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Theorem 19.8 Suppose that to each oriented isomorphism class of \( {2n} \) -dimensional oriented real vector bundles \( {\zeta }^{2n} \) over \( M \) we have associated a class \( \widehat{e}\left( {\zeta }^{2n}\right) \in \) \( {H}^{2n}\left( M\right) \) that satisfies\n\n(i) \( {f}^{ \bullet }\left( {\widehat{e}\left... | Proof. Given a complex line bundle \( L \) over \( M \), we can define \( c\left( L\right) = \widehat{e}\left( {L}_{\mathbf{R}}\right) \) . Then \( {f}^{ \bullet }c\left( L\right) = c\left( {{f}^{ \bullet }L}\right) \), and the argument used at the beginning of the proof of Theorem 18.9 shows that \( \mathrm{c}\left( L... | Yes |
Proposition 19.9 For an oriented \( {2k} \) -dimensional real vector bundle \( \zeta ,{p}_{k}\left( \zeta \right) = \) \( e{\left( \zeta \right) }^{2} \) . | Proof. We give \( \zeta \) a metric \( \langle \) , \( \rangle \) and chose a compatible metric connection \( \nabla \) . Then \( e\left( \zeta \right) \) is represented locally by \( {\left( -1\right) }^{k}/{\left( 2\pi \right) }^{k}\operatorname{Pf}\left( {{F}^{\nabla }\left( \mathbf{e}\right) }\right) \) where \( \m... | Yes |
Theorem 20.2 For any complex \( n \) -dimensional vector bundle \( \xi \) over \( M,{H}^{ \bullet }\left( {P\left( \xi \right) }\right) \) is a free \( {H}^{ \bullet }\left( M\right) \) -module with basis\n\n\[ 1,{c}_{1}\left( {H\left( \xi \right) }\right) ,\ldots ,{c}_{1}{\left( H\left( \xi \right) \right) }^{n - 1}. ... | Proof of Theorem 18.10. Starting with \( \xi \) over \( M \) with \( {\dim }_{\mathbb{C}}\xi = n + 1 \), we consider the composition\n\n\[ P\left( {\xi }_{n - 1}\right) \overset{{\pi }_{n - 1}}{ \rightarrow }\cdots \rightarrow P\left( {\xi }_{1}\right) \overset{{\pi }_{1}}{ \rightarrow }P\left( \xi \right) \overset{{\p... | Yes |
Lemma 20.4 The map \[ \psi : Q \times {\mathbf{C}}^{ \bullet } \rightarrow G{L}_{2}^{ + }\left( \mathbf{R}\right) ;\;\left( {A, a + \imath b}\right) \mapsto A \cdot \left( \begin{matrix} a & - b \\ b & a \end{matrix}\right) \] is a homeomorphism. | Proof. Let \( B \in G{L}_{2}^{ + }\left( \mathbf{R}\right) \) with transpose \( {B}^{ * } \) . Then \( B{B}^{ * } \) is positive definite, and by the spectral theorem it has a unique square root \( {\left( B{B}^{ * }\right) }^{1/2} \) which commutes with \( B{B}^{ * } \) . This gives the polar decomposition \[ B = {\le... | Yes |
Lemma 20.5 The space \( Q \) is contractible by the homotopy\n\n\[ F : Q \times \left\lbrack {0,1}\right\rbrack \rightarrow Q;\;F\left( {A, t}\right) = {A}^{t} \] | Proof. This is again a consequence of the spectral decomposition. A matrix \( A \in Q \) has positive eigenvalues \( \lambda ,{\lambda }^{-1} \) with say \( \lambda \geq 1 \) . Then \( \lambda \) depends continuously on \( A \) . The case \( \lambda = 1 \) occurs only for the identity matrix \( I \) . If \( A \neq I \)... | Yes |
Lemma 20.8 The map \( {i}_{ \bullet } : {H}_{c}^{q}\left( {W}_{m}\right) \rightarrow {H}^{q}\left( {\mathbf{{CP}}}^{m - 1}\right) \) is an isomorphism except in possibly two cases: for \( q = 0 \), and, if \( m \) is even, for \( q = m \) . In fact \( {H}_{c}^{0}\left( {W}_{m}\right) = 0 \) , and for \( m \) even there... | Proof. The exact sequence of Proposition 13.11 for the pair \( \left( {{\mathbf{{CP}}}^{m - 1},{\mathbf{{RP}}}^{m - 1}}\right) \) takes the form\n\n\[ \cdots \overset{{j}^{ * }}{ \rightarrow }{H}^{q - 1}\left( {\mathbf{{RP}}}^{m - 1}\right) \overset{\delta }{ \rightarrow }{H}_{c}^{q}\left( {W}_{m}\right) \overset{{i}_{... | Yes |
Proposition 20.9 The cohomology \( {H}^{{2p} - 1}\left( {W}_{m}\right) = 0 \) for all \( p \), and\n\n\[ \n{H}^{2p}\left( {W}_{m}\right) \cong \left\{ \begin{array}{ll} {\mathbf{R}}^{2} & \text{ if }{2p} - m - 2 \\ \mathbf{R} & \text{ if }{2p} \neq m - 2\text{ and }0 \leq {2p} \leq {2m} - 4 \\ 0 & \text{ if }{2p} \geq ... | Proof. We apply Poincaré duality to the oriented \( \left( {{2m} - 2}\right) \) -dimensional manifold \( {\mathrm{{CP}}}^{m - 1} \) and to \( {W}_{m} \) . Lemma 13.6 gives a commutative diagram\n\n\[ \n\begin{matrix} {H}^{p}\left( {\mathbf{{CP}}}^{m - 1}\right) \xrightarrow[]{{H}^{p}\left( i\right) }{H}^{p}\left( {W}_{... | Yes |
Theorem 21.3 (Thom isomorphism) Let \( \\xi \) be an oriented \( m \) -dimensional real vector bundle over a compact manifold \( M \) . There is a unique class \( U \\in {H}_{c}^{m}\\left( E\\right) \) with integral I over each fiber \( {\\xi }_{p} \), and the map\n\n\[ \n\\Phi : {H}^{q}\\left( M\\right) \\overset{ \\c... | Proof. The exact sequence (2) shows that the orientation class \( u \\in {H}^{m}\\left( {S\\left( {\\xi \\oplus 1}\\right) }\\right) \) has the form \( u = {i}_{ \\bullet }\\left( U\\right) \) for a uniquely determined \( U \\in {H}_{c}^{m}\\left( E\\right) \) . The first statement now follows from Theorem 21.2. The ho... | Yes |
Lemma 21.5 Let \( s : M \rightarrow E \) be an arbitrary smooth section of \( E \) . Then \( \widetilde{e}\left( \xi \right) = \) \( {s}^{ * }\left( U\right) \) . | Proof. Since \( s \) is a proper map, it induces a homomorphism\n\n\[ \n{H}_{\mathrm{c}}^{ \bullet }\left( E\right) \rightarrow {H}_{\mathrm{c}}^{ \bullet }\left( M\right) = {H}^{ \bullet }\left( M\right) \n\] \n\ncf. Chapter 13. A closed form \( \omega \in {\Omega }_{\mathrm{c}}^{m}\left( E\right) \) that represents \... | Yes |
Lemma 21.6 Let \( f : N \rightarrow M \) be a smooth map of compact manifolds and \( \xi \) an oriented vector bundle over \( M \) . Then \( \bar{e}\left( {{f}^{ \bullet }\xi }\right) = {f}^{ \bullet }\widehat{e}\left( \xi \right) \) . | Proof. We have the pull-back diagram\n\n\n\nwhere \( E \) and \( {E}^{\prime } \) are the total spaces of \( \xi \) and \( {f}^{ \bullet }\xi \) respectively. The map \( \widehat{f} \) is proper, so the class \( U \i... | Yes |
Example 21.10 Let \( H \) be the canonical complex line bundle over \( {\mathrm{{CP}}}^{1},{H}^{ * } \) its dual bundle, and \( \xi = {\left( {H}^{ \bullet }\right) }_{\mathrm{R}} \) the underlying oriented real bundle. The bundle \( H \) is a subbundle of the trivial 2-dimensional bundle | \[ E\left( H\right) = \{ \left( {L;\mathbf{z}}\right) \mid L \in {\mathbf{{CP}}}^{\mathbf{1}},\mathbf{z} \in L\} \subset {\mathbf{{CP}}}^{\mathbf{2}} \times {\mathbf{C}}^{\mathbf{2}} \] (cf. Example 17.2). Dually there is an epimorphism from the dual product bundle onto \( {H}^{ \bullet } \), and we let \( s : {\mathrm... | Yes |
Theorem 21.11 For oriented vector bundles over compact manifolds, \( \dot{e}\left( \xi \right) = e\left( \xi \right) \) . | Proof. We have already seen in (3) above that \( \widehat{e}\left( {\xi }^{2m}\right) = {a}^{m}e\left( {\xi }^{2m}\right) \) for some constant \( a \) ; it remains to be shown that \( a = 1 \) . For \( {\xi }^{2} = {\left( {H}^{ \bullet }\right) }_{\mathrm{R}} \), the previous example shows that \( I\left( {\dot{e}\lef... | Yes |
Theorem 21.12 For any oriented compact smooth manifold \( M \) ,\n\n\[ I\left( {e\left( {\tau }_{M}\right) }\right) = \chi \left( M\right) \] | Proof. We simply apply Theorem 21.9 to \( \xi = {\tau }_{M} \), taking for \( s \) a gradient-like vector field \( X \) w.r.t. some Morse function; cf. Definition 12.7. The proof of Lemma 12.8 shows that \( X \) is transversal to the zero section, and that the sum in (5) is equal to \( \operatorname{Index}\left( X\righ... | Yes |
Theorem 2.5.1 Let \( f\left( {{\mathbf{z}}_{1},{\mathbf{z}}_{2}}\right) \) and \( g\left( {{\mathbf{z}}_{1},{\mathbf{z}}_{2}}\right) \) be two probability densities, where the support of \( f \) is a subset of the support of \( g \) . Then,\n\n\[ \n{\operatorname{var}}_{g}\left\{ \frac{f\left( {{\mathbf{Z}}_{1},{\mathb... | Proof: It is easy to see that\n\n\[ \n\frac{{f}_{1}\left( {\mathbf{z}}_{1}\right) }{{g}_{1}\left( {\mathbf{z}}_{1}\right) } = \int \frac{f\left( {{\mathbf{z}}_{1},{\mathbf{z}}_{2}}\right) }{{g}_{1}\left( {\mathbf{z}}_{1}\right) {g}_{2 \mid 1}\left( {{\mathbf{z}}_{2} \mid {\mathbf{z}}_{1}}\right) }{g}_{2 \mid 1}\left( {... | Yes |
Theorem 6.6.1 Suppose the Markov chain resulting from a data augmentation scheme is in stationarity. Then,\n\n\[ \operatorname{cov}\left\{ {h\left( {x}_{1}^{\left( 0\right) }\right), h\left( {x}_{1}^{\left( 1\right) }\right) }\right\} = {\operatorname{var}}_{\pi }\left\{ {{E}_{\pi }\left\{ {h\left( {x}_{1}\right) \mid ... | Proof: Without loss of generality, we assume that \( {E}_{\pi }h\left( {x}_{1}\right) = 0 \) . Then,\n\n\[ \operatorname{cov}\left\{ {h\left( {x}_{1}^{\left( 0\right) }\right), h\left( {x}_{1}^{\left( 1\right) }\right) }\right\} = E\left\{ {h\left( {x}_{1}^{\left( 0\right) }\right) h\left( {x}_{1}^{\left( 1\right) }\ri... | Yes |
Lemma 6.6.1 Let \( {\mathbf{x}}^{\left( 0\right) } \) and \( {\mathbf{x}}^{\left( 1\right) } \) be two consecutive realizations of the random-scan Gibbs sampler under stationarity, and let i be the random variable representing which index is updated at stage one, taking values on \( I = \{ 1,\ldots, d\} \) with distrib... | Proof: From the definition of the scan, it is understood that \[ E\left\{ {h\left( {\mathbf{x}}^{\left( 0\right) }\right) h\left( {\mathbf{x}}^{\left( 1\right) }\right) }\right\} = E\left\lbrack {E\left\lbrack {E\left\{ {h\left( {\mathbf{x}}^{\left( 0\right) }\right) h\left( {\mathbf{x}}^{\left( 1\right) }\right) \mid ... | Yes |
Theorem 6.6.2 Let \( {\mathbf{x}}^{\left( 0\right) },{\mathbf{x}}^{\left( 1\right) },\ldots \), be consecutive samples generated by the random scan under stationarity, and let \( \mathbf{i} \) be the random variable representing the random index in the updating scheme. For \( h\left( \mathbf{x}\right) \in {L}_{0}^{2}\l... | Proof: The expression is derived by repeatedly applying Lemma 6.6.1 and the Markov property. The monotonicity is a simple property of conditional expectations. \( \diamondsuit \) | No |
Theorem 8.3.1 Suppose \( \Gamma \) is a locally compact group of transformations on \( \mathcal{X} \) and \( L \) is its left-Haar measure. Let \( \pi \) be an arbitrary probability measure on space \( \mathcal{X} \) . Suppose \( \mathbf{x} \sim \pi \left( \mathbf{x}\right) \) and \( \gamma \) is drawn from \( \Gamma \... | The standard Gibbs sampler can be realized by applying this theorem to those \( \Gamma \) ’s corresponding to the translation group along each coordinate direction. An easy extension of the standard Gibbs sampler is to let \( \Gamma \) be the translation group along an arbitrary direction; that is,\n\n\[ \n\Gamma = \le... | Yes |
Corollary 8.3.1 Suppose \( \Gamma = \{ \gamma \} \) is a locally compact transformation group on \( \mathcal{X} \) . Then each orbit of \( \Gamma \) can be treated as a fiber; that is, we can define \( {\mathcal{X}}_{\alpha } = \{ \mathbf{x} \in \mathcal{X} : \;\mathbf{x} = \gamma \left( \alpha \right) \} \), where \( ... | ## Proof: A direct consequence of Theorem 8.3.1. \( \diamond \) | No |
Lemma 11.1.1 Suppose \( \mathbf{x} \sim \pi \) and \( \mathbf{y} \) is any fixed point in a d-dimensional space. Let \( e = \left( {\mathbf{x} - \mathbf{y}}\right) /\parallel \mathbf{x} - \mathbf{y}\parallel \) be a unit vector. If \( r \) is drawn from distribution in (11.1), then \( {\mathbf{x}}^{\prime } = \mathbf{y... | Proof: Without loss of generality, we need only to show the case when \( \mathbf{y} = 0 \) . Then, the Markovian move is of the form\n\n\[ \n{\mathbf{x}}^{\prime } = {r}^{ * }\mathbf{x},\;{r}^{ * } \neq 0 \n\]\n\nand is a scale group transformation of \( \mathbf{x} \) . Now, the question becomes this: What distribution... | No |
Example 1. Simple random walk on a line. Suppose the \( {Z}_{n} \) are i.i.d. Bernoulli random variables (coin tosses), with \( P\left( {{Z}_{i} = 1}\right) = 1 - P\left( {{Z}_{i} = }\right. \) \( - 1) = p \) . Let \( {S}^{\left( 0\right) } = 0 \) and let \( {S}^{\left( t\right) } = {Z}_{1} + \cdots + {Z}_{t} \) (the d... | As \( t \rightarrow \infty \), however, \( {S}^{\left( t\right) } \) does not “converge” to a stable distribution and will drift to \( \pm \infty \) (according to whether \( p < {0.5}, = {0.5} \), or \( > {0.5} \) ). | No |
Let \( {\mathbf{x}}^{\left( 0\right) } = \left( {-1, - 1,\ldots , - 1}\right) \) be a vector of length \( d \) . We let \( {\mathbf{x}}^{\left( t + 1\right) } \) be generated recursively as follows: Randomly pick a coordinate of \( {\mathbf{x}}^{\left( t\right) } \) and negate its current value. Then, the sequence \( {... | As \( t \rightarrow \infty \), the simple-random-walk chain stabilizes to the uniform distribution. In other words, when \( t \) is large enough, the chance that you will guess correctly which vertex of the cube is occupied by \( {\mathbf{x}}^{\left( t\right) } \) is roughly \( {2}^{-d} \) (if I only tell you that \( t... | Yes |
Theorem 12.3.1 Suppose the state space \( \mathcal{X} \) of a Markov chain is finite. The transition function of this chain is irreducible and aperiodic; then, \( {A}^{\left( n\right) }\left( {{\mathbf{x}}_{0},\mathbf{y}}\right) = P\left( {{\mathbf{x}}^{\left( n\right) } = \mathbf{y} \mid {\mathbf{x}}^{\left( 0\right) ... | Proof: First, it can be seen that the irreducibility and aperiodicity conditions guarantee the existence of an integer \( {n}_{0} \) such that \( {A}^{\left( {n}_{0}\right) }\left( {\mathbf{x},\mathbf{y}}\right) > 0 \) for all \( \mathbf{x},\mathbf{y} \in \mathcal{X} \) ; that is, we are able to find a large enough \( ... | Yes |
Theorem 12.5.1 (Poincaré inequality) The second largest eigenvalue of an irreducible Markov chain \( A \) satisfies\n\n\[ \n{\beta }_{1} \leq 1 - {\kappa }^{-1} \n\]\n\nwhere \( \kappa \) is defined by (12.11). | Proof: Because of (12.9), we need only to show that \( \operatorname{var}\left( \phi \right) \leq \gamma \mathcal{E}\left( {\phi ,\phi }\right) \) . Define for any edge \( \phi \left( e\right) = \phi \left( \mathbf{w}\right) - \phi \left( \mathbf{v}\right) \), where \( e \) is the directed edge connecting from \( \math... | Yes |
For the simple random walk on a connected graph \( G = \left( {V, E}\right) \), the second largest eigenvalue is bounded by \[ {\beta }_{1} \leq 1 - \frac{2\left| E\right| }{{d}_{ * }^{2}{\gamma }_{ * }b} \] where \( {d}_{ * },{\gamma }_{ * } \), and \( b \) are as defined earlier. | Simple random walk on a cube. A cube on an \( d \) -dimensional space (denoted as \( {\mathbb{Z}}_{2}^{d} \) ) can be coded by the set of all \( {2}^{d} \) binary strings of length \( d,\mathbf{x} = \left( {{x}_{1},\ldots ,{x}_{d}}\right) \), where \( {x}_{j} = 0 \) or 1 . A simple random walk on it can be implemented ... | Yes |
Theorem 12.5.2 (Cheeger’s inequality) Let \( {\beta }_{1} \) be the second largest eigenvalue of the Markov chain transition matrix \( A \) . Then,\n\n\[ 1 - {2h} \leq {\beta }_{1} \leq 1 - \frac{{h}^{2}}{2} \] | Since the proof of the theorem is a bit complicated, we refer the interested reader to Diaconis and Stroock (1991) for more details. | No |
Theorem 12.5.3 (Jerrum and Sinclair) The second largest eigenvalue of a finite-state reversible and irreducible Markov chain satisfies\n\n\[ \n{\beta }_{1} \leq 1 - 1/8{\eta }^{2} \n\]\n\nwhere \( \eta \) is defined in (12.13). | For the simple random walk on \( {\mathbb{Z}}_{2}^{d} \), we can define a path from \( \mathbf{x} = \) \( \left( {{x}_{1},\ldots ,{x}_{d}}\right) \) to \( \mathbf{y} = \left( {{y}_{1},\ldots ,{y}_{d}}\right) \) as the one that changes one coordinate of \( \mathbf{x} \) a time from left to right:\n\n\[ \n\left( {{x}_{1}... | Yes |
Lemma 12.6.1 Suppose \( {\mathbf{x}}^{\left( 0\right) } \sim \pi \) . For any \( h, g \in {L}_{0}^{2}\left( \pi \right) \), we have\n\n\[ \operatorname{cov}\left( {h\left( {\mathbf{x}}^{\left( n\right) }\right), g\left( {\mathbf{x}}^{\left( 0\right) }\right) }\right) = {\operatorname{cov}}_{\pi }\left\{ {{F}^{k}h\left(... | Proof: Since \( {E}_{\pi }h\left( \mathbf{x}\right) = {E}_{\pi }g\left( \mathbf{x}\right) = 0 \), we have\n\n\[ \operatorname{cov}\left( {h\left( {{\mathbf{x}}^{\left( n\right) }, g\left( {\mathbf{x}}^{\left( 0\right) }\right) = E\left\{ {h\left( {\mathbf{x}}^{\left( n\right) }\right) g\left( {\mathbf{x}}^{\left( 0\rig... | No |
Lemma 12.6.2 If the Markov chain is reversible, then \( F = B \) . | Proof: Since the reversibility of a Markov chain implies the detailed balance condition, \( \pi \left( \mathbf{x}\right) A\left( {\mathbf{x},\mathbf{y}}\right) = \pi \left( \mathbf{y}\right) A\left( {\mathbf{y},\mathbf{x}}\right) \), we have\n\n\[ \n{Bh}\left( \mathbf{y}\right) = \int h\left( \mathbf{x}\right) \frac{\p... | Yes |
Theorem 12.6.1 For a reversible Markov chain, if \( {\mathbf{x}}^{\left( 0\right) } \sim \pi \), then \( \forall g \in \) \( {L}_{0}^{2}\left( \pi \right) , \)\n\n\[ \operatorname{cov}\left\{ {g\left( {\mathbf{x}}^{\left( 0\right) }\right), g\left( {\mathbf{x}}^{\left( 2m\right) }\right) }\right\} = {\mathbb{E}}_{\pi }... | Proof: The first conclusion follows from lemma 12.6.1 with \( k = m \) and lemma 12.6.2. The monotonicity of even-lag autocorrelations follows from inequality\n\n\[ \operatorname{var}\left\lbrack {E\{ g\left( \mathbf{x}\right) \mid \mathbf{y}\} }\right\rbrack \leq \operatorname{var}\{ g\left( \mathbf{x}\right) \} \]\n\... | Yes |
Lemma 12.6.3 The spectral radius of operators \( F \) and \( B \) are equal. If this radius is less than 1 , then the chain converges to its stationary distribution geometrically in \( {\chi }^{2} \) -distance, provided that the starting density \( {P}_{0}\left( {d\mathbf{x}}\right) \) has a finite \( {\chi }^{2} \) -d... | Proof: It is well known that the adjoint operators in a Hilbert space have the same norms and spectral radii (i.e., equality \( \begin{Vmatrix}{F}^{n}\end{Vmatrix} = \begin{Vmatrix}{B}^{n}\end{Vmatrix} \) holds for all \( n \) ). Hence, we have the equality between the two spectral radii:\n\n\[ \n{r}_{F} = \mathop{\lim... | Yes |
Theorem 12.7.2 Under the same conditions as in Theorem 12.7.1,\n\n\[ \sqrt{m}\left\lbrack {{\bar{h}}_{m} - {E}_{\pi }\left( h\right) }\right\rbrack \rightarrow N\left\lbrack {0,\sigma {\left( h\right) }^{2}}\right\rbrack \]\n\nweakly (i.e., in distribution) for any initial distribution on \( {\mathbf{x}}^{\left( 0\righ... | In practice, the variance term \( \sigma {\left( h\right) }^{2} \) needs to be estimated from the Monte Carlo samples based on (5.13) of page 125; that is\n\n\[ \sigma {\left( h\right) }^{2} = {\sigma }^{2}\left\lbrack {1 + 2\mathop{\sum }\limits_{{j = 1}}^{\infty }{\rho }_{j}}\right\rbrack \equiv 2{\tau }_{\text{int }... | No |
Lemma 13.2.1 The marginal chains \( \left\{ {x}_{1}^{\left( t\right) }\right\} \) and \( \left\{ {x}_{2}^{\left( t\right) }\right\} \) constructed in data augmentation are mutually interleaving. | Consider the forward operator for the marginal chain \( \left\{ {x}_{1}^{\left( t\right) }\right\} \) . From (13.3), we have\n\n\[ \n{F}_{1}h\left( {x}_{1}\right) = \int h\left( {x}_{1}^{\left( 1\right) }\right) {A}_{1}\left( {{x}_{1},{x}_{1}^{\left( 1\right) }}\right) d{x}_{1}^{\left( 1\right) } \n\]\n\n\[ \n= {E}_{\p... | Yes |
Theorem 13.2.1 Suppose we can either group the first two components together or integrate out the first component so as to result in a RSGS with \( d - 1 \) components. We also assume that the scheduling probability \( {\alpha }_{i} \) remains unchanged for \( i = 3,\ldots, d \) . Then, the collapsing sampler converges... | Proof: We directly compare (13.4) for the three samplers. For grouping,\n\n\[ \n{\begin{Vmatrix}{F}_{g}h\end{Vmatrix}}^{2} = \left( {{\alpha }_{1} + {\alpha }_{2}}\right) \operatorname{var}\left\lbrack {E\left\{ {h\left( \mathbf{x}\right) \mid {\mathbf{x}}_{\left\lbrack -1, - 2\right\rbrack }}\right\} }\right\rbrack \n... | Yes |
Lemma 13.3.1 (Tierney) Suppose \( {A}_{1} \) and \( {A}_{2} \) have the same invariant distribution \( \pi \) and satisfy \( {A}_{1} \succcurlyeq {A}_{2} \) . Then, the corresponding forward operators, \( {F}_{1} \) and \( {F}_{2} \), satisfy\n\n\[ \left\langle {\left( {{F}_{2} - {F}_{1}}\right) f, f}\right\rangle \geq... | Proof: Here, we only prove the case when the state space is finite. Please refer to Tierney (1998) for a more general proof. Suppose the total number of states is \( N \) . In this case, we can express the target distribution as the vector \( \mathbf{\pi } = \left( {{\pi }_{1},\ldots ,{\pi }_{N}}\right) \) and the tran... | No |
Theorem 13.3.1 (Peskun) Suppose \( {A}_{1} \) and \( {A}_{2} \) are reversible transition kernels with the same invariant distribution and \( {A}_{1} \succcurlyeq {A}_{2} \) . Then, for all \( f \in {L}_{0}^{2}\left( \pi \right) \) (i.e., mean zero functions), we have \( v\left( {f,{A}_{1}}\right) \leq v\left( {f,{A}_{... | Proof: It is easy to see that for any transition matrix \( A \) ,\n\n\[ v\left( {f, A}\right) = \left\langle {f,\left\{ {I + {\left( I - A\right) }^{-1}}\right\} f}\right\rangle \]\n\nNote that operator \( \left( {I - A}\right) \) is invertible only in the restricted space \( {L}_{0}^{2}\left( \pi \right) \) , but not ... | Yes |
Theorem 13.3.2 For the same proposal transition \( T\left( {\mathbf{x},\mathbf{y}}\right) \), the acceptance function suggested by Metropolis et al. dominates that proposed by Barker in terms of asymptotic efficiency (13.5). | Proof: The transition function of the Metropolis algorithm is\n\n\[ \n{A}_{M}\left( {\mathbf{x},\mathbf{y}}\right) = T\left( {\mathbf{x},\mathbf{y}}\right) \min \{ 1, r\left( {\mathbf{x},\mathbf{y}}\right) \} \text{ for }\mathbf{x} \neq \mathbf{y}, \]\n\nwhere\n\n\[ \nr\left( {\mathbf{x},\mathbf{y}}\right) = \frac{\pi ... | Yes |
Theorem 13.3.3 Suppose that \( \mathbf{x} = \left( {{x}_{1},\ldots ,{x}_{d}}\right) \), where \( {x}_{i} \) takes \( {m}_{i} < \) \( \infty \) possible values, and that \( \pi \left( \mathbf{x}\right) \) is the distribution of interest. Then, the Metropolized Gibbs sampler defined in Section 6.3.2 for discrete random v... | Proof: Suppose that in the random-scan Gibbs sampler, we choose each component with probability \( {\alpha }_{i} \) . Then, all the nonzero elements of the transition matrix \( {P}_{1} \) of the random scan Gibbs sampler are of the form\n\n\[ \n{P}_{1}\left( {\mathbf{x},\mathbf{y}}\right) = {\alpha }_{i}\pi \left( {{y}... | Yes |
Theorem 13.3.4 The Metropolis transition with a mixture proposal dominates the corresponding mixture of Metropolis transitions; that is,\n\n\[ \n{A}^{ * } \succcurlyeq \mathop{\sum }\limits_{i}{\alpha }_{i}{A}_{i} \n\] | Proof: Because of the simple inequality\n\n\[ \n\min \left( {{A}_{1},{B}_{1}}\right) + \min \left( {{A}_{2},{B}_{2}}\right) \leq \min \left( {{A}_{1} + {A}_{2},{B}_{1} + {B}_{2}}\right) \n\]\nwe have that\n\n\[ \n\mathop{\sum }\limits_{i}{\alpha }_{i}{A}_{i}\left( {\mathbf{x},\mathbf{y}}\right) = \mathop{\sum }\limits_... | Yes |
Theorem 13.4.1 For the Metropolized independence sampler, all the eigenvalues of its transition matrix are \( 1 > {\lambda }_{1} \geq {\lambda }_{2} \geq \cdots \geq {\lambda }_{m - 1} \geq 0 \), where \( {\lambda }_{k} = \mathop{\sum }\limits_{{i = k}}^{m}\left( {{p}_{i} - {\pi }_{i}/{w}_{k}}\right) = {E}_{\pi }\{ 1/w... | Proof: Since \( A = G + \mathbf{e}{\mathbf{p}}^{T}, A{\widetilde{\mathbf{v}}}_{k} = G{\widetilde{\mathbf{v}}}_{k} + \mathbf{e}\left( {{\mathbf{p}}^{T}{\widetilde{\mathbf{v}}}_{k}}\right) \) . It is further noted that\n\n\[ {\mathbf{p}}^{T}{\widetilde{\mathbf{v}}}_{k} = {S}_{\pi }\left( k\right) {\pi }_{k} + {p}_{k}{S}_... | Yes |
Lemma 13.6.1 Let \( g\left( \text{ }\right) \) be the invariant distribution of \( T \) and let \( g\left( {\mathbf{x},\mathbf{y}}\right) = \) \( g\left( \mathbf{x}\right) T\left( {\mathbf{x},\mathbf{y}}\right) \) . Then,\n\n\[ \n{e}_{0} = {E}_{g}\left\{ {\log \frac{g\left( {\mathbf{y},\mathbf{x}}\right) }{g\left( {\ma... | Proof: By definition, we have\n\n\[ \n{e}_{0} = {E}_{g}\left\{ {\log \frac{g\left( {\mathbf{y},\mathbf{x}}\right) }{g\left( {\mathbf{x},\mathbf{y}}\right) }}\right\} \leq \log {E}_{g}\left\{ \frac{g\left( {\mathbf{y},\mathbf{x}}\right) }{g\left( {\mathbf{x},\mathbf{y}}\right) }\right\} = 0.\n\]\n\nWe used the Jensen's ... | Yes |
Theorem 13.6.1 Suppose the sample space \( \mathcal{X} \) of \( \mathbf{x} \) is finite and the proposal transition \( T\left( {\mathbf{x},\mathbf{y}}\right) \) is nonreversible. Then, the process \( \left( {{\mathbf{x}}^{\left( t\right) },\log {w}^{\left( t\right) }}\right) \) induced by the Q-type move is positive re... | Case (v): Mixing different types of moves \( \left( {\theta = 1}\right) \) . Suppose in each iteration that we make a \( Q \) -type move with probability \( \alpha \) and a Metropolis move with probability \( 1 - \alpha \) . When \( w \) is sufficiently large, there will be no rejection for the \( Q \) -type moves, thu... | Yes |
Corollary 2.3.1. The convex hull of a finite subset \( \left\{ {{b}_{0},\ldots ,{b}_{m}}\right\} \) of \( {R}^{n} \) consists of all the vectors of the form \( {\lambda }_{0}{b}_{0} + \cdots + {\lambda }_{m}{b}_{m} \), with \( {\lambda }_{0} \geq 0,\ldots ,{\lambda }_{m} \geq 0,{\lambda }_{0} + \cdots + {\lambda }_{m} ... | Proof. Every convex combination of elements selected from \( \left\{ {{b}_{0},\ldots ,{b}_{m}}\right\} \) can be expressed as a convex combination of \( {b}_{0},\ldots ,{b}_{m} \) by including the unneeded vectors \( {b}_{i} \) with zero coefficients. | No |
Corollary 2.5.1. Let \( {b}_{i} \in {R}^{n} \) for \( i \in I \), where \( I \) is an arbitrary index set. Then\n\n\[ K = \\left\\{ {x \in {R}^{n} \mid \\left\\langle {x,{b}_{i}}\\right\\rangle \\leq 0, i \in I}\\right\\} \n\]\n\nis a convex cone. | Proof. As in Corollary 2.1.1. | No |
Corollary 2.6.3. Let \( C \) be a convex set, and let\n\n\[ K = \{ {\lambda x} \mid \lambda > 0, x \in C\} . \]\n\nThen \( K \) is the smallest convex cone which includes \( C \) . | Proof. This follows from the preceding corollary. Namely, every positive linear combination of elements of \( C \) is a positive scalar multiple of a convex combination of elements of \( C \) and hence is an element of K. \( \parallel \) | Yes |
Corollary 3.4.1. The orthogonal projection of a convex set \( C \) on a subspace \( L \) is another convex set. | Proof. The orthogonal projection mapping onto \( L \) is a linear transformation, the one which assigns to each point \( x \) the unique \( y \in L \) such that \( \left( {x - y}\right) \bot L \) . | No |
Corollary 4.7.2. If \( f \) is a positively homogeneous proper convex function, then \( f\left( {-x}\right) \geq - f\left( x\right) \) for every \( x \) . | Proof. \( f\left( x\right) + f\left( {-x}\right) \geq f\left( {x - x}\right) = f\left( 0\right) \geq 0 \) . | Yes |
Corollary 6.5.2. Let \( {C}_{1} \) be a convex set. Let \( {C}_{2} \) be a convex set contained in \( \mathrm{{cl}}{C}_{1} \) but not entirely contained in the relative boundary of \( {C}_{1} \) . Then \( \mathrm{{ri}}{C}_{2} \subset \mathrm{{ri}}{C}_{1} \) . | Proof. The hypothesis implies ri \( {C}_{2} \) has a point in common with ri \( \left( {\mathrm{{cl}}{C}_{1}}\right) = \mathrm{{ri}}{C}_{1} \), for otherwise the relative boundary \( \mathrm{{cl}}{C}_{1} \smallsetminus \mathrm{{ri}}{C}_{1} \), which is a closed set, would contain \( \mathrm{{ri}}{C}_{2} \) and its clos... | Yes |
Corollary 6.6.1. For any convex set \( C \) and any real number \( \lambda \) , ri \( \left( {\lambda C}\right) = \lambda \) ri \( C \) . | Proof. Take \( A : x \rightarrow {\lambda x} \) . | No |
Corollary 6.6.2. For any convex sets \( {C}_{1} \) and \( {C}_{2} \) in \( {R}^{n} \) ,\n\n\[ \n\operatorname{ri}\left( {{C}_{1} + {C}_{2}}\right) = \operatorname{ri}{C}_{1} + \operatorname{ri}{C}_{2}, \]\n\n\[ \n\operatorname{cl}\left( {{C}_{1} + {C}_{2}}\right) \supset \operatorname{cl}{C}_{1} + \operatorname{cl}{C}_... | Proof. \( {C}_{1} + {C}_{2} = A\left( {{C}_{1} \oplus {C}_{2}}\right) \), where \( A \) is the addition linear transformation from \( {R}^{2n} \) to \( {R}^{n} \), i.e. \( A : \left( {{x}_{1},{x}_{2}}\right) \rightarrow {x}_{1} + {x}_{2} \) . | Yes |
Corollary 6.8.1. Let \( C \) be a non-empty convex set in \( {R}^{n} \), and let \( K \) be the convex cone in \( {R}^{n + 1} \) generated by \( \{ \left( {1, x}\right) \mid x \in C\} \) . Then \( \mathrm{{ri}}K \) consists of the pairs \( \left( {\lambda, x}\right) \) such that \( \lambda > 0 \) and \( x \in \lambda \... | Proof. Apply the theorem with \( {R}^{m} = R,{R}^{p} = {R}^{n} \) . | No |
Corollary 7.4.2. If \( f \) is a proper convex function such that \( \operatorname{dom}f \) is an affine set (which is true in particular iff is finite throughout \( {R}^{n} \) ), then \( f \) is closed. | Proof. Here \( \operatorname{dom}f \) has no relative boundary points, so \( \operatorname{cl}f \) agrees with \( f \) everywhere. | No |
Corollary 8.4.1. Let \( C \) be a closed convex set, and let \( M \) be an affine set such that \( M \cap C \) is non-empty and bounded. Then \( {M}^{\prime } \cap C \) is bounded for every affine set \( {M}^{\prime } \) parallel to \( M \) . | Proof. We have \( {0}^{ + }{M}^{\prime } = {0}^{ + }M \) by definition of \ | No |
Corollary 8.6.2. A convex function \( f \) is constant on any affine set \( M \) where it is finite and bounded above. | Proof. Redefining \( f \) to be \( + \infty \) outside \( M \) if necessary, we can assume that \( M = \operatorname{dom}f \) . Then \( f \) is closed (Corollary 7.4.2). By the preceding corollary, \( f \) is constant along every line in \( M \) . Since \( M \) contains the line through any two of its (different) point... | Yes |
Corollary 8.7.1. Let \( f \) be a closed proper convex function. If the level set \( \{ x \mid f\left( x\right) \leq \alpha \} \) is non-empty and bounded for one \( \alpha \), it is bounded for every \( \alpha \) . | Proof. Apply Theorem 8.4. | No |
Corollary 9.1.1. Let \( {C}_{1},\ldots ,{C}_{m} \) be non-empty convex sets in \( {R}^{n} \) satisfying the following condition: if \( {z}_{1},\ldots ,{z}_{m} \) are vectors such that \( {z}_{i} \in {0}^{ + }\left( {\operatorname{cl}{C}_{i}}\right) \) and \( {z}_{1} + \cdots + {z}_{m} = 0 \), then actually \( {z}_{i} \... | Proof. Let \( C \) be the direct sum \( {C}_{1} \oplus \cdots \oplus {C}_{m} \) in \( {R}^{mn} \), and let \( A \) be the linear transformation\n\n\[ \left( {{x}_{1},\ldots ,{x}_{m}}\right) \rightarrow {x}_{1} + \cdots + {x}_{m},\;{x}_{i} \in {R}^{n}. \]\n\nThen \( {AC} = {C}_{1} + \cdots + {C}_{m} \) . Since\n\n\[ \op... | Yes |
Corollary 9.1.3. Let \( {K}_{1},\ldots ,{K}_{m} \) be non-empty convex cones in \( {R}^{n} \) satisfying the following condition: if \( {z}_{i} \in \operatorname{cl}{K}_{i} \) for \( i = 1,\ldots, m \) and \( {z}_{1} + \cdots + {z}_{m} = 0 \), then \( {z}_{i} \) belongs to the lineality space of \( \operatorname{cl}{K}... | Proof. Take \( {C}_{i} = {K}_{i} \) in Corollary 9.1.1. | No |
Corollary 11.4.1. Let \( {C}_{1} \) and \( {C}_{2} \) be non-empty disjoint closed convex sets in \( {R}^{n} \) having no common directions of recession. Then there exists a hyperplane separating \( {C}_{1} \) and \( {C}_{2} \) strongly. | Proof. We have \( 0 \notin \left( {{C}_{1} - {C}_{2}}\right) \) since \( {C}_{1} \) and \( {C}_{2} \) are disjoint. But \( \operatorname{cl}\left( {{C}_{1} - {C}_{2}}\right) = {C}_{1} - {C}_{2} \) under the recession condition by Corollary 9.1.2. | Yes |
Corollary 11.4.2. Let \( {C}_{1} \) and \( {C}_{2} \) be non-empty convex sets in \( {R}^{n} \) whose closures are disjoint. If either set is bounded, there exists a hyperplane separating \( {C}_{1} \) and \( {C}_{2} \) strongly. | Proof. Apply the first corollary to cl \( {C}_{1} \) and cl \( {C}_{2} \), one of which has no directions of recession at all. | No |
Corollary 11.5.1. Let \( S \) be any subset of \( {R}^{n} \) . Then \( \mathrm{{cl}} \) (conv \( S \) ) is the intersection of all the closed half-spaces containing \( S \) . | Proof. A closed half-space contains \( C = \operatorname{cl} \) (conv \( S \) ) if and only if it contains \( S \) . | No |
Corollary 11.5.2. Let \( C \) be a convex subset of \( {R}^{n} \) other than \( {R}^{n} \) itself. Then there exists a closed half-space containing \( C \) . In other words, there exists some \( b \in {R}^{n} \) such that the linear function \( \langle \cdot, b\rangle \) is bounded above on \( C \) . | Proof. The hypothesis implies that \( \operatorname{cl}C \neq {R}^{n} \) (for otherwise \( {R}^{n} = \) ri \( \left( {\operatorname{cl}C}\right) \subset C \) ). By the theorem, a point belongs to \( \operatorname{cl}C \) if and only if it belongs to every closed half-space containing cl \( C \), so the collection of cl... | No |
Corollary 11.7.1. A non-empty closed convex cone in \( {R}^{n} \) is the intersection of the homogeneous closed half-spaces which contain it (a homogeneous half-space being one with the origin on its boundary). | Proof. Use the theorem to refine the proof of Theorem 11.5. | No |
Corollary 11.7.3. Let \( K \) be a convex cone in \( {R}^{n} \) other than \( {R}^{n} \) itself. Then \( K \) is contained in some homogeneous closed half-space of \( {R}^{n} \) . In other words, there exists some vector \( b \neq 0 \) such that \( \langle x, b\rangle \leq 0 \) for every \( x \in K \) . | Proof. Like Corollary 11.5.2. | No |
Corollary 12.1.1. If \( f \) is any function from \( {R}^{n} \) to \( \left\lbrack {-\infty ,\infty }\right\rbrack \), then cl (conv \( f \) ) is the pointwise supremum of the collection of all affine functions on \( {R}^{n} \) majorized by \( f \) . | Proof. Since \( \operatorname{cl}\left( {\operatorname{conv}f}\right) \) is the greatest closed convex function majorized by \( f \), the affine functions \( h \) such that \( h \leq \operatorname{cl}\left( {\operatorname{conv}f}\right) \) are the same as those such that \( h \leq f \) . | Yes |
For any convex function \( f \) on \( {R}^{n} \), one actually has\n\n\[ \n{f}^{ * }\left( {x}^{ * }\right) = \sup \left\{ {\left\langle {x,{x}^{ * }}\right\rangle - f\left( x\right) \mid x \in \operatorname{ri}\left( {\operatorname{dom}f}\right) }\right\} .\n\] | The supremum gives \( {g}^{ * }\left( {x}^{ * }\right) \), where \( g \) is the function which agrees with \( f \) on \( \mathrm{{ri}}\left( {\operatorname{dom}f}\right) \) but is \( + \infty \) elsewhere. We have \( \operatorname{cl}g = \operatorname{cl}f \) (Corollary 7.3.4), and hence \( {g}^{ * } = {f}^{ * } \) by ... | Yes |
Corollary 12.3.1. A closed convex function \( f \) is symmetric with respect to a given set \( G \) of orthogonal linear transformations if and only if \( {f}^{ * } \) is symmetric with respect to \( G \) . | Proof. Specializing Theorem 12.3 to the case where \( h = f, a = 0 = \) \( {a}^{ * },\alpha = 0 \), we see that \( {fA} = f \) implies \( {f}^{ * }{A}^{* - 1} = {f}^{ * } \) . When \( A \) is orthogonal, \( {A}^{* - 1} = A \) by definition. Thus if \( {fA} = f \) for every \( A \in G \), then \( {f}^{ * }A = {f}^{ * } ... | Yes |
Corollary 13.3.1. Let \( f \) be a closed convex function on \( {R}^{n} \). In order that \( {f}^{ * } \) be finite everywhere, so that \( \operatorname{dom}{f}^{ * } = {R}^{n} \), it is necessary and sufficient that \( f \) be co-finite. | Proof. We have \( \operatorname{dom}{f}^{ * } = {R}^{n} \) if and only if \( \operatorname{dom}{f}^{ * } \) is not contained in any closed half-space of \( {R}^{n} \) (Corollary 11.5.2). This is equivalent to the condition that \( {\delta }^{ * }\left( {x \mid \operatorname{dom}{f}^{ * }}\right) < + \infty \) only for ... | Yes |
Corollary 13.3.4. Let \( f \) be a closed proper convex function. Let \( {x}^{ * } \) be a fixed vector and let \( g\left( x\right) = f\left( x\right) - \left\langle {x,{x}^{ * }}\right\rangle \) . Then\n\n(a) \( {x}^{ * } \in \operatorname{cl}\left( {\operatorname{dom}{f}^{ * }}\right) \) if and only if \( \left( {g{0... | Proof. Let \( C = \left( {\operatorname{dom}{f}^{ * }}\right) - {x}^{ * } \) . Clearly \( {x}^{ * } \in \operatorname{cl}\left( {\operatorname{dom}{f}^{ * }}\right) \) if and only if \( 0 \in \mathrm{{cl}}C \), and so forth. We have \( {g}^{ * }\left( {y}^{ * }\right) = {f}^{ * }\left( {{y}^{ * } + {x}^{ * }}\right) \)... | Yes |
Corollary 13.4.1. Closed proper convex functions conjugate to each other have the same rank. | Proof. This is immediate from the formulas in the theorem and the definition of rank. | No |
Corollary 13.4.2. Let \( f \) be a closed proper convex function. Then \( \operatorname{dom}{f}^{ * } \) has a non-empty interior if and only if there are no lines along which \( f \) is (finite and) affine. | Proof. The dimension of \( {f}^{ * } \) is \( n \) if and only if the lineality of \( f \) is 0 . | No |
Corollary 14.5.1. Let \( C \) be a closed convex set containing the origin. Then \( {C}^{ \circ } \) is bounded if and only if \( 0 \in \) int \( C \) . Dually, \( C \) is bounded if and only if \( 0 \in \) int \( {C}^{ \circ } \) . | Proof. We have \( {C}^{ \circ } \) bounded if and only if the support function \( \gamma \left( {\cdot \mid C}\right) \) of \( {C}^{ \circ } \) is finite everywhere (Corollary 13.2.2.). On the other hand, \( \gamma \left( {\cdot \mid C}\right) \) is finite everywhere if and only if \( 0 \in \) int \( C \) (Corollary 6.... | Yes |
Corollary 15.3.1. A closed proper convex function \( f \) is positively homogeneous of degree \( p \), where \( 1 < p < \infty \), if and only if it is of the form\n\n\[ f\left( x\right) = \left( {1/p}\right) k{\left( x\right) }^{p} \]\n\nfor a certain closed gauge \( k \). For such an \( f \), the conjugate of \( f \)... | Proof. If \( f \) is positively homogeneous of degree \( p \), then \( f \) is gauge-like. The corollary follows from the fact that the function \( g\left( \zeta \right) = \left( {1/p}\right) {\zeta }^{p},\zeta \geq 0 \) , satisfies the conditions of the theorem and has \( {g}^{ + }\left( {\zeta }^{ * }\right) = \left(... | Yes |
Corollary 15.5.1. If \( f \) is any non-negative closed convex function vanishing at the origin, one has \( {f}^{{ * }^{ \circ }} = {f}^{\circ * } \) . | Proof. \( {f}^{\circ * } = {g}^{* * } = g = {f}^{* \circ } \) . | No |
Corollary 16.1.1. For any non-empty convex set \( C \), one has \( {\delta }^{ * }\left( {{x}^{ * } \mid {\lambda C}}\right) = \lambda {\delta }^{ * }\left( {{x}^{ * } \mid C}\right) ,0 \leq \lambda < \infty . \) | Proof. Take \( f\left( x\right) = \delta \left( {x \mid C}\right) \). | No |
Corollary 16.2.1. Let \( A \) be a linear transformation from \( {R}^{n} \) to \( {R}^{m} \) . Let \( g \) be a proper convex function on \( {R}^{m} \) . In order that there exist no vector \( {y}^{ * } \in {R}^{m} \) such that\n\n\[ \n{A}^{ * }{y}^{ * } = 0,\;\left( {{g}^{ * }{0}^{ + }}\right) \left( {y}^{ * }\right) ... | Proof. For the subspace \( L = \left\{ {{Ax} \mid x \in {R}^{n}}\right\} \) one has\n\n\[ \n{L}^{ \bot } = \left\{ {{y}^{ * } \mid {A}^{ * }{y}^{ * } = 0}\right\}\n\]\n\nApply the lemma to \( L \) and \( g \) . | No |
Corollary 16.3.1. Let \( A \) be a linear transformation from \( {R}^{n} \) to \( {R}^{m} \). For any convex set \( C \) in \( {R}^{n} \), one has\n\n\[ \n{\delta }^{ * }\left( {{y}^{ * } \mid {AC}}\right) = {\delta }^{ * }\left( {{A}^{ * }{y}^{ * } \mid C}\right) ,\;\forall {y}^{ * } \in {R}^{m}.\n\]\n\nFor any convex... | Proof. Take \( f\left( x\right) = \delta \left( {x \mid C}\right), g\left( y\right) = \delta \left( {y \mid D}\right) \). | No |
Corollary 16.4.1. Let \( {C}_{1},\ldots ,{C}_{m} \) be non-empty convex sets in \( {R}^{n} \) . Then\n\n\[ \n{\delta }^{ * }\left( {\cdot \mid {C}_{1} + \cdots + {C}_{m}}\right) = {\delta }^{ * }\left( {\cdot \mid {C}_{1}}\right) + \cdots + {\delta }^{ * }\left( {\cdot \mid {C}_{m}}\right) ,\n\]\n\n\[ \n{\delta }^{ * }... | Proof. Take \( {f}_{i} = \delta \left( {\cdot \mid {C}_{i}}\right) \) . \( \parallel \) | No |
Corollary 16.4.2. Let \( {K}_{1},\ldots ,{K}_{m} \) be non-empty convex cones in \( {R}^{n} \). Then \[ {\left( {K}_{1} + \cdots + {K}_{m}\right) }^{ \circ } = {K}_{1}^{ \circ } \cap \cdots \cap {K}_{m}^{ \circ }, \] \[ {\left( \mathrm{{cl}}{K}_{1} \cap \cdots \cap \mathrm{{cl}}{K}_{m}\right) }^{ \circ } = \mathrm{{cl}... | Proof. Apply the theorem to \( {f}_{i} = \delta \left( {\cdot \mid {K}_{i}}\right) \). One has \( {f}_{i}^{ * } = \delta \left( {\cdot \mid {K}_{i}^{ \circ }}\right) \), as explained at the beginning of §14. | No |
Corollary 19.3.1. Let \( A \) be a linear transformation from \( {R}^{n} \) to \( {R}^{m} \). For each polyhedral convex function \( f \) on \( {R}^{n} \), the convex function \( {Af} \) is polyhedral on \( {R}^{m} \), and the infimum in its definition, if finite, is attained. For each polyhedral convex function \( g \... | Proof. The image of epi \( f \) under the linear transformation \( \left( {x,\mu }\right) \rightarrow \left( {{Ax},\mu }\right) \) is a polyhedral convex set, and it equals epi \( \left( {Af}\right) \). The inverse image of epi \( g \) under this same transformation is a polyhedral convex set, and it equals epi \( \lef... | Yes |
Corollary 20.3.1. Let \( {C}_{1} \) and \( {C}_{2} \) be non-empty convex sets in \( {R}^{n} \) such that \( {C}_{1} \) is polyhedral, \( {C}_{2} \) is closed and \( {C}_{1} \cap {C}_{2} = \varnothing \) . Suppose that \( {C}_{1} \) and \( {C}_{2} \) have no common directions of recession, except for directions in whic... | Proof. According to Theorem 11.4, strong separation is possible if \( 0 \notin \operatorname{cl}\left( {{C}_{1} - {C}_{2}}\right) \) . We have \( 0 \notin {C}_{1} - {C}_{2} \), of course, since \( {C}_{1} \) and \( {C}_{2} \) are disjoint. The direction hypothesis implies by the present theorem that \( {C}_{1} + \left(... | Yes |
Corollary 22.3.1 (Farkas’ Lemma). An inequality \( \left\langle {{a}_{0}, x}\right\rangle \leq 0 \) is a consequence of the system\n\n\[ \left\langle {{a}_{i}, x}\right\rangle \leq 0,\;i = 1,\ldots, m \]\n\nif and only if there exist non-negative real numbers \( {\lambda }_{1},\ldots ,{\lambda }_{m} \) such that\n\n\[ ... | Proof. The hypothesis of Theorem 22.3 is satisfied, because the zero vector satisfies \( \left\langle {{a}_{i}, x}\right\rangle \leq 0 \) for \( i = 1,\ldots, m \) . | No |
Corollary 23.5.3. Let \( C \) be a non-empty closed convex set. Then, for each vector \( {x}^{ * },\partial {\delta }^{ * }\left( {{x}^{ * } \mid C}\right) \) consists of the points \( x \) (if any) where the linear function \( \left\langle {\cdot ,{x}^{ * }}\right\rangle \) achieves its maximum over \( C \) . | Proof. Take \( f = \delta \left( {\cdot \mid C}\right) \) in the theorem, so that \( {f}^{ * } \) is the support function \( {\delta }^{ * }\left( {\cdot \mid C}\right) \) . Invoke the equivalence of \( \left( {\mathrm{a}}^{ * }\right) \) and (b). | No |
Corollary 26.3.1. Let \( f \) be a closed proper convex function. Then \( \partial f \) is a one-to-one mapping if and only if \( f \) is strictly convex on \( \operatorname{int}\left( {\operatorname{dom}f}\right) \) and essentially smooth. | Proof. We have \( {\left( \partial f\right) }^{-1} = \partial {f}^{ * } \) by Corollary 23.5.1. Thus, by Theorem 26.1, \( \partial f \) is one-to-one if and only if \( f \) and \( {f}^{ * } \) are both essentially smooth. Since \( f \) is the conjugate of \( {f}^{ * } \), the essential smoothness of \( {f}^{ * } \) is ... | Yes |
Corollary 28.1.1. Let \( \left( P\right) \) be an ordinary convex program, and let \( \left( {{\lambda }_{1},\ldots ,{\lambda }_{m}}\right) \) be a Kuhn-Tucker vector for \( \left( P\right) \) . Assume that the functions \( {f}_{i} \) are all closed. If the infimum of \[ h = {f}_{0} + {\lambda }_{1}{f}_{1} + \cdots + {... | Proof. The hypothesis implies that \( h \) and the objective function \( f \) of \( \left( P\right) \) are closed. Suppose that the infimum of \( h \) is attained at a unique point \( \bar{x} \) . The corollary will follow from the theorem if we show that \( \left( P\right) \) has at least one optimal solution, i.e. th... | No |
The Lagrangian of \( \left( P\right) \) is given simply by \[ L\left( {{v}_{1}^{ * }, x}\right) = - {v}_{1}^{ * } + \mathop{\sum }\limits_{{k = 1}}^{n}\left\lbrack {{f}_{0k}\left( {\xi }_{k}\right) + {v}_{1}^{ * }{\xi }_{k}}\right\rbrack \] for every \( {v}_{1}^{ * } \in R \) and \( x \in {R}^{n} \), where \( {v}_{1}^{... | Computing the minimum of the convex function \( - g \) is, of course, a relatively easy matter, since only one real variable \( {v}_{1}^{ * } \) is involved and the conjugate functions \( {f}_{0k}^{ * } \) are fairly simple to determine. Thus the decomposition principle allows us in this example to replace a problem in... | Yes |
Corollary 29.1.2. Let \( F \) be any convex bifunction from \( {R}^{m} \) to \( {R}^{n} \) . Suppose that the optimal value in the convex program \( \left( P\right) \) associated with \( F \) is finite. A Kuhn-Tucker vector then fails to exist for \( \left( P\right) \) if and only if there exists a vector \( u \in {R}^... | Proof. Apply Theorem 23.3 to inf \( F \) . | No |
Corollary 30.2.2. Let \( F \) be a closed convex bifunction from \( {R}^{m} \) to \( {R}^{n} \) , and let \( \left( P\right) \) be the convex program associated with \( F \) . Then the optimal value inf \( {F0} \) in \( \left( P\right) \) and the optimal value \( \sup {F}^{ * }0 \) in \( \left( {P}^{ * }\right) \) sati... | Proof. According to Theorem 30.2,\n\n\[ \left( {\mathrm{{cl}}\left( {\inf F}\right) }\right) \left( 0\right) = - {\left( {F}^{ * }0\right) }^{ * }\left( 0\right) = - \mathop{\inf }\limits_{{u}^{ * }}\left\{ {\left\langle {0,{u}^{ * }}\right\rangle - \left( {{F}^{ * }0}\right) \left( {u}^{ * }\right) }\right\} \]\n\n\[ ... | Yes |
Corollary 30.2.3. Let \( F \) be a closed convex bifunction from \( {R}^{m} \) to \( {R}^{n} \) , and let \( \left( P\right) \) be the associated convex program. Except in the case where neither \( \left( P\right) \) nor \( \left( {P}^{ * }\right) \) is consistent, one has\n\n\[ \mathop{\liminf }\limits_{{u \rightarrow... | Proof. From what we know in general about the closure operation for convex functions, the formula\n\n\[ \left( {\mathrm{{cl}}\left( {\inf F}\right) }\right) \left( 0\right) = \mathop{\liminf }\limits_{{u \rightarrow 0}}\left( {\inf F}\right) \left( u\right) \]\n\nholds except in cases where the left side is \( - \infty... | Yes |
Corollary 31.4.1. Let \( f \) be a closed proper convex function on \( {R}^{n} \) . One has\n\n\[ \inf \{ f\left( x\right) \mid x \geq 0\} = - \inf \left\{ {{f}^{ * }\left( {x}^{ * }\right) \mid {x}^{ * } \geq 0}\right\} \]\n\nif either of the following conditions holds:\n\n(a) There exists a vector \( x \in \operatorn... | Proof. Take \( K \) to be the non-negative orthant of \( {R}^{n} \) . | No |
Corollary 31.4.2. Let \( f \) be a closed proper convex function on \( {R}^{n} \) , and let \( L \) be a subspace of \( {R}^{n} \) . One has\n\n\[ \inf \left\{ {f\left( x\right) \mid x \in L}\right\} = - \inf \left\{ {{f}^{ * }\left( {x}^{ * }\right) \mid {x}^{ * } \in {L}^{ \bot }}\right\} \]\n\nif either of the follo... | Proof. Take \( K = L \) .\n\nIf \( f\left( x\right) = h\left( {z + x}\right) - \left\langle {{z}^{ * }, x}\right\rangle \), where \( z \) and \( {z}^{ * } \) are given vectors and \( h \) is any closed proper convex function, then\n\n\[ {f}^{ * }\left( {x}^{ * }\right) = {h}^{ * }\left( {{z}^{ * } + {x}^{ * }}\right) -... | Yes |
Corollary 32.2.1. Let \( f \) be a convex function, and let \( C \) be any closed convex set which is not merely an affine set or half of an affine set. The supremum of f relative to \( C \) is then the same as the supremum of f relative to the relative boundary of \( C \), and the former is attained only when the latt... | Proof. Here \( C \) is the convex hull of its relative boundary by Theorem 18.4. | No |
Corollary 32.3.1. Let \( f \) be a convex function, and let \( C \) be a closed convex set contained in \( \operatorname{dom}f \) . Suppose that \( C \) contains no lines. Then, if the supremum of \( f \) relative to \( C \) is attained at all, it is attained at some extreme point of \( C \) . | Proof. If \( C \) contains no lines, then \( L = \{ 0\} \) and \( C \cap {L}^{ \bot } = C \) . | No |
Corollary 33.1.1. If \( K \) is any concave-convex function on \( {R}^{m} \times {R}^{n} \) , then \( {\mathrm{{cl}}}_{1}K \) and \( {\mathrm{{cl}}}_{2}K \) are concave-convex functions such that \( {\mathrm{{cl}}}_{1}K \) is concave-closed and \( {\operatorname{cl}}_{2}K \) is convex-closed. (Similarly for convex-conc... | Proof. According to the theorem, \( \left( {{\mathrm{{cl}}}_{2}K}\right) \left( {u,{x}^{ * }}\right) \) is of the form \( \left\langle {{Fu},{x}^{ * }}\right\rangle \) for a certain convex bifunction \( F \), where \( \left\langle {{Fu},{x}^{ * }}\right\rangle \) is concave in \( u \) and closed convex in \( {x}^{ * } ... | No |
Corollary 35.8.1. Let \( K \) be a concave-convex function on \( {R}^{m} \times {R}^{n} \) , and let \( \left( {u, v}\right) \) be a point at which \( K \) is finite. A necessary and sufficient condition for \( K \) to be differentiable at \( \left( {u, v}\right) \) is that \( K \) be finite on a neighborhood of \( \le... | Proof. This follows by Theorem 25.2. | No |
Corollary 37.2.1. In the notation of the theorem, one has \( 0 \in \) int \( {D}^{ * } \) if and only if the convex functions \( K\left( {u, \cdot }\right) \) for \( u \in \mathrm{{ri}}C \) have no common direction of recession. Similarly, one has \( 0 \in \) int \( {C}^{ * } \) if and only if the convex functions \( -... | Proof. One has \( 0 \notin \) int \( {D}^{ * } \) if and only if there exists a vector \( w \neq 0 \) such that \( {\delta }^{ * }\left( {w \mid {D}^{ * }}\right) \leq 0 \), i.e. (according to the preceding proof)\n\n\[ K\left( {u, v + w}\right) - K\left( {u, v}\right) \leq 0,\;\forall v \in D,\;\forall u \in \operator... | Yes |
Corollary 37.6.1. Let \( K \) be a closed proper concave-convex function on \( {R}^{m} \times {R}^{n} \) with effective domain \( C \times D \) . If \( C \) and \( D \) are bounded, \( K \) has a saddle-point and a finite saddle-value. | Proof. As for Corollary 37.3.1. | No |
Corollary 38.7.1. Let \( F \) be a proper convex bifunction from \( {R}^{m} \) to \( {R}^{n} \). Let \( f \) be a proper convex function on \( {R}^{m} \) such that \( \operatorname{ri}\left( {\operatorname{dom}f}\right) \) meets ri \( \left( {\operatorname{dom}F}\right) \). Then \( \left\langle {f,{F}^{ * }{x}^{ * }}\r... | Proof. Given any \( {x}^{ * } \), apply the theorem with \( g = \left\langle {\cdot ,{x}^{ * }}\right\rangle \) . | No |
Corollary 38.7.2. Let \( F \) be a proper convex bifunction from \( {R}^{m} \) to \( {R}^{n} \) , and let \( G \) be a proper convex bifunction from \( {R}^{n} \) to \( {R}^{p} \) . Assume that ri \( \left( {\operatorname{dom}{F}_{ * }}\right) \) and \( \operatorname{ri}\left( {\operatorname{dom}G}\right) \) have a poi... | Proof. The first and last of these inner products are equal simply by Corollary 33.2.1, since \( {F}^{ * }{G}^{ * } = {\left( GF\right) }^{ * } \) by Theorem 38.5. On the other hand, the first equality is valid by the preceding corollary if \( \mathrm{{ri}}\left( {\operatorname{dom}{Fu}}\right) \) meets ri (dom \( G \)... | No |
Corollary 39.7.1. Let \( A \) be a closed convex process from \( {R}^{m} \) to \( {R}^{n} \), and let \( C \) be a non-empty closed convex set in \( {R}^{m} \). If no non-zero vector in \( {A}^{-1}0 \) belongs to the recession cone of \( C \) (which is true in particular if \( C \) is bounded), then \( {AC} \) is close... | Proof. Make \( A \) supremum oriented, and apply the theorem with \( f = \delta \left( {\cdot \mid C}\right) \). The set \( K = \operatorname{dom}{f}^{ * } \) is the barrier cone of \( C \), and its polar is the recession cone of \( C \) (Corollary 14.2.1). If the convex cones \( K \) and\n\n\[ \operatorname{dom}{A}^{*... | Yes |
The n-dimensional non-negative orthant, \( {\mathcal{R}}_{ + }^{n} = \left\{ {x \in {\mathcal{R}}^{n}}\right. \) : \( x \geq 0\} \), is a convex cone. | The dual of the cone is also \( {\mathcal{R}}_{ + }^{n} \) ; it is self-dual. | No |
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