 11.5.2). The proof is not as simple as one might expect, and is based on a geometric version of the Hahn-Banach theorem. | Yes |
Proposition 51.11. Let \( C \) be any nonempty convex set in \( {\mathbb{R}}^{n} \) . For any \( x \in \mathbf{{relint}}\left( C\right) \) and any \( y \in \bar{C} \), we have \( \left( {1 - \lambda }\right) x + {\lambda y} \in \mathbf{{relint}}\left( C\right) \) for all \( \lambda \) such that \( 0 \leq \lambda < 1 \)... | Proposition 51.11 is proven in Rockafellar [136] (Theorem 6.1). The proof is not difficult but quite technical. | No |
Proposition 51.12. For any proper convex function \( f \) on \( {\mathbb{R}}^{n} \), we have\n\n\[ \operatorname{relint}\left( {\operatorname{epi}\left( f\right) }\right) = \left\{ {\left( {x,\mu }\right) \in {\mathbb{R}}^{n + 1} \mid x \in \operatorname{relint}\left( {\operatorname{dom}\left( f\right) }\right), f\left... | Proof. Proposition 51.12 is proven in Rockafellar [136] (Lemma 7.3). By working in the affine hull of \( \mathbf{{epi}}\left( f\right) \), the statement of Proposition 51.12 is equivalent to\n\n\[ \operatorname{int}\left( {\mathbf{{epi}}\left( f\right) }\right) = \left\{ {\left( {x,\mu }\right) \in {\mathbb{R}}^{m + 1}... | Yes |
Theorem 51.13. Let \( f \) be a proper convex function on \( {\mathbb{R}}^{n} \) . For any \( x \in \operatorname{relint}\left( {\operatorname{dom}\left( f\right) }\right) \) , there is a nonvertical supporting hyperplane \( \mathcal{H} \) to \( \mathbf{{epi}}\left( f\right) \) at \( \left( {x, f\left( x\right) }\right... | Proof. By Proposition 51.13, for any \( x \in \operatorname{relint}\left( {\operatorname{dom}\left( f\right) }\right) \), we have \( \left( {x,\mu }\right) \in \operatorname{relint}\left( {\operatorname{epi}\left( f\right) }\right) \) for all \( \mu \in \mathbb{R} \) such that \( f\left( x\right) < \mu \) . Since by de... | Yes |
Proposition 51.14. Let \( f : {\mathbb{R}}^{n} \rightarrow \mathbb{R} \cup \{ - \infty , + \infty \} \) be a convex function. For any \( x \in {\mathbb{R}}^{n} \) , if \( f\left( x\right) \) is finite, then the function\n\n\[ \lambda \mapsto \frac{f\left( {x + {\lambda u}}\right) - f\left( x\right) }{\lambda } \]\n\nis... | Proposition 51.14 is proven in Rockafellar [136] (Theorem 23.1). The proof is not difficult but not very informative. | Yes |
Let \( f : {\mathbb{R}}^{n} \rightarrow \mathbb{R} \cup \{ - \infty , + \infty \} \) be a convex function. For any \( x \in {\mathbb{R}}^{n} \) , if \( f\left( x\right) \) is finite, then a vector \( u \in {\mathbb{R}}^{n} \) is a subgradient to \( f \) at \( x \) if and only if\n\n\[ \n{f}^{\prime }\left( {x;y}\right)... | Sketch of proof. Proposition 51.15 is proven in Rockafellar [136] (Theorem 23.2). We prove the inequality. If we write \( z = x + {\lambda y} \) with \( \lambda > 0 \), then the subgradient inequality implies\n\n\[ \nf\left( {x + {\lambda u}}\right) \geq f\left( x\right) + \langle z - x, u\rangle = f\left( x\right) + \... | Yes |
If \( f \) is the celebrated ReLU function (ramp function) from deep learning defined so that \[ \operatorname{ReLU}\left( x\right) = \max \{ x,0\} = \left\{ \begin{array}{ll} 0 & \text{ if }x < 0 \\ x & \text{ if }x \geq 0 \end{array}\right. \] then \( \partial \operatorname{ReLU}\left( 0\right) = \left\lbrack {0,1}\r... | The function ReLU is differentiable for \( x \neq 0 \) , with \( \operatorname{ReL}{\mathrm{U}}^{\prime }\left( x\right) = 0 \) if \( x < 0 \) and \( \operatorname{ReL}{\mathrm{U}}^{\prime }\left( x\right) = 1 \) if \( x > 0 \). | Yes |
Proposition 51.16. Let \( f : {\mathbb{R}}^{n} \rightarrow \mathbb{R} \cup \{ - \infty , + \infty \} \) be a convex function. For any \( x \in {\mathbb{R}}^{n} \), if \( f\left( x\right) \) is finite and if \( f \) is subdifferentiable at \( x \), then \( f \) is proper. If \( f \) is not subdifferentiable at \( x \), ... | Proposition 51.16 is proven in Rockafellar [136] (Theorem 23.3). It confirms that improper convex functions are rather pathological objects, because if a convex function is subdifferentiable for some \( x \) such that \( f\left( x\right) \) is finite, then \( f \) must be proper. This is because if \( f\left( x\right) ... | Yes |
Theorem 51.17. Let \( f : {\mathbb{R}}^{n} \rightarrow \mathbb{R} \cup \{ + \infty \} \) be a proper convex function. For any \( x \notin \operatorname{dom}\left( f\right) \) , we have \( \partial f\left( x\right) = \varnothing \) . For any \( x \in \operatorname{relint}\left( {\operatorname{dom}\left( f\right) }\right... | Theorem 51.17 is proven in Rockafellar [136] (Theorem 23.4). | Yes |
Consider the proper convex function defined on \( {\mathbb{R}}^{2} \) given by\n\n\[ f\left( {x, y}\right) = \max \{ g\left( x\right) ,\left| y\right| \}\]\n\nwhere\n\n\[ g\left( x\right) = \left\{ \begin{array}{ll} 1 - \sqrt{x} & \text{ if }x \geq 0 \\ + \infty & \text{ if }x < 0 \end{array}\right.\] | It is easy to see that \( \operatorname{dom}\left( f\right) = \left\{ {\left( {x, y}\right) \in {\mathbb{R}}^{2} \mid x \geq 0}\right\} \), but\n\n\( \operatorname{dom}\left( {\partial f}\right) = \left\{ {\left( {x, y}\right) \in {\mathbb{R}}^{2} \mid x \geq 0}\right\} - \{ \left( {0, y}\right) \mid - 1 < y < 1\} \), ... | Yes |
Theorem 51.18. Let \( f \) be a convex function on \( {\mathbb{R}}^{n} \), and let \( x \in {\mathbb{R}}^{n} \) such that \( f\left( x\right) \) is finite. If \( f \) is differentiable at \( x \) then \( \partial f\left( x\right) = \left\{ {\nabla {f}_{x}}\right\} \) (where \( \nabla {f}_{x} \) is the gradient of \( f ... | The first direction is easy to prove. Indeed, if \( f \) is differentiable at \( x \), then\n\n\[ {f}^{\prime }\left( {x;y}\right) = \left\langle {y,\nabla {f}_{x}}\right\rangle \;\text{ for all }y \in {\mathbb{R}}^{n}, \]\n\nso by Proposition 51.15, a vector \( u \) is a subgradient at \( x \) iff\n\n\[ \left\langle {... | Yes |
Theorem 51.20. Let \( f \) be a proper convex function on \( {\mathbb{R}}^{n} \), and let \( D \) be the set of vectors where \( f \) is differentiable. Then \( D \) is a dense subset of \( \operatorname{int}\left( {\operatorname{dom}\left( f\right) }\right) \), and its complement in \( \operatorname{int}\left( {\opera... | Theorem 51.20 is proven in Rockafellar [136] (Theorem 25.5). | Yes |
Proposition 51.22. Let \( {f}_{1},\ldots ,{f}_{n} \) be proper convex functions on \( {\mathbb{R}}^{n} \), and let \( f = {f}_{1} + \cdots + {f}_{n} \) . For \( x \in {\mathbb{R}}^{n} \), we have\n\n\[ \partial f\left( x\right) \supseteq \partial {f}_{1}\left( x\right) + \cdots + \partial {f}_{n}\left( x\right) \]\n\nI... | Proposition 51.22 is proven in Rockafellar [136] (Theorem 23.8). | Yes |
Proposition 51.23. Let \( f \) be the function given by \( f\left( x\right) = h\left( {Ax}\right) \) for all \( x \in {\mathbb{R}}^{n} \), where \( h \) is a proper convex function on \( {\mathbb{R}}^{m} \) and \( A \) is an \( m \times n \) matrix. Then\n\n\[ \partial f\left( x\right) \supseteq {A}^{\top }\left( {\par... | Proposition 51.23 is proven in Rockafellar [136] (Theorem 23.9). | Yes |
Proposition 51.24. Let \( f \) be a proper convex function on \( {\mathbb{R}}^{n} \), and let \( x \in {\mathbb{R}}^{n} \) be a vector such that \( f \) is subdifferentiable at \( x \) but \( f \) does not achieve its minimum at \( x \) . Then the normal cone \( {N}_{C}\left( x\right) \) at \( x \) to the sublevel set ... | Proposition 51.24 is proven in Rockafellar [136] (Theorem 23.7). | Yes |
Proposition 51.31. Let \( f : {\mathbb{R}}^{n} \rightarrow \mathbb{R} \cup \{ + \infty \} \) be any proper convex function. For any \( \epsilon > 0 \) , if \( {h}_{x} \) is given by\n\n\[ \n{h}_{x}\left( y\right) = f\left( {x + y}\right) - f\left( x\right) ,\;\text{ for all }y \in {\mathbb{R}}^{n},\n\]\n\nthen\n\n\[ \n... | Proof. We have\n\n\[ \n{h}_{x}^{ * }\left( y\right) = \mathop{\sup }\limits_{{z \in {\mathbb{R}}^{n}}}\left( {\langle y, z\rangle - {h}_{x}\left( z\right) }\right)\n\]\n\n\[ \n= \mathop{\sup }\limits_{{z \in {\mathbb{R}}^{n}}}\left( {\langle y, z\rangle - f\left( {x + z}\right) + f\left( x\right) }\right)\n\]\n\n\[ \n=... | Yes |
Proposition 51.32. Let \( f \) be a closed and proper convex function, and let \( x \in {\mathbb{R}}^{n} \) such that \( f\left( x\right) \) is finite. Then | \[ {f}^{\prime }\left( {x;y}\right) = \mathop{\lim }\limits_{{\epsilon \downarrow 0}}{\delta }^{ * }\left( {y \mid {\partial }_{\epsilon }f\left( x\right) }\right) = \mathop{\lim }\limits_{{\epsilon \downarrow 0}}{I}_{{\partial }_{\epsilon }f\left( x\right) }^{ * }\left( y\right) \;\text{ for all }y \in {\mathbb{R}}^{n... | No |
Proposition 51.33. Let \( f \) be a proper convex function over \( {\mathbb{R}}^{n} \) . A vector \( x \in {\mathbb{R}}^{n} \) belongs to the minimum set of \( f \) iff\n\n\[ 0 \in \partial f\left( x\right) \]\n\niff \( f\left( x\right) \) is finite and\n\n\[ {f}^{\prime }\left( {x;y}\right) \geq 0\;\text{ for all }y \... | Of course, if \( f \) is differentiable at \( x \), then \( \partial f\left( x\right) = \left\{ {\nabla {f}_{x}}\right\} \), and we obtain the well-known condition \( \nabla {f}_{x} = 0 \) . | Yes |
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