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Theorem 6.23. (Dubovitskii-Milyutin) Let \( {\left\{ {K}_{i}\right\} }_{1}^{k} \) be open convex cones, and \( {K}_{k + 1} \) a convex cone in \( {\mathbb{R}}^{n} \) . Then\n\n\[ \n{ \cap }_{i = 1}^{k + 1}{K}_{i} = \varnothing \n\]\n\n(6.7)\n\nif and only if there exist\n\n\[ \n{l}_{i} \in {K}_{i}^{ * },{\left\{ {l}_{i... | This finite-dimensional version of the Dubovitskii-Milyutin theorem follows immediately from Lemma 6.22. The theorem is also true in infinite-dimensional topological vector spaces; see Section 6.7. | Yes |
Lemma 6.27. Let \( E \) be a real vector space. A set \( H \subset E \) is a hyperplane if and only if \( H \) is a proper maximal affine subset of \( E \) . | Proof. Clearly, a hyperplane \( {H}_{\left( \ell ,\alpha \right) } \) is a proper affine subset of \( E \) . The max-imality of \( H \) holds: if \( a \in E \smallsetminus H \), then \( \ell \left( a\right) \neq 0 \), so that if \( x \in E \), we have \( \ell \left( x\right) = \ell \left( {\left( {\ell \left( x\right) ... | Yes |
Lemma 6.30. If \( A \) and \( B \) are two nonempty, disjoint convex sets in a vector space \( E \), then there exist complementary convex sets \( C \) and \( D \) in \( E \) such that \( A \subseteq C \) and \( B \subseteq D \) . | Proof. We introduce a relation \( \preccurlyeq \) on the set \( \mathcal{C} \) of disjoint convex subsets \( \left( {C, D}\right) \subseteq E \times E \) such that \( A \subseteq C \) and \( B \subseteq D \) by the inclusion relation, that is, we declare \( \left( {C, D}\right) \preccurlyeq \left( {{C}^{\prime },{D}^{\... | Yes |
Theorem 6.32. Let \( C \) and \( D \) be nonempty convex sets in a vector space \( E \) such that \( \operatorname{ai}\left( C\right) \neq \varnothing \) . Then there exists a hyperplane \( H \) separating \( C \) and \( D \) if and only if \( \operatorname{ai}\left( C\right) \cap D = \varnothing \), in which case \( \... | Proof. Suppose that the hyperplane \( H \) separates \( C \) and \( D \), such that \( C \subseteq {\bar{H}}^{ + } \) and \( D \subseteq {\bar{H}}^{ - } \) . The set \( C \) cannot lie on \( H \), since \( \operatorname{aff}\left( C\right) = E \) ; hence there exists a point \( y \in C \cap {H}^{ + } \) . We must have ... | Yes |
Theorem 6.34. Let \( C \) be a nonempty convex set in a vector space \( E \) . If \( M \) is an affine set such that \( \operatorname{rai}\left( C\right) \cap M = \varnothing \), then there exists a hyperplane \( H \) extending \( M \) such that \( \operatorname{rai}\left( C\right) \cap H = \varnothing \) . | Proof. The proof of the theorem is the same as the proof of Theorem 6.17 except that we replace \( \operatorname{ri}\left( C\right) \) in that proof by \( \operatorname{rai}\left( C\right) \) and invoke Theorem 6.33 instead of Theorem 6.15. | Yes |
Lemma 6.36. Let \( E \) be a real vector space, and \( {\left\{ {\ell }_{i}\right\} }_{1}^{k} \) a set of linear functionals on \( E \) . The system of strict linear inequalities\n\n\[ \n{\ell }_{i}\left( x\right) < {\alpha }_{i},\;i = 1,\ldots, k \n\]\n\n(6.10)\n\nis inconsistent if and only if there exist nonnegative... | Proof. Both sets of conditions cannot hold simultaneously, because if \( x \) satisfies (6.10), then we have the contradiction\n\n\[ \n0 < \mathop{\sum }\limits_{{i = 1}}^{k}{\lambda }_{i}\left( {{\alpha }_{i} - {\ell }_{i}\left( x\right) }\right) = \mathop{\sum }\limits_{{i = 1}}^{k}{\lambda }_{i}{\alpha }_{i} \leq 0.... | Yes |
Theorem 6.38. (Dubovitskii-Milyutin) Let \( {\left\{ {C}_{i}\right\} }_{1}^{k}, k > 1 \), be nonempty convex sets in a vector space \( E \), such that \( {\left\{ {C}_{i}\right\} }_{1}^{k - 1} \) are algebraically open, that is, \( \operatorname{ai}\left( {C}_{i}\right) = {C}_{i}, i = 1,\ldots, k - 1 \) . The following... | Proof. We first prove that (a) implies (b). Define the sets \[ {K}_{1} \mathrel{\text{:=}} \left\{ {\left( {{x}_{k},{x}_{k},\ldots ,{x}_{k}}\right) : {x}_{k} \in {C}_{k}}\right\} \] \[ {K}_{2} \mathrel{\text{:=}} {C}_{1} \times \cdots \times {C}_{k - 1}. \] We have, by elementary arguments, \[ \text{ai}{K}_{2} = \text{... | Yes |
Theorem 6.39. (Dubovitskii-Milyutin) Let \( {\left\{ {C}_{i}\right\} }_{1}^{k}, k > 1 \), be nonempty convex sets in a topological vector space \( E \), such that \( {\left\{ {C}_{i}\right\} }_{1}^{k - 1} \) are open, that is, \( \operatorname{int}\left( {C}_{i}\right) = {C}_{i}, i = 1,\ldots, k - 1 \) . The following ... | Proof. By virtue of Theorem 5.20, \( \operatorname{int}\left( {C}_{i}\right) = \operatorname{ai}\left( {C}_{i}\right) \) for \( i = 1,\ldots, k - 1 \), so it follows from Theorem 6.38 that we need to prove only that if (a) is true, then the linear functionals \( {\left\{ {\ell }_{i}\right\} }_{1}^{k} \) in (b) are cont... | Yes |
Corollary 6.42. Let \( C \) be a nonempty convex set in a vector space \( E \), such that \( \operatorname{rai}\left( C\right) \neq \varnothing \) . If \( M \subset E \) is an affine set satisfying \( \operatorname{rai}\left( C\right) \cap M = \varnothing \), then there exists a hyperplane \( H \supseteq M \) extending... | Proof. We assume without any loss of generality that \( 0 \in \operatorname{rai}\left( C\right) \) . Define the linear subspace \( L \mathrel{\text{:=}} \operatorname{span}M \) ; then \( M \) is a hyperplane in \( L \) given by the formula \( M = \{ x \in L : g\left( x\right) = 1\} \), where \( g \) is a linear functio... | Yes |
Lemma 7.2. A finitely generated cone is a closed set. | Proof. Let \( K \) be a finitely generated cone:\n\n\[ K = \left\{ {\mathop{\sum }\limits_{{j = 1}}^{k}{t}_{j}{a}_{j} : {t}_{j} \geq 0, j = 1,\ldots, k}\right\} . \]\n\nBy Carathéodory’s theorem (Theorem 4.21, p. 94), any point \( x \in K \) can be written as\n\n\[ x = \mathop{\sum }\limits_{{j = 1}}^{k}{\delta }_{j}{b... | Yes |
Lemma 7.3. The dual of finitely generated cone \( K = \overline{\operatorname{cone}}\left( {{a}_{1},\ldots ,{a}_{k}}\right) \) is the polyhedral cone \( L = { \cap }_{j = 1}^{k}\left\{ {x : \left\langle {{a}_{j}, x}\right\rangle \leq 0}\right\} \) . | Proof. Clearly, we have\n\n\[ \n{K}^{ * } = \left\{ {x : \left\langle {\mathop{\sum }\limits_{{j = 1}}^{k}{t}_{j}{a}_{j}, x}\right\rangle \leq 0\text{ for all }{t}_{j} \geq 0}\right\} \supseteq { \cap }_{1}^{k}\left\{ {x : \left\langle {{a}_{j}, x}\right\rangle \leq 0}\right\} .\n\]\n\nIf \( x \in {K}^{ * } \), choosin... | Yes |
Theorem 7.4. Let \( {a}_{1},\ldots ,{a}_{k} \in E \) . The finitely generated cone\n\n\[ K = \overline{\operatorname{cone}}\left( {{a}_{1},\ldots ,{a}_{k}}\right) \]\n\nand the polyhedral cone\n\n\[ L = \left\{ {x : \left\langle {{a}_{j}, x}\right\rangle \leq 0, j = 1,\ldots, k}\right\} \]\n\nare polars of each other, ... | Proof. It follows from Lemma 7.3 that \( {K}^{ * } = L \) . Since \( K \) is closed by Lemma 7.2, Theorem 6.19 implies that \( K = {\left( {K}^{ * }\right) }^{ * } = {L}^{ * } \) . | Yes |
Corollary 7.5. Let \( A \) be an \( n \times k \) matrix. Then the cones \( K = \{ {Av} : v \geq 0\} \) and \( L = \left\{ {x : {A}^{T}x \leq 0}\right\} \) are polars of each other. | Proof. Let \( A = \left\lbrack {{a}_{1}\ldots ,{a}_{k}}\right\rbrack \), where \( \left\{ {a}_{i}\right\} \) are the columns of \( A \) . Then \( K = \) \( \overline{\operatorname{cone}}\left( {{a}_{1},\ldots ,{a}_{k}}\right) \) and \( L = \left\{ {x : \left\langle {{a}_{j}, x}\right\rangle \leq 0, j = 1,\ldots, k}\rig... | Yes |
Theorem 7.6. (Farkas’s lemma, homogeneous version) Let \( {a}_{1},\ldots ,{a}_{k} \) be given vectors in \( E \) . The following statements are equivalent:\n\n(a) If \( x \in E \) satisfies the inequalities \( \left\langle {{a}_{i}, x}\right\rangle \leq 0, i = 1,\ldots, k \), then it also satisfies the inequality \( \l... | Proof. This is essentially a restatement of Theorem 7.4. Define\n\n\[ K = \left\{ {x : \left\langle {{a}_{i}, x}\right\rangle \leq 0, i = 1,\ldots, k}\right\} .\n\nPart (a) is equivalent to the statement, \( b \in {K}^{ * } \), whereas part (b) states that \( b \in \overline{\operatorname{cone}}\left( {{a}_{1},\ldots ,... | Yes |
Corollary 7.7. Let \( {c}_{1},\ldots ,{c}_{k},{a}_{1},\ldots ,{a}_{l} \) be given vectors in \( E \) . The following statements are equivalent:\n\n(a)\n\n\[ \left\lbrack {\left\langle {{c}_{i}, x}\right\rangle = 0, i = 1,\ldots, k,\left\langle {{a}_{j}, x}\right\rangle \leq 0, j = 1,\ldots, l}\right\rbrack \; \Rightarr... | Proof. The equality \( \left\langle {{c}_{i}, x}\right\rangle = 0 \) is equivalent to the inequalities \( \left\langle {{c}_{i}, x}\right\rangle \leq 0 \) and \( \left\langle {-{c}_{i}, x}\right\rangle \leq 0 \) . By Farkas’s lemma, part (a) is equivalent to\n\n\[ b \in \overline{\operatorname{cone}}\left( {{c}_{1},\ld... | Yes |
Theorem 7.8. Every finitely generated cone \( K \) is a convex polyhedral cone, and vice versa. | Proof. We first show that every finitely generated cone \( K \) is a polyhedral cone. Let \( K = \overline{\operatorname{cone}}\left( {{a}_{1},\ldots ,{a}_{k}}\right) \subseteq E \) be a finitely generated cone. We claim that \( K \) is a polyhedral cone using induction on \( k \) . If \( k = 1 \), then \( K = \{ {ta} ... | Yes |
Theorem 7.13. (Minkowski-Weyl) A nonempty set \( P \subseteq E \) is a convex polyhedron if and only if there exist vectors \( {\left\{ {v}_{i}\right\} }_{1}^{k} \) and \( {\left\{ {d}_{j}\right\} }_{1}^{l} \) such that\n\n\[ P = \operatorname{co}\left( {{v}_{1},\ldots ,{v}_{k}}\right) + \overline{\operatorname{cone}}\... | Proof. Let \( P \) be a convex polyhedron, say in the form (7.3). We show that \( P \) has the form (7.4). Define the polyhedral cone\n\n\[ K \mathrel{\text{:=}} \left\{ {\left( {x, t}\right) \in E \times \mathbb{R} : \left\langle {{a}_{j}, x}\right\rangle \leq {\alpha }_{j}t, t \geq 0, j = 1,\ldots, m}\right\} \]\n\n\... | Yes |
Lemma 7.14. If\n\n\[ P = \\left\\{ {x \\in E : \\left\\langle {{a}_{j}, x}\\right\\rangle \\leq {\\alpha }_{j}, j = 1,\\ldots, m}\\right\\} \]\n\nis a nonempty polyhedron, then\n\n\[ \\overline{K\\left( P\\right) } = \\left\\{ {\\left( {x, t}\\right) : x \\in E, t \\geq 0,\\left\\langle {{a}_{j}, x}\\right\\rangle \\le... | Proof. We claim that\n\n\[ \\operatorname{rec}\\left( P\\right) = \\left\\{ {d : \\left\\langle {{a}_{j}, d}\\right\\rangle \\leq 0, j = 1,\\ldots, m}\\right\\} \]\n\nLet \( {x}_{0} \\in P \) . If \( d \\in \\operatorname{rec}P \), then \( {x}_{0} + {td} \\in P \) for \( t > 0 \), that is, \( \\left\\langle {{a}_{j},{x... | Yes |
Lemma 7.15. Let\n\n\[ P = \operatorname{co}\left( {{v}_{1},\ldots ,{v}_{k}}\right) + \overline{\operatorname{cone}}\left( {{d}_{1},\ldots ,{d}_{l}}\right) \]\n\n\[ = \left\{ {\mathop{\sum }\limits_{{i = 1}}^{k}{\lambda }_{i}{v}_{i} + \mathop{\sum }\limits_{{j = 1}}^{l}{\mu }_{j}{d}_{j} : \mathop{\sum }\limits_{{i = 1}}... | Proof. We claim that \( \operatorname{rec}\left( P\right) = \overline{\operatorname{cone}}\left\{ {{d}_{j} : j = 1,\ldots, l}\right\} = : L \) . Clearly, we have that each \( {d}_{j} \) belongs to \( \operatorname{rec}\left( P\right) \), proving the inclusion \( L \subseteq \operatorname{rec}\left( P\right) \) . Conver... | Yes |
For a given vector \( 0 \neq a \in E \), consider the open half-space\n\n\[ C \mathrel{\text{:=}} \{ d \in E : \langle a, d\rangle < 0\} .\n\]\nThe dual cone \( {C}^{ * } \) is given by\n\n\[ {C}^{ * } = \overline{\operatorname{cone}}\left( a\right) = \{ {ta} : t \geq 0\} . | Proof. It is easily verified that \( {C}^{ * } = {\left( \bar{C}\right) }^{ * } \) for any set \( C \), and it is equally easy to see that \( \bar{C} = \{ d \in E : \langle a, d\rangle \leq 0\} \) . It then follows from Theorem 7.4 that \( {C}^{ * } = \overline{\operatorname{cone}}\left( a\right) . | No |
Theorem 7.19. (Motzkin's transposition theorem, affine version) Let \( A, B \), and \( C \) be matrices with the same number of rows. The linear system\n\n\[ \n{A}^{T}x < a,{B}^{T}x \leq b,{C}^{T}x = c \n\]\n\n(7.6)\n\nis inconsistent if and only if the system\n\n\[ \n{Ay} + {Bz} + {Cw} = 0,\langle a, y\rangle + \langl... | Proof. The system (7.6) is consistent if and only if the homogeneous system\n\n\[ \nt > 0,{A}^{T}x < {ta},{B}^{T}x \leq {tb},{C}^{T}x = {tc} \n\]\n\nin the variables \( \left( {x, t}\right) \), that is, the system\n\n\[ \n\left( {0, - 1}\right) \left( \begin{array}{l} x \\ t \end{array}\right) < 0,\;\left\lbrack {{A}^{... | Yes |
Theorem 7.20. (Farkas’s lemma, affine version) Let \( {\left\{ {a}_{i}\right\} }_{1}^{m},{a}_{i} \in E \) , \( {\left\{ {\alpha }_{i}\right\} }_{1}^{m},{\alpha }_{i} \in \mathbb{R} \), be given vectors and scalars. Suppose that the linear inequalities\n\n\[ \left\langle {{a}_{i}, x}\right\rangle \leq {\alpha }_{i},\;i ... | Proof. Define \( A = \left\lbrack {{a}_{1},\ldots ,{a}_{m}}\right\rbrack, a = {\left( {\alpha }_{1},\ldots ,{\alpha }_{m}\right) }^{T} \) . Then (a) is equivalent to the inconsistency of the system\n\n\[ {A}^{T}x \leq a,\; - {c}^{T}x < - \gamma . \]\n\nBy the affine version of Motzkin's transposition theorem (Theorem 7... | Yes |
Theorem 7.22. Let \( f \) be a Gâteaux differentiable function. Consider the optimization problem\n\n\[ \n\min \;f\left( x\right) \n\]\n\n\[ \n\text{s.t.}\left\langle {{a}_{i}, x}\right\rangle \geq {\beta }_{i},\;i = 1,\ldots, m\text{.} \n\]\n\nIf \( {x}^{ * } \) is a local minimizer of \( f \), then there exist nonneg... | Proof. The variational inequality for this problem is\n\n\[ \n\left\lbrack {\left\langle {{a}_{i}, x}\right\rangle \geq {\beta }_{i},\;i = 1,\ldots, m}\right\rbrack \; \Rightarrow \;\left\lbrack {\left\langle {\nabla f\left( {x}^{ * }\right), x}\right\rangle \geq \left\langle {\nabla f\left( {x}^{ * }\right) ,{x}^{ * }... | Yes |
Theorem 7.23. (Tucker [255]) If \( L \subseteq {\mathbb{R}}^{n} \) is a linear subspace and \( {L}^{ \bot } \) its orthogonal complement, then there exist vectors \( {x}^{ * } \in L \) and \( {y}^{ * } \in {L}^{ \bot } \) such that \( {x}^{ * } \) and \( {y}^{ * } \) are strictly complementary, that is, \( {x}^{ * } \g... | Proof. Write \( L = \{ x : {Ax} = 0\} \), where \( A \) is an \( m \times n \) matrix. Then \( {L}^{T} = \) \( {A}^{T}\left( {\mathbb{R}}^{m}\right) \) is the range of \( {A}^{T} \) . For each index \( i,1 \leq i \leq n \), it follows from the homogeneous version of Motzkin’s transposition theorem (Theorem 7.17) that e... | Yes |
Lemma 8.1. Assume that the linear program \( \left( P\right) \) has a nonempty feasible region. Then \( \left( P\right) \) has a solution if and only if its objective function \( \langle c, x\rangle \) is bounded from above on the constraint set. | Proof. Denote by\n\n\[ C \mathrel{\text{:=}} \left\{ {x : \left\langle {{a}_{i}, x}\right\rangle \leq {b}_{i}, i = 1,\ldots, m}\right\} \]\n\nthe constraint set of \( \left( P\right) \), and suppose that the objective function is bounded from above by a constant \( M < \infty \) . By Theorem 7.13, \( C \) has a represe... | Yes |
Theorem 8.2. Let the linear program \( \left( P\right) \) have a solution. A feasible point \( {x}^{ * } \) is a solution to \( \left( P\right) \) if and only if there exist multipliers \( {\left\{ {y}_{i}^{ * }\right\} }_{1}^{m} \) such that\n\n\[ \mathop{\sum }\limits_{{i = 1}}^{m}{y}_{i}^{ * }{a}_{i} = c,\;{y}_{i}^{... | Proof. Let \( {x}^{ * } \) and \( {y}^{ * } \) satisfy (8.1). If \( x \) is feasible, then\n\n\[ \langle c, x\rangle = \left\langle {\mathop{\sum }\limits_{{i = 1}}^{m}{y}_{i}^{ * }{a}_{i}, x}\right\rangle = \mathop{\sum }\limits_{{i = 1}}^{m}{y}_{i}^{ * }\left\langle {{a}_{i}, x}\right\rangle \leq \mathop{\sum }\limit... | Yes |
Corollary 8.4. Suppose that the linear program \( \left( P\right) \) has a solution. A feasible point \( {x}^{ * } \) is a solution to \( \left( P\right) \) if and only if there exist multipliers \( {\left\{ {y}_{i}^{ * }\right\} }_{1}^{m} \) satisfying the conditions\n\n\[ \mathop{\sum }\limits_{{i = 1}}^{m}{y}_{i}^{ ... | This follows immediately from the proof of the theorem above, since the condition \( \left\langle {c,{x}^{ * }}\right\rangle = \left\langle {b,{y}^{ * }}\right\rangle \) is equivalent to\n\n\[ \left\langle {b,{y}^{ * }}\right\rangle - \left\langle {c,{x}^{ * }}\right\rangle = \mathop{\sum }\limits_{{i = 1}}^{m}{y}_{i}^... | Yes |
Theorem 8.5. (Weak duality theorem for linear programming) If \( x \) is a feasible solution to \( \left( P\right) \) and \( y \) is a feasible solution to \( \left( D\right) \), then\n\n\[ \langle c, x\rangle \leq \langle b, y\rangle \] | Proof. This follows from the observation, already used above, that\n\n\[ \langle b, y\rangle - \langle c, x\rangle = \langle b, y\rangle - \mathop{\sum }\limits_{{i = 1}}^{m}{y}_{i}\left\langle {{a}_{i}, x}\right\rangle = \mathop{\sum }\limits_{{i = 1}}^{m}{y}_{i}\left( {{b}_{i} - \left\langle {{a}_{i}, x}\right\rangle... | Yes |
Theorem 8.6. (Strong duality theorem for linear programming) If \( \\left( P\\right) \) has an optimal solution \( {x}^{ * } \), then \( \\left( D\\right) \) has an optimal solution \( {y}^{ * } \) and the optimal objective values of \( \\left( P\\right) \) and \( \\left( D\\right) \) are the same, that is,\n\n\[ \n\\l... | This follows immediately from Theorem 8.2. | No |
Lemma 9.3. If \( {x}^{ * } \in \mathcal{F}\left( P\right) \) is a local minimum of \( \left( P\right) \), then\n\n\[ \mathcal{F}\mathcal{D}\left( {x}^{ * }\right) \cap \mathcal{S}\mathcal{D}\left( {f;{x}^{ * }}\right) = \varnothing . \]\n | Proof. The lemma is obvious: if the intersection is not empty, then there exists a sequence of feasible points \( {x}_{n} \rightarrow {x}^{ * } \) such that \( f\left( {x}_{n}\right) < f\left( {x}^{ * }\right) \), which contradicts our assumption that \( {x}^{ * } \) is a local minimizer of \( \left( P\right) \) . | Yes |
Theorem 9.4. (Fritz John) If a point \( {x}^{ * } \) is a local minimizer of \( \left( P\right) \) , then there exist multipliers \( \left( {\lambda ,\mu }\right) \mathrel{\text{:=}} \left( {{\lambda }_{0},{\lambda }_{1},\ldots ,{\lambda }_{r},{\mu }_{1},\ldots ,{\mu }_{m}}\right) \), not all zero, \( \left( {{\lambda ... | Proof. Because of the \ | No |
If the vectors\n\n\[ \n\\left\{ \\nabla {g}_{i}\\left( {x}^{ * }\\right), i \\in I\\left( {x}^{ * }\\right) ,\\nabla {h}_{j}\\left( {x}^{ * }\\right), j = 1,\\ldots, m\\right\} \n\]\n\nare linearly independent, then \( {\\lambda }_{0} > 0 \) and we have\n\n\[ \n\\nabla f\\left( {x}^{ * }\\right) + \\mathop{\\sum }\\lim... | Proof. If \( {\\lambda }_{0} = 0 \), then\n\n\[ \n\\mathop{\\sum }\\limits_{{i \\in I\\left( {x}^{ * }\\right) }}{\\lambda }_{i}\\nabla {g}_{i}\\left( {x}^{ * }\\right) + \\mathop{\\sum }\\limits_{{j = 1}}^{m}{\\mu }_{j}\\nabla {h}_{j}\\left( {x}^{ * }\\right) = 0. \n\]\n\nThe linear independence hypothesis of the vect... | Yes |
Theorem 9.7. Let \( {x}^{ * } \) be a feasible solution to the optimization problem \( \left( P\right) \) in (9.1), satisfying the FJ conditions (9.2) and (9.3), where we write (9.2) in the form\n\n\[ \n{\lambda }_{0}\nabla f\left( {x}^{ * }\right) + \mathop{\sum }\limits_{{i \in I\left( {x}^{ * }\right) }}{\lambda }_{... | Proof. Suppose that \( {x}^{ * } \) is not a local minimizer of \( \left( P\right) \) . Then there exists a feasible sequence of points \( {x}_{k} \rightarrow {x}^{ * } \) satisfying \( f\left( {x}_{k}\right) < f\left( {x}^{ * }\right) \) . Writing \( {x}_{k} = {x}^{ * } + {t}_{k}{d}_{k} \) with \( {t}_{k} > 0,\begin{V... | Yes |
Theorem 9.9. Let \( {x}^{ * } \) be an FJ point for problem (P) in (9.1). The KKT conditions\n\n\[ \nabla f\left( {x}^{ * }\right) + \mathop{\sum }\limits_{{i \in I\left( {x}^{ * }\right) }}{\lambda }_{i}\nabla {g}_{i}\left( {x}^{ * }\right) + \mathop{\sum }\limits_{{j = 1}}^{m}{\mu }_{0}\nabla {h}_{j}\left( {x}^{ * }\... | Proof. The equivalence of (9.12) and (9.11) follows immediately from the homogeneous version of Motzkin's transposition theorem. | No |
Corollary 9.10. (Concave and linear constraints) Let \( {x}^{ * } \) be a local minimizer of problem \( \left( P\right) \) in (9.1). The KKT conditions hold at \( {x}^{ * } \) if the active constraints \( {\left\{ {g}_{i}\right\} }_{i \in I\left( {x}^{ * }\right) } \) are concave functions in a convex neighborhood of \... | Proof. Let \( d \) satisfy the conditions\n\n\[ \left\langle {\nabla {g}_{i}\left( {x}^{ * }\right), d}\right\rangle \leq 0, i \in I\left( {x}^{ * }\right) ,\;\left\langle {\nabla {h}_{j}\left( {x}^{ * }\right), d}\right\rangle = 0, j = 1,\ldots, m.\]\n\nThe point \( x\left( t\right) = {x}^{ * } + {td} \) is feasible f... | Yes |
Theorem 9.11. (Mangasarian-Fromovitz [193]) Let \( {x}^{ * } \) be an FJ point for problem \( \left( P\right) \) in (9.1). If the gradients \( {\left\{ \nabla {h}_{j}\left( {x}^{ * }\right) \right\} }_{1}^{m} \) of the equality constraints are linearly independent and there exists a direction \( d \) satisfying the con... | Proof. On the one hand, since (9.13) is consistent, the homogeneous version of Motzkin’s transposition theorem implies that in any solution \( 0 \leq \lambda \mathrel{\text{:=}} \left( {{\lambda }_{i} : }\right. \) \( \left. {i \in I\left( {x}^{ * }\right) }\right) \) and \( \mu : \left( {{\mu }_{1},\ldots ,{\mu }_{m}}... | Yes |
Corollary 9.12. (Slater [243]) Let the functions \( {\left\{ {g}_{i}\right\} }_{1}^{r} \) in (9.1) be convex, and the functions \( {\left\{ {h}_{j}\right\} }_{1}^{r} \) affine. Let \( {x}^{ * } \) be a local minimizer of problem \( \left( P\right) \) . If there exists a feasible point \( {x}_{0} \), strictly feasible f... | Proof. Let \( {h}_{j}\left( x\right) = \left\langle {{a}_{j}, x - {x}_{0}}\right\rangle, j = 1,\ldots, m \) . If \( {\left\{ {a}_{j}\right\} }_{1}^{m} \) is linearly dependent, then we can choose a linearly independent subset of it, say \( {\left\{ {a}_{j}\right\} }_{1}^{k} \), such that \( \operatorname{span}\left\{ {... | Yes |
\[ \min {x}^{2} + 4{y}^{2} + {16}{z}^{2} \] \[ \text{s.t.}{xy} = 1\text{.} \] | Since the objective function is coercive, there exist global minimizer(s) to the problem. The constraint function \( h\left( {x, y, z}\right) = {xy} - 1 = 0 \) has the gradient \( \nabla h\left( {x, y, z}\right) = \left( {y, x,0}\right) \neq 0 \) on the constraint set \( {xy} = 1 \) . It follows from Corollary 9.6 that... | Yes |
Let \( Q \) be an \( n \times n \) symmetric matrix, \( c \in {\mathbb{R}}^{n} \), and \( \Delta > 0 \) . Consider the problem\n\n\[ \min \;q\left( x\right) \mathrel{\text{:=}} \frac{1}{2}\langle {Qx}, x\rangle + \langle c, x\rangle \]\n\n\[ \text{s.t.}\parallel x\parallel \leq \Delta \text{.} \] | Since the constraint function has a nonzero gradient at every point of the feasible region, we may assume that \( {\lambda }_{0} = 1 \) . We change the constraint to \( \parallel x{\parallel }^{2} \leq {\Delta }^{2} \), and write the Lagrangian function\n\n\[ L = \frac{1}{2}\langle {Qx}, x\rangle + \langle c, x\rangle ... | Yes |
Lemma 9.19. Let \( {x}^{ * } \) be a local minimizer of \( \left( P\right) \) satisfying the \( {KKT} \) conditions with multipliers \( {\lambda }^{ * },{\mu }^{ * } \) . If \( d \in {\mathbb{R}}^{n} \) is a feasible direction at \( {x}^{ * } \) with the property that there exists a sequence of feasible points \( {x}_{... | Proof. Let \( d \) and \( \left\{ {x}_{k}\right\} \) satisfy the assumptions of the lemma. Defining \( {d}_{k} = \) \( {x}_{k} - {x}^{ * } \), we have\n\n\[ \text{0} \]\n\n\[ \leq f\left( {x}_{k}\right) - f\left( {x}^{ * }\right) \]\n\n\[ = L\left( {{x}_{k},{\lambda }^{ * },{\mu }^{ * }}\right) - L\left( {{x}^{ * },{\l... | Yes |
Theorem 9.20. Let \( {x}^{ * } \) be a local minimizer of \( \left( P\right) \) satisfying the \( {KKT} \) conditions with multipliers \( {\lambda }^{ * },{\mu }^{ * } \) . If the active gradient vectors,\n\n\[ \nabla {g}_{i}\left( {x}^{ * }\right), i \in I\left( {x}^{ * }\right) ,\nabla {h}_{j}\left( {x}^{ * }\right),... | Proof. Since the active gradients at \( {x}^{ * } \) are linearly independent, it follows from Lyusternik’s theorem that \( M \) coincides with the set of tangent directions to the set\n\n\[ \left\{ {x : {g}_{i}\left( x\right) = 0, i \in I\left( {x}^{ * }\right) ,{h}_{j}\left( x\right) = 0, j = 1,\ldots, m}\right\} \]\... | Yes |
Theorem 9.21. Let \( {x}^{ * } \) be a feasible point for \( \left( P\right) \) that satisfies the \( {KKT} \) conditions with multipliers \( {\lambda }^{ * },{\mu }^{ * } \) . If\n\n\[ \left\langle {{\nabla }_{x}^{2}L\left( {{x}^{ * },{\lambda }^{ * },{\mu }^{ * }}\right) d, d}\right\rangle > 0 \]\n\nfor all \( d \neq... | Proof. Suppose that (9.23) is not satisfied, and let \( {\epsilon }_{k} \) be a sequence positive numbers converging to zero. Then there exists a sequence of feasible points \( {x}_{k} \rightarrow {x}^{ * } \) such that \( f\left( {x}_{k}\right) < f\left( {x}^{ * }\right) + {\epsilon }_{k}{\begin{Vmatrix}{x}_{k} - {x}^... | Yes |
Corollary 9.22. Let \( {x}^{ * } \) be a feasible point satisfying the KKT conditions with multipliers \( {\lambda }^{ * },{\mu }^{ * } \) . If \( {\lambda }_{i}^{ * } > 0 \) for all \( i \in I\left( {x}^{ * }\right) \) (this is called the strict complementarity condition) and the Hessian \( {\nabla }_{x}^{2}L\left( {{... | The corollary follows immediately from Theorem 9.21. | No |
Lemma 9.23. Let \( {x}^{ * } \) be a KKT point of the quadratic program\n\n\[ \min \{ f\left( x\right) \mathrel{\text{:=}} \frac{1}{2}\langle {Qx}, x\rangle + \langle c, x\rangle : {Ax} \leq b\} . \]\n\n\( \left( P\right) \)\n\nIf \( \langle {Qd}, d\rangle \geq 0 \) for all directions \( d \) satisfying the condition t... | Proof. The individual linear constraints of the program are \( \left\langle {{a}_{i}, x}\right\rangle \leq {b}_{i} \) where \( \left\{ {a}_{i}\right\} \) are the rows of \( A \) . Let \( x \) be a feasible point in a small enough neighborhood of \( {x}^{ * } \) and define \( d = x - {x}^{ * } \) . We have\n\n\[ f\left(... | Yes |
Theorem 10.2. (Courant-Fischer) Let \( A \) be an \( n \times n \) symmetric real matrix with eigenvalues\n\n\[ \n{\lambda }_{1}\left( A\right) \leq {\lambda }_{2}\left( A\right) \leq \cdots \leq {\lambda }_{n}\left( A\right)\n\]\n\narranged in ascending order. Then\n\n\[ \n{\lambda }_{k}\left( A\right) = \mathop{\max ... | Proof. We will prove only the first equality, since the second one follows from the first applied to the matrix \( - A \) . Let \( {u}_{i} \) be the eigenvector corresponding to \( {\lambda }_{i} \) obtained in problem \( \left( {P}_{i}\right) \) . Denote by \( {L}_{k}^{ * } \) the linear span of \( {\left\{ {u}_{i}\ri... | Yes |
Corollary 10.3. (Weyl) Let \( A, B \) be \( n \times n \) symmetric real matrix, and \( {\left\{ {\lambda }_{i}\left( A\right) \right\} }_{1}^{n},{\left\{ {\lambda }_{i}\left( B\right) \right\} }_{1}^{n} \), and \( {\left\{ {\lambda }_{i}\left( A + B\right) \right\} }_{1}^{n} \) the eigenvalues of \( A, B \), and \( A ... | Proof. We prove only the first inequality; the second one is proved similarly:\n\n\[ \n{\lambda }_{i}\left( {A + B}\right) = \mathop{\max }\limits_{{L}_{i - 1}}\mathop{\min }\limits_{{x \in {L}_{i - 1}^{ \bot },\parallel x\parallel = 1}}\langle \left( {A + B}\right) x, x\rangle \n\]\n\n\[ \n\geq \mathop{\max }\limits_{... | Yes |
Corollary 10.5. Let \( A \) be an \( n \times n \) symmetric real matrix, and \( {A}_{k}{ak} \times \) \( k \) principal submatrix of \( A \) that is obtained by deleting \( n - k \) rows and the corresponding columns of \( A \) . If \( {\left\{ {\lambda }_{i}\left( A\right) \right\} }_{1}^{n} \) and \( {\left\{ {\lamb... | Proof. We have\n\n\[{\lambda }_{i}\left( {A}_{k}\right) = \mathop{\max }\limits_{{M}_{i - 1}}\mathop{\min }\limits_{{x \in {M}_{i - 1}^{ \bot },\parallel x\parallel = 1}}\left\langle {{A}_{k}x, x}\right\rangle\]\n\n\[\leq \mathop{\max }\limits_{{{L}_{n - k + i - 1}x \in {L}_{n - k + i - 1}^{ \bot }}}\mathop{\min }\limi... | Yes |
Theorem 10.6. (Singular-value decomposition of a matrix) Let \( A \) be an \( m \times n \) real matrix. There exist an orthogonal \( m \times m \) matrix \( U \), an orthogonal \( n \times n \) matrix \( V \), and an \( m \times n \) matrix \( \sum \) whose only nonzero elements are the diagonal entries \( {\sum }_{ii... | Proof. We consider the optimization problem \[ \min - \langle {Ax}, y\rangle \] \[ \text{s. t.}\parallel x{\parallel }^{2} - 1 = 0\text{,} \] \[ \parallel y{\parallel }^{2} - 1 = 0. \] Since the feasible set is compact, there exists a global minimizer \( \left( {{x}^{ * },{y}^{ * }}\right) = \left( {{v}_{1},{u}_{1}}\ri... | Yes |
Example 10.8. (Broyden's method)\n\nThis problem appears in Broyden's method for approximating a root of a nonlinear system of equations \( G\\left( x\\right) = 0 \), where \( G : {\\mathbb{R}}^{n} \\rightarrow {\\mathbb{R}}^{n} \) is a nonlinear map. It is the problem\n\n\\[ \n\\min \\parallel X{\\parallel }_{F}^{2}\n... | This norm is a Euclidean norm that comes from the trace inner product\n\n\\[\n\\langle X, Y\\rangle = \\operatorname{tr}\\left( {{X}^{T}Y}\\right) = \\mathop{\\sum }\\limits_{{i, j = 1}}^{n}{x}_{ij}{y}_{ij}\n\\]\n\non \( {\\mathbb{R}}^{n \\times n} \), the vector space of \( n \\times n \) matrices. Thus,\n\n\\[\n\\par... | Yes |
Theorem 11.1. Let \( L : {C}_{1} \times {C}_{2} \rightarrow \mathbb{R} \) be a continuous function where \( {C}_{1} \subset {\mathbb{R}}^{n} \) and \( {C}_{2} \subset {\mathbb{R}}^{m} \) are compact convex sets. If \( L \) is a convex-concave function, that is, \( x \mapsto L\left( {x, y}\right) \) is convex for fixed ... | \[ \mathop{\min }\limits_{{x \in {C}_{1}}}\mathop{\max }\limits_{{y \in {C}_{2}}}L\left( {x, y}\right) = \mathop{\max }\limits_{{y \in {C}_{2}}}\mathop{\min }\limits_{{x \in {C}_{1}}}L\left( {x, y}\right) . \] | Yes |
Theorem 11.2. (Weak duality theorem) If (P) and (D) are the associated primal and dual programs with \( L \), then\n\n\[ \sup \left( D\right) \leq \inf \left( P\right) \]\n\nthat is,\n\n\[ \mathop{\sup }\limits_{{y \in \mathcal{B}}}\mathop{\inf }\limits_{{x \in \mathcal{A}}}L\left( {x, y}\right) \leq \mathop{\inf }\lim... | Proof. It is easy to see that\n\n\[ \mathop{\inf }\limits_{{x \in \mathcal{A}}}L\left( {x, y}\right) \leq L\left( {u, y}\right) \text{ for all }u \in \mathcal{A}, y \in \mathcal{B}. \]\n\nThis implies that\n\n\[ \mathop{\sup }\limits_{{y \in \mathcal{B}}}\mathop{\inf }\limits_{{x \in \mathcal{A}}}L\left( {x, y}\right) ... | Yes |
Corollary 11.3. Let \( L \) be a Lagrangian function, and let \( \left( P\right) \) and \( \left( D\right) \) be the associated primal and dual programs with \( L \) . If the primal minimization problem \( \left( P\right) \) is unbounded, then the dual maximization problem \( \left( D\right) \) is infeasible. | Proof. \( \left( P\right) \) is unbounded if and only if \( \inf \left( P\right) = - \infty \) . Then \( \sup \left( D\right) = - \infty \) as well, which means that \( \left( D\right) \) is has an empty feasible region. | Yes |
Theorem 11.5. (Saddle point theorem) Let \( L : \mathcal{A} \times \mathcal{B} \rightarrow \mathbb{R} \) and \( \left( {{x}^{ * },{y}^{ * }}\right) \in \) \( \mathcal{A} \times \mathcal{B} \) . The following conditions are equivalent:\n\n(a) \( \left( {{x}^{ * },{y}^{ * }}\right) \) is a saddle point of \( L\left( {x, ... | Proof. Suppose that (a) holds. We have\n\n\[ \mathop{\sup }\limits_{{y \in \mathcal{B}}}L\left( {{x}^{ * }, y}\right) = \mathop{\max }\limits_{{y \in \mathcal{B}}}L\left( {{x}^{ * }, y}\right) = L\left( {{x}^{ * },{y}^{ * }}\right) = \mathop{\min }\limits_{{x \in \mathcal{A}}}L\left( {x,{y}^{ * }}\right) = \mathop{\inf... | Yes |
Corollary 11.6. The set of saddle points of \( L : \mathcal{A} \times \mathcal{B} \rightarrow \mathbb{R} \) is a direct product \( {\mathcal{A}}_{0} \times {\mathcal{B}}_{0} \), where \( {\mathcal{A}}_{0} \subseteq \mathcal{A} \) and \( {\mathcal{B}}_{0} \subseteq \mathcal{B} \) . | Proof. This result follows immediately from Theorem 11.5. An independent proof runs as follows: suppose \( \left( {{x}_{1}^{ * },{y}_{1}^{ * }}\right) \) and \( \left( {{x}_{2}^{ * },{y}_{2}^{ * }}\right) \) are saddle points of \( L \) . We need to show that the points \( \left( {{x}_{1}^{ * },{y}_{2}^{ * }}\right) \)... | Yes |
A point \( \left( {{x}^{ * },{\lambda }^{ * },{\mu }^{ * }}\right) \) is a saddle point of the Lagrangian \( L \) , that is,\n\n\[ L\left( {{x}^{ * },\lambda ,\mu }\right) \leq L\left( {{x}^{ * },{\lambda }^{ * },{\mu }^{ * }}\right) \leq L\left( {x,{\lambda }^{ * },{\mu }^{ * }}\right) \text{ for all }x \in C,0 \leq \... | Proof. Theorem 11.5 implies immediately that \( \left( {{x}^{ * },{\lambda }^{ * },{\mu }^{ * }}\right) \) is a saddle point if and only if \( \left( i\right) - \left( {iii}\right) \) hold, and if \( \left( {{x}^{ * },{\lambda }^{ * },{\mu }^{ * }}\right) \) is a saddle point, then \( \left( {iv}\right) \) holds, that ... | Yes |
Corollary 11.9. If the functions \( f,{\left\{ {g}_{i}\right\} }_{1}^{r} \), and \( {\left\{ {h}_{j}\right\} }_{1}^{m} \) are differentiable, and \( C = {\mathbb{R}}^{n} \), then the KKT conditions hold,\n\n\[ 0 = {\nabla }_{x}L\left( {{x}^{ * },{\lambda }^{ * },{\mu }^{ * }}\right) = \nabla f\left( {x}^{ * }\right) + ... | Proof. The inequality \( L\left( {{x}^{ * },{\lambda }^{ * },{\mu }^{ * }}\right) \leq L\left( {x,{\lambda }^{ * },{\mu }^{ * }}\right) \) shows that \( {x}^{ * } \) minimizes \( L\left( {x,{\lambda }^{ * },{\mu }^{ * }}\right) \) over \( {\mathbb{R}}^{n} \) . This immediately implies the corollary. | No |
Lemma 11.11. Let \( {\lambda }^{ * } \in {\mathbb{R}}_{ + }^{r} \) and \( {\mu }^{ * } \in {\mathbb{R}}^{m} \) be arbitrary. If \( {x}^{ * } \in C \) is a global minimizer of the function\n\n\[ L\left( {x,{\lambda }^{ * },{\mu }^{ * }}\right) = f\left( x\right) + \mathop{\sum }\limits_{{i = 1}}^{r}{\lambda }_{i}^{ * }{... | Proof. If \( x \) satisfies the constraints of \( \left( P\right) \), then\n\n\[ f\left( {x}^{ * }\right) = L\left( {{x}^{ * },{\lambda }^{ * },{\mu }^{ * }}\right) - \mathop{\sum }\limits_{{i = 1}}^{r}{\lambda }_{i}^{ * }{g}_{i}\left( {x}^{ * }\right) - \mathop{\sum }\limits_{{j = 1}}^{m}{\mu }_{j}^{ * }{h}_{j}\left( ... | Yes |
Lemma 11.13. Let \( l \) be a nonnegative affine function on a convex set \( C \subseteq \) \( {\mathbb{R}}^{n} \). If \( l\left( {x}_{0}\right) = 0 \) at some point in \( {x}_{0} \in \operatorname{ri}\left( C\right) \), then \( l \) is identically zero on \( C \). | Proof. Suppose that \( l\left( x\right) > 0 \) at some point \( x \in C \). Since \( {x}_{0} \in \operatorname{ri}\left( C\right) \), there exists \( t > 1 \) such that the point \( {x}_{1} = x + t\left( {{x}_{0} - x}\right) = \left( {1 - t}\right) x + t{x}_{0} \) is in \( C \). This gives the contradiction \( 0 \leq l... | Yes |
Theorem 11.14. (Convex transposition theorem) Let \( C \subseteq {\mathbb{R}}^{n} \) be a nonempty convex set, \( {\left\{ {g}_{i}\right\} }_{1}^{r} \) convex functions, \( {\left\{ {h}_{j}\right\} }_{1}^{m} \) affine functions on \( {\mathbb{R}}^{n} \) , such that \( \operatorname{dom}\left( {g}_{i}\right) \) contain ... | Proof. The statements (a) and (b) cannot both be true: if \( \bar{x} \in C \) satisfies (a) and \( \left( {\lambda ,\mu }\right) \) satisfies (b), then we have a contradiction\n\n\[ \n0 \leq \mathop{\sum }\limits_{{i = 1}}^{r}{\lambda }_{i}{g}_{i}\left( \bar{x}\right) + \mathop{\sum }\limits_{{j = 1}}^{m}{\mu }_{j}{h}_... | Yes |
Theorem 11.15. (Strong duality theorem of convex programming) Suppose that the convex program \( \left( P\right) \) in (11.4) has a finite infimum, that is, \( - \infty < \inf \left( P\right) < \infty \), and the conditions \( C \subseteq \operatorname{dom}\left( f\right), C \subseteq \operatorname{dom}\left( {g}_{i}\r... | Proof. Define \( {f}^{ * } \mathrel{\text{:=}} \inf \left( P\right) \), and consider the system of constraints\n\n\[ f\left( x\right) - {f}^{ * } < 0 \]\n\n\[ {g}_{i}\left( x\right) < 0,\;i = 1,\ldots, r \]\n\n\[ {h}_{j}\left( x\right) \leq 0,\;j = 1,\ldots, m \]\n\n\[ x \in C\text{.} \]\n\nTheorem 11.14 applies to the... | Yes |
Let the convex program \( \left( P\right) \) have only linear constraints;\n\n\[ \min \;f\left( x\right) \]\n\n\[ \text{s.t.}\;{h}_{j}\left( x\right) \leq 0,\;j = 1,\ldots, m\text{,} \]\n\n\( \left( P\right) \)\n\n\[ x \in {\mathbb{R}}^{n} \]\n\nand an objective function \( f \) with \( \operatorname{dom}\left( f\right... | Furthermore, if \( \left( P\right) \) has an optimal solution \( {x}^{ * } \), then \( \left( {{x}^{ * },{\mu }^{ * }}\right) \) is a saddle point of the Lagrangian function\n\n\[ L\left( {x,\mu }\right) = f\left( x\right) + \mathop{\sum }\limits_{{j = 1}}^{m}{\mu }_{j}{h}_{j}\left( x\right) .\n\] | Yes |
Theorem 11.18. Let \( \left( P\right) \) be the convex quadratic program above. The dual program is also a convex quadratic program.\n\nIf \( \left( P\right) \) is unbounded, then \( \left( D\right) \) is infeasible.\n\nIf \( \left( P\right) \) is feasible and bounded from below, then both \( \left( P\right) \) and \( ... | Proof. If \( \left( P\right) \) is unbounded, then Corollary 11.3 implies that \( \left( D\right) \) is infeasible. If \( \left( P\right) \) is feasible and bounded from below, then Corollary 11.17 on page 287 implies that the dual program \( \left( D\right) \) has an optimal solution \( \left( {{\lambda }^{ * },{\mu }... | Yes |
Lemma 11.20. Let \( {a}_{1},\ldots ,{a}_{m} \) be real numbers. We have\n\n\[ \max \left\{ {{a}_{1},\ldots ,{a}_{m}}\right\} = \mathop{\max }\limits_{{\lambda \in {\Delta }_{m - 1}}}\mathop{\sum }\limits_{{i = 1}}^{m}{\lambda }_{i}{a}_{i} \]\n\nwhere \( {\Delta }_{m - 1} \) is the standard unit simplex in \( {\mathbb{R... | Proof. We may assume that \( {a}_{1} = \max \left\{ {{a}_{1},\ldots ,{a}_{m}}\right\} \) without any loss of generality. If \( \lambda \in {\sum }_{m} \), we clearly have\n\n\[ \mathop{\sum }\limits_{{i = 1}}^{m}{\lambda }_{i}{a}_{i} \leq \mathop{\sum }\limits_{{i = 1}}^{m}{\lambda }_{i}{a}_{1} = {a}_{1} \]\n\nThis pro... | Yes |
Lemma 11.21. If \( i \notin I\left( {x}^{ * }\right) \), then \( {\lambda }_{i}^{ * } = 0 \) . | Proof. We have\n\n\[ \n{f}^{ * } \mathrel{\text{:=}} \mathop{\max }\limits_{{i \in I\left( {x}^{ * }\right) }}{f}_{i}\left( {x}^{ * }\right) = \mathop{\max }\limits_{{1 \leq i \leq m}}{f}_{i}\left( {x}^{ * }\right) = \mathop{\sum }\limits_{{i = 1}}^{m}{\lambda }_{i}^{ * }{f}_{i}\left( {x}^{ * }\right) \n\]\n\n\[ \n= \l... | Yes |
Theorem 11.23. Let \( E \) and \( F \) be two finite-dimensional Euclidean spaces. Let \( A : E \rightarrow F \) be a linear operator, \( {A}^{ * } : F \rightarrow E \) its adjoint, \( c \in E, b \in F \) , and \( K \subset E \) a regular convex cone in \( E \) . Consider the conic programming pairs\n\n\[ \min \langle ... | Proof. Consider the Lagrangian function \( L : K \times F \rightarrow \mathbb{R} \) ,\n\n\[ L\left( {x, y}\right) \mathrel{\text{:=}} \langle c, x\rangle + \langle b - {Ax}, y\rangle = \langle b, y\rangle + \left\langle {c - {A}^{ * }y, x}\right\rangle ,\] and observe that \( \left( P\right) \) can be written as the mi... | Yes |
Corollary 11.24. Consider the conic programming pairs\n\n\\( \\min \\;\\langle c, x\\rangle \\)\n\\[\n\\begin{array}{ll} \\max & \\mathop{\\sum }\\limits_{{i = 1}}^{m}{b}_{i}{y}_{i} \\\\ \\text{ s. t. } & \\mathop{\\sum }\\limits_{{i = 1}}^{m}{y}_{i}{a}_{i} + s = c, \\\\ & s \\in {K}^{ * }. \\end{array}\n\\]\n\n\\( \\l... | The corollary follows immediately from Theorem 11.23 by defining the linear operator \\( A : E \\rightarrow {\\mathbb{R}}^{m} \\) by\n\n\\[\n{Ax} \\mathrel{\\text{:=}} {\\left( \\left\\langle {a}_{1}, x\\right\\rangle ,\\ldots ,\\left\\langle {a}_{1}, x\\right\\rangle \\right) }^{T},\n\\]\n\nand noting that\n\n\\[\n\\l... | Yes |
Theorem 11.26. (Hoffman’s lemma) There exists a constant \( c\left( A\right) > 0 \) , which depends only on the matrix \( A \) defining the polyhedron \( P = \{ z : {Az} \leq b\} \) , such that\n\n\[ d\left( {x, P}\right) \leq c\left( A\right) \begin{Vmatrix}{\left( Ax - b\right) }^{ + }\end{Vmatrix}\text{ for all }x \... | Proof. In its broad outlines, the idea of the proof is simple. The most important idea is the second equation below, which follows from the fact that \( \parallel v\parallel = \mathop{\max }\limits_{{\parallel z\parallel \leq 1}}\langle v, z\rangle \) for a vector in \( {\mathbb{R}}^{n} \) . Assuming that equality hold... | Yes |
Lemma 12.1. Let \( K \subset {\mathbb{R}}^{n} \) be a compact set. Then \( 0 \in \operatorname{co}\left( K\right) \) if and only if\n\n\[ \{ h : \langle x, h\rangle < 0\text{ for all }x \in K\} = \varnothing . \] | Proof. If \( 0 \in \operatorname{co}\left( K\right) \), then \( \mathop{\sum }\limits_{i}{\lambda }_{i}{x}_{i} = 0 \) for some \( {x}_{i} \in K \) and \( 0 \leq \lambda \neq 0 \) . Then \( \mathop{\sum }\limits_{i}{\lambda }_{i}\left\langle {{x}_{i}, h}\right\rangle = 0 \) and we cannot have \( \left\langle {{x}_{i}, h... | Yes |
Theorem 12.2. Consider the semi-infinite program\n\n\\[ \n\\mathop{\\min }\\limits_{{x \\in X}}\\mathop{\\max }\\limits_{{y \\in Y}}\\;f\\left( {x, y}\\right)\n\\]\n\n\\[\n\\text{s.t.}g\\left( {x, z}\\right) \\leq 0,\\;z \\in Z\\text{,}\n\\]\n\nwhere \\( f\\left( {x, y}\\right) \\) and \\( {\\nabla }_{x}f\\left( {x, y}... | Proof. Define the functions\n\n\\[\n\\varphi \\left( x\\right) \\mathrel{\\text{:=}} \\mathop{\\max }\\limits_{{y \\in Y}}f\\left( {x, y}\\right)\n\\]\n\n\\[\n\\gamma \\left( x\\right) \\mathrel{\\text{:=}} \\mathop{\\max }\\limits_{{z \\in Z}}g\\left( {x, z}\\right)\n\\]\n\n\\[\n\\vartheta \\left( x\\right) \\mathrel{... | Yes |
Lemma 12.6. If \( K \subset {\mathbb{R}}^{n} \) is a convex body, then there exists an ellipsoid of minimum volume circumscribing \( K \) . | Proof. We claim that the feasible sublevel sets in the problem (12.6) are compact; then the Weierstrass theorem implies that (12.6) has an optimal solution. Being a convex body, \( K \) contains a ball of radius \( r \), which is then contained in every ellipsoid \( E\left( {X, c}\right) \supseteq K \) containing \( K ... | Yes |
Theorem 12.7. Let \( K \subset {\mathbb{R}}^{n} \) be a convex body. There exists an ellipsoid of minimum volume circumscribing \( K \). If \( E\left( {X, c}\right) \) is such an ellipsoid, then there exist a multiplier vector \( \lambda = \left( {{\lambda }_{1},\ldots ,{\lambda }_{k}}\right) > 0,0 \leq k \leq n\left( ... | Proof. The existence of a minimum-volume ellipsoid has already been proved in Lemma 12.6. Let \( E\left( {X, c}\right) \) be such an ellipsoid. Since the constraints in (12.6) are indexed by \( y \in K \), a compact set, Theorem 12.4 applies. Therefore, there exist a nonzero multiplier vector \( \left( {{\lambda }_{0},... | Yes |
Theorem 12.9. Let \( K \) be a convex body in \( {\mathbb{R}}^{n} \). The minimum-volume ellipsoid circumscribing \( K \) is unique. Moreover, the optimality conditions (12.7) are necessary and sufficient for an ellipsoid \( E\left( {X, c}\right) \) to be the minimum-volume ellipsoid circumscribing \( K \). | Proof. The necessity of the conditions (12.7) has already been proved in Theorem 12.7. To prove the sufficiency, we assume, without loss of generality, that \( E\left( {I,0}\right) = B \) satisfies the optimality conditions (12.8) for some set of multipliers \( \left\{ {\lambda }_{i}\right\} \). Let \( E \supseteq K \)... | Yes |
Theorem 12.11. Let \( K \) be a convex body in \( {\mathbb{R}}^{n} \) and \( E\left( {X, c}\right) = {E}^{K} \) its optimal circumscribing ellipsoid. The ellipsoid with the same center \( c \) but shrunk by a factor \( n \) is contained in \( K \) . If \( K \) is symmetric, that is, \( K = - K \), then the ellipsoid wi... | Proof. Without loss of generality, we assume that \( {E}^{K} = E\left( {I,0}\right) = B \) . The theorem states that \( {n}^{-1}B \subseteq K \) . Let\n\n\[ P = \operatorname{co}\left( {\left\{ {u}_{i}\right\} }_{1}^{k}\right) \]\n\nbe the convex hull of the contact points. We claim that the stronger statement \( {n}^{... | Yes |
Theorem 12.12. Let \( K \subset {\mathbb{R}}^{n} \) be a convex body. There exists an ellipsoid of maximum-volume inscribed in \( K \) . If \( E\left( {X, c}\right) \) is such an ellipsoid, then there exist a multiplier vector \( \lambda = \left( {{\lambda }_{1},\ldots ,{\lambda }_{k}}\right) > 0,0 \leq k \leq n\left( ... | Proof. The existence of a maximum-volume ellipsoid inscribed in \( K \) has already been proved above. Let \( E\left( {X, c}\right) \) denote this ellipsoid, and define \( Y = \) \( {X}^{-1} \) . Since the constraints in (12.11) are indexed by \( \parallel d\parallel = 1 \), Theorem 12.4 applies. Therefore, there exist... | Yes |
Theorem 12.13. Let \( K \) be a convex body in \( {\mathbb{R}}^{n} \). The maximum-volume ellipsoid inscribed in \( K \) is unique. Furthermore, the optimality conditions (12.12) are necessary and sufficient for an ellipsoid \( E\left( {X, c}\right) \) to be the maximum-volume ellipsoid inscribed in \( K \). | The proof uses Lemma 12.8. It is omitted, since it is very similar to the proof of Theorem 12.9. | No |
Theorem 12.14. Let \( K \) be a convex body in \( {\mathbb{R}}^{n} \) and let \( E\left( {X, c}\right) = {E}_{K} \) be its optimal inscribed ellipsoid. The ellipsoid with the same center \( c \) but blown up by a factor \( n \) contains \( K \) . If \( K = - K \) is symmetric, then the ellipsoid with the same center \(... | Proof. The proof here is similar to the proof of Theorem 12.11. Without loss of generality, we assume that \( {E}_{K} = E\left( {I,0}\right) = B \) . The first part of the theorem follows if we can prove the claim that\n\n\[ K \subseteq {P}^{ * } \subseteq {nB} \]\n\nSince \( 1 = {s}_{B}\left( {u}_{i}\right) = {s}_{K}\... | Yes |
Lemma 12.17. Let \( {\left\{ {x}_{i}\right\} }_{1}^{l} \subset {\mathbb{R}}^{n} \) and \( {\left\{ {y}_{i}\right\} }_{1}^{l} \subset {\mathbb{R}}^{m} \) be points satisfying (12.18). Let \( x \in {\mathbb{R}}^{n} \) be given. There exists a point \( y \in {\mathbb{R}}^{m} \) such that\n\n\[ \n\begin{Vmatrix}{y - {y}_{i... | Proof. If \( x = {x}_{j} \) for some \( j \), then the lemma holds with the choice of \( y = {y}_{j} \) ; thus we consider the case that \( x \) is distinct from \( {\left\{ {x}_{i}\right\} }_{1}^{l} \). Consider the minimax problem\n\n\[ \n\mathop{\min }\limits_{{v \in {\mathbb{R}}^{m}}}\mathop{\max }\limits_{{1 \leq ... | No |
Theorem 12.18. If \( S \subset {\mathbb{R}}^{m} \) and \( f : S \rightarrow {\mathbb{R}}^{n} \) is Lipschitz continuous, then \( f \) has an extension to a function \( \widetilde{f} : {\mathbb{R}}^{m} \rightarrow {\mathbb{R}}^{n} \) having the same Lipschitz constant as \( f \) . | Proof. Without loss of generality, we may assume that \( f \) is nonexpansive, that is, \( \begin{Vmatrix}{f\left( {x}_{1}\right) - f\left( {x}_{2}\right) }\end{Vmatrix} \leq \begin{Vmatrix}{{x}_{1} - {x}_{2}}\end{Vmatrix} \) for all \( {x}_{1},{x}_{2} \in S \) . By Zorn’s lemma, \( f \) has a maximal extension to a no... | Yes |
Theorem 13.1. (Radon) If \( A \subseteq {\mathbb{R}}^{n} \) is an affinely dependent set, then \( A \) can be partitioned into two sets \( B, C \) such that \( \operatorname{co}\left( B\right) \cap \operatorname{co}\left( C\right) \neq \varnothing \) . | Proof. Pick an affinely dependent set \( {\left\{ {x}_{i}\right\} }_{1}^{k} \) in \( A \) ; then there exists \( \lambda \mathrel{\text{:=}} \) \( \left( {{\lambda }_{1},\ldots ,{\lambda }_{k}}\right) \neq 0 \) such that\n\n\[ \mathop{\sum }\limits_{{i = 1}}^{k}{\lambda }_{i}{x}_{i} = 0\;\text{ and }\;\mathop{\sum }\li... | Yes |
Theorem 13.2. (Helly) Let \( {\left\{ {A}_{i}\right\} }_{i = 1}^{k} \) be a finite collection of convex sets in \( {\mathbb{R}}^{n} \) . If the intersection of any \( n + 1 \) sets from this collection is nonempty, then \( { \cap }_{i = 1}^{k}{A}_{i} \neq \varnothing . \) | Proof. The theorem is trivially true for \( k \leq n + 1 \), so we consider the case \( k > n + 1 \) . The proof is by induction on \( k \) . Suppose that the theorem has been proved for \( k - 1 \), and let \( {\left\{ {A}_{i}\right\} }_{1}^{k} \) be a collection of convex sets satisfying the hypothesis of the theorem... | Yes |
Example 13.3. Let \( K \) and \( {\left\{ {C}_{i}\right\} }_{i = 1}^{k}, k > n + 1 \), be convex sets in \( {\mathbb{R}}^{n} \). (a) If the intersection of every \( n + 1 \) of the sets \( {C}_{i} \) contains a translated copy of \( K \), then \( { \cap }_{1}^{k}{C}_{i} \) must also contain a translated copy of \( K \)... | To prove (a), define the sets \( {D}_{i} \mathrel{\text{:=}} \left\{ {x \in {\mathbb{R}}^{n} : K \subseteq x + {C}_{i}}\right\} \) ; it is easy to verify that the set \( {D}_{i} \) is convex. Since \( x \in {D}_{i} \) means that \( - x + K \subseteq {C}_{i} \), the statement \( x \in { \cap }_{j = 1}^{n + 1}{D}_{{i}_{j... | Yes |
Corollary 13.4. Let \( {\left\{ {A}_{\alpha }\right\} }_{\alpha \in \mathcal{A}} \) be any collection of closed convex sets in \( {\mathbb{R}}^{n} \) such that some finite intersection of sets from this collection is bounded and nonempty. If the intersection of any \( n + 1 \) sets from this collection is nonempty, the... | Proof. If \( \mathcal{F} \) is any finite subset of \( \mathcal{A} \), it follows from Theorem 13.2 that \( { \cap }_{\alpha \in \mathcal{F}}{A}_{\alpha } \neq \varnothing \) . Let \( {A}_{0} \mathrel{\text{:=}} { \cap }_{i \in {\mathcal{F}}_{0}}{A}_{i} \neq \varnothing \) be a bounded set, and define \( {\widehat{A}}_... | Yes |
Theorem 13.5. (Kirchberger) Let \( S \) and \( T \) be two finite subsets of \( {\mathbb{R}}^{n} \) . The sets \( S \) and \( T \) can be strictly separated if and only if every subset of \( S \) and \( T \) , consisting of at most \( n + 2 \) points can be strictly separated. | Proof. We may assume that \( \left| {S \cup T}\right| \geq n + 2 \) . For each \( s \in S \) and each \( t \in T \) , define the open half-spaces in \( {\mathbb{R}}^{n + 1} \)\n\n\[ \n{I}_{s} \mathrel{\text{:=}} \left\{ {\left( {{\lambda }_{0},\lambda }\right) \in \mathbb{R} \times {R}^{n} : \langle s,\lambda \rangle >... | Yes |
Theorem 13.6. Consider the problem\n\n\[ \rho \mathrel{\text{:=}} \inf \{ f\left( x\right) : g\left( {x, y}\right) \leq 0, y \in Y\} \]\n\n\( \left( P\right) \)\n\nwhere \( x \) belongs to a convex set in \( {\mathbb{R}}^{n}, f\left( x\right) \) and \( g\left( {x, y}\right) \) are lower semicontinuous convex functions ... | Proof. We assume that \( \left| Y\right| > n \) ; otherwise the theorem is trivial. For each subset \( Z \subset Y \), define the corresponding feasible set \( {F}_{Z} \mathrel{\text{:=}} \{ x : g\left( {x, y}\right) \leq 0, y \in Z\} \) ; for convenience, set \( {F}_{y} \mathrel{\text{:=}} {F}_{\{ y\} } \). Define\n\n... | Yes |
Consider the problem \[ \min \left\{ {\parallel x{\parallel }_{\infty } : \langle b, x\rangle = \beta }\right\} \] where \( b = \left( {{b}_{0},{b}_{1},\ldots ,{b}_{n}}\right) \), and \( {b}_{i} \neq 0 \) for all \( i \) . The optimal solution \( {x}^{ * } \) is unique, and \( {x}^{ * } \) and the optimal objective val... | Proof. Clearly, we can assume that \( \beta \geq 0 \) . We have \[ \beta = \mathop{\sum }\limits_{{i = 0}}^{n}{b}_{i}{x}_{i} \leq \mathop{\sum }\limits_{{i = 0}}^{n}\left| {b}_{i}\right| \cdot \left| {x}_{i}\right| \leq \parallel x{\parallel }_{\infty }\left( {\mathop{\sum }\limits_{{i = 0}}^{n}\left| {b}_{i}\right| }\... | Yes |
Lemma 13.8. Let \( {\left\{ \left( {x}_{i},{y}_{i}\right) \right\} }_{i = 0}^{n} \) be \( n + 1 \) given points in the plane such that\n\n\[ \n{x}_{0} < {x}_{1} < \cdots < {x}_{n} \n\]\n\nand consider the minimax problem\n\n\[ \n\mathop{\min }\limits_{{p \in {\mathcal{P}}_{n - 1}}}\mathop{\max }\limits_{i}\left| {{y}_{... | Proof. Writing \( {u}_{i} = {y}_{i} - p\left( {x}_{i}\right) \) for the discrepancies, problem (13.4) becomes the optimization problem\n\n\[ \n\min \parallel u{\parallel }_{\infty } \n\]\n\n(13.5)\n\n\[ \n\text{s.t.}{Xa} = y - u \n\]\n\nin the decision variables \( a \) and \( u \), where\n\n\[ \nX = \left\lbrack \begi... | Yes |
Theorem 13.9. Let \( f\left( x\right) \) be a bounded (not necessarily continuous) function on the interval \( \left\lbrack {a, b}\right\rbrack \), and \( n \) a given positive integer. The problem\n\n\[ \min \left\{ {\parallel f - p{\parallel }_{\infty } : p \in {\mathcal{P}}_{n - 1}}\right\} \]\n\nhas an optimal solu... | Proof. We write our problem as the semi-infinite linear program (13.3) in the decision variables \( \left( {z, a}\right) = \left( {z,{a}_{0},\ldots ,{a}_{n - 1}}\right) \in {\mathbb{R}}^{n + 1} \) . We can clearly assume that \( \left| z\right| \leq \mathop{\sup }\limits_{{x \in \left\lbrack {a, b}\right\rbrack }}\left... | Yes |
Theorem 13.10. (Chebyshev’s theorem) Let \( f\left( x\right) \) be a continuous function on the interval \( \left\lbrack {a, b}\right\rbrack \), and \( n \) a given positive integer. There exists a unique polynomial \( {p}^{ * } \) of degree at most \( n - 1 \) minimizing the norm \( \parallel f - p{\parallel }_{\infty... | Proof. Theorem 13.9 guarantees everything except the uniqueness of \( {p}^{ * } \) . Extend the function \( \rho \left( {{x}_{0},{x}_{1},\ldots ,{x}_{n}}\right) \) to all of \( {\left\lbrack a, b\right\rbrack }^{n + 1} \) by defining it to be zero when \( {\left\{ {x}_{i}\right\} }_{0}^{n} \) are not all distinct. Note... | Yes |
Lemma 13.11. Let \( f\left( x\right) \) be a continuous function on the interval \( \left\lbrack {a, b}\right\rbrack \), and \( n \) a given positive integer. Let \( {p}^{ * } \in {\mathcal{P}}_{n - 1} \) be a polynomial of degree at most \( n - 1 \) such that there exist \( n + 1 \) distinct points \( {x}_{0} < {x}_{1... | Proof. Suppose that there exists \( p \in {\mathcal{P}}_{n - 1} \) satisfying \( \parallel f - p{\parallel }_{\infty } < \mu \) . Then the polynomial \( r \mathrel{\text{:=}} p - {p}^{ * } \in {\mathcal{P}}_{n - 1} \) has the property that \[ r\left( {x}_{i}\right) = \left( {f\left( {x}_{i}\right) - {p}^{ * }\left( {x}... | Yes |
Theorem 13.12. (Bárány) Let \( {\left\{ {A}_{i}\right\} }_{i = 1}^{n + 1} \) be \( n + 1 \) nonempty sets in \( {\mathbb{R}}^{n} \) . If \( x \in {\mathbb{R}}^{n} \) belongs to the convex hull of each set \( {A}_{i} \), then there exists a point \( {x}_{i} \in {A}_{i} \) such that \( x \) belongs to the convex hull of ... | Proof. We may assume, without loss of generality, that \( x = 0 \), and by virtue of Theorem 4.13 that each \( {A}_{i} \) contains at most \( n + 1 \) affinely independent points. Consider the collection \( \mathcal{A} \) of all colorful sets \( A = \left\{ {{x}_{1},\ldots ,{x}_{n + 1}}\right\} \) with \( {x}_{i} \in {... | Yes |
Lemma 13.14. The directional derivative of a convex function \( f \) is sublinear, that is,\n\n\[ \n{f}^{\prime }\left( {x;{d}_{1} + {d}_{2}}\right) \leq {f}^{\prime }\left( {x;{d}_{1}}\right) + {f}^{\prime }\left( {x;{d}_{2}}\right) .\n\] | Proof. We have\n\n\[ \nf\left( {x + \frac{t}{2}\left( {{d}_{1} + {d}_{2}}\right) }\right) = f\left( {\frac{1}{2}\left( {x + t{d}_{1}}\right) + \frac{1}{2}\left( {x + t{d}_{2}}\right) }\right) \n\]\n\n\[ \n\leq \frac{1}{2}f\left( {x + t{d}_{1}}\right) + \frac{1}{2}f\left( {x + t{d}_{2}}\right) \n\]\n\nwhich implies\n\n\... | Yes |
Lemma 13.15. A sublinear function \( f : K \rightarrow \mathbb{R} \) is strictly convex if and only if\n\n\[ f\left( y\right) > {f}^{\prime }\left( {x;y}\right) \text{for all}x, y\text{not positively collinear.} \] | Proof. Let \( x, y \) be not positively collinear. If \( f \) is sublinear, then \( f\left( {x + {ty}}\right) < \) \( f\left( x\right) + f\left( {ty}\right) = f\left( x\right) + {tf}\left( y\right) \), and this implies for any \( {t}_{0} > 0 \) ,\n\n\[ {f}^{\prime }\left( {x;y}\right) = \mathop{\inf }\limits_{{t > 0}}\... | Yes |
Lemma 13.16. Let \( f : K \rightarrow \lbrack 0,\infty ) \) be a convex function on a convex cone \( K \subseteq {\mathbb{R}}^{n},0 \in K \), such that \( f\left( x\right) > 0 \) for \( x \neq 0 \) . If \( f \) is homogeneous of degree \( p \), then \( g\left( x\right) = f{\left( x\right) }^{1/p} \) is a sublinear func... | Proof. Since \( g \) is homogeneous of degree one, it suffices to prove the inequality \[ g\left( {x + y}\right) \leq g\left( x\right) + g\left( y\right) \] (13.6) because we then have \[ g\left( {\left( {1 - t}\right) x + {ty}}\right) \leq g\left( {\left( {1 - t}\right) x}\right) + g\left( {ty}\right) = \left( {1 - t}... | Yes |
Corollary 13.17. (Minkowski’s inequality) Let \( p > 1 \) . Then\n\n\[ \parallel x + y{\parallel }_{p} \leq \parallel x{\parallel }_{p} + \parallel y{\parallel }_{p}\text{ for all }x, y \in {\mathbb{R}}^{n}, \]\n\nwith equality holding if and only if the vectors \( x \) and \( y \) are proportional, \( y = {\alpha x} \... | Proof. The function \( t \mapsto {t}^{p} \) is strictly convex for \( t \geq 0 \), since its derivative \( p{t}^{p - 1} \) is strictly increasing. It follows that the function \( f\left( x\right) = \mathop{\sum }\limits_{{i = 1}}^{n}{x}_{i}^{p} \) is convex on \( {\mathbb{R}}_{ + }^{n} \) (it is a sum of convex functio... | Yes |
Theorem 13.18. Let \( f : C \rightarrow \mathbb{R} \) be a convex function that is bounded from above on a closed convex set \( C \subseteq {\mathbb{R}}^{n} \) . Then,\n\n(a) The function \( f \) is constant along any direction on the lineality space \( {L}_{C} \) of \( C \) .\n\n(b) The function \( f \) is nonincreasi... | Proof. Let \( M = \mathop{\sup }\limits_{{x \in C}}f\left( x\right) < \infty \) . Pick a point \( x \in C \), and consider the convex function \( g\left( l\right) \mathrel{\text{:=}} f\left( {l + x}\right) \) on the lineality subspace \( {L}_{C} \) of \( C \) . Since \( l = \left( {1/t}\right) \left( {tl}\right) + \lef... | Yes |
Corollary 13.19. Let \( f : C \rightarrow \mathbb{R} \) be a convex function bounded from above on a closed convex set \( C \subseteq {\mathbb{R}}^{n} \). If \( C \) is a convex polyhedron, then \( f \) attains a maximum on \( C \). The same result is true if ext \( \left( C\right) \) is compact and \( f \) is upper se... | Proof. If \( C \) is a convex polyhedron, then Theorem 7.13 implies that \( f \) attains its maximum on \( {\left\{ {v}_{i}\right\} }_{1}^{k} \). The rest of the corollary follows immediately from Theorems 13.18 and 2.3. | No |
Lemma 13.21. Let \( K \) be a pointed convex cone that decomposes into the direct sum (13.7). If \( x \in {K}_{i} \) is a sum \( x = {x}_{1} + \cdots + {x}_{k} \) of elements \( {x}_{j} \in K \) , then each \( {x}_{j} \in {K}_{i} \) . | Proof. We have \( 0 = {\Pi }_{{\widehat{E}}_{i}}x = {\Pi }_{{\widehat{E}}_{i}}{x}_{1} + \cdots + {\Pi }_{{\widehat{E}}_{i}}{x}_{k} \) . Each term \( {\widehat{x}}_{j} \mathrel{\text{:=}} {\Pi }_{{\widehat{E}}_{i}}{x}_{j} \) belongs to \( {\widehat{K}}_{i} \subseteq K \), so that \( {\widehat{x}}_{j} \in K \) and \( - {... | Yes |
Theorem 13.22. Let \( K \subseteq E \) be a decomposable pointed convex cone. The irreducible decompositions of \( K \) are identical modulo indexing, that is, the set of cones \( {\left\{ {K}_{i}\right\} }_{i = 1}^{m} \) is unique. Moreover, the subspaces \( {E}_{i} \) corresponding to the nonzero cones \( {K}_{i} \) ... | Proof. Suppose that \( K \) admits two irreducible decompositions\n\n\[ K = {\bigoplus }_{i = 1}^{m}{K}_{i} \subseteq {\bigoplus }_{i = 1}^{m}{E}_{i}\;\text{ and }\;K = {\bigoplus }_{j = 1}^{q}{C}_{j} \subseteq {\bigoplus }_{j = 1}^{q}{F}_{j}. \]\n\nNote that each nonzero summand in either decomposition of \( K \) must... | Yes |
Lemma 14.2. Let \( x \in U \) be a noncritical point of \( f \), and \( d \in {\mathbb{R}}^{n} \) a nonzero vector.\n\nIf \( \langle \nabla f\left( x\right), d\rangle < 0 \) (d makes an obtuse angle with the gradient \( \nabla f\left( x\right) \) ), then \( d \) is a descent direction of \( f \) at \( x \) .\n\nConvers... | Proof. Since \( f \) is Gâteaux differentiable,\n\n\[ f\left( {x + {td}}\right) = f\left( x\right) + t\langle \nabla f\left( x\right), d\rangle + o\left( t\right) .\n\]\n\n(14.2)\n\nThus, if \( d \) satisfies \( \langle \nabla f\left( x\right), d\rangle < 0 \), then \( f\left( {x + {td}}\right) < f\left( x\right) \) fo... | Yes |
Theorem 14.4. Let \( {\left\{ {x}_{k}\right\} }_{0}^{\infty } \) be the sequence of vectors generated by a descent method\n\n\[ \n{x}_{k + 1} = {x}_{k} + {\alpha }_{k}{d}_{k} \]\n\nwhere \( {d}_{k} \) satisfies (14.5), and \( {\alpha }_{k} \) is chosen by the Armijo rule with parameters \( s,\beta ,\sigma \) . If \( \m... | Proof. By taking a further subsequence if necessary, we may assume that \( {d}_{{k}_{i}} \) converges to some direction vector \( {d}^{ * },\begin{Vmatrix}{d}^{ * }\end{Vmatrix} = 1 \) . Since the \( i \) th step is successful in (14.3), we have\n\n\[ \nf\left( {x}_{{k}_{i}}\right) - f\left( {{x}_{{k}_{i}} + {\alpha }_... | Yes |
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