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Example 5.4 Minimizing a separable function subject to an equality constraint. We consider the problem\n\n\\[ \n\\text{ minimize }\\;{f}_{0}\\left( x\\right) = \\mathop{\\sum }\\limits_{{i = 1}}^{n}{f}_{i}\\left( {x}_{i}\\right) \n\\]\n\n\\[ \n\\text{subject to}\\;{a}^{T}x = b\\text{,}\n\\]\n\nwhere \\( a \\in {\\mathb... | The Lagrangian is\n\n\\[ \nL\\left( {x,\\nu }\\right) = \\mathop{\\sum }\\limits_{{i = 1}}^{n}{f}_{i}\\left( {x}_{i}\\right) + \\nu \\left( {{a}^{T}x - b}\\right) = - {b\\nu } + \\mathop{\\sum }\\limits_{{i = 1}}^{n}\\left( {{f}_{i}\\left( {x}_{i}\\right) + \\nu {a}_{i}{x}_{i}}\\right) ,\n\\]\n\nwhich is also separable... | Yes |
Consider the unconstrained geometric program\n\n\\[ \n\\text{minimize}\\log \\left( {\\mathop{\\sum }\\limits_{{i = 1}}^{m}\\exp \\left( {{a}_{i}^{T}x + {b}_{i}}\\right) }\\right) \\text{.}\n\\] | We first reformulate it by introducing new variables and equality constraints:\n\n\\[ \n\\text{ minimize }\\;{f}_{0}\\left( y\\right) = \\log \\left( {\\mathop{\\sum }\\limits_{{i = 1}}^{m}\\exp {y}_{i}}\\right)\n\\]\n\n\\[ \n\\text{subject to}\\{Ax} + b = y\\text{,}\n\\]\n\nwhere \\( {a}_{i}^{T} \\) are the rows of \\... | Yes |
We consider the unconstrained norm approximation problem\n\n\\[ \n\\text{minimize}\\parallel {Ax} - b\\parallel \\text{,}\n\\]\n\n\\( \\left( {5.63}\\right) \\)\n\nwhere \\( \\parallel \\cdot \\parallel \\) is any norm. | Once again we reformulate the problem as\n\n\\[ \n\\begin{array}{ll} \\text{ minimize } & \\parallel y\\parallel \\\\ \\text{ subject to } & {Ax} - b = y. \\end{array}\n\\]\n\nThe Lagrange dual problem is, following (5.61),\n\n\\[ \n\\text{maximize}\\;{b}^{T}\\nu\n\\]\n\n\\[ \n\\text{subject to}\\parallel \\nu {\\paral... | Yes |
Example 5.7 Inequality constrained geometric program. The inequality constrained geometric program\n\n\[ \n\\begin{array}{ll} \\text{ minimize } & \\log \\left( {\\mathop{\\sum }\\limits_{{k = 1}}^{{K}_{0}}{e}^{{a}_{0k}^{T}x + {b}_{0k}}}\\right) \\\\\n\\text{ subject to } & \\log \\left( {\\mathop{\\sum }\\limits_{{k =... | is of the form (5.65) with \( {f}_{i} : {\\mathbf{R}}^{{K}_{i}} \\rightarrow \\mathbf{R} \) given by \( {f}_{i}\\left( y\\right) = \\log \\left( {\\mathop{\\sum }\\limits_{{k = 1}}^{{K}_{i}}{e}^{{y}_{k}}}\\right) \). The conjugate of this function is\n\n\[ \n{f}_{i}^{ * }\\left( \\nu \\right) = \\left\\{ \\begin{array}... | Yes |
Example 5.8 We consider again the minimum norm problem\n\n\[ \n\\text{minimize}\\parallel {Ax} - b\\parallel \\text{,}\n\]\n\nwhere \( \\parallel \\cdot \\parallel \) is some norm. We reformulate this problem as\n\n\[ \n\\text{minimize}\\;\\left( {1/2}\\right) \\parallel y{\\parallel }^{2}\n\]\n\n\[ \n\\text{subject to... | The dual of the reformulated problem is\n\n\[ \n\\text{maximize}\\; - \\left( {1/2}\\right) \\parallel \\nu {\\parallel }_{ * }^{2} + {b}^{T}\\nu\n\]\n\n\[ \n\\text{subject to}\\{A}^{T}\\nu = 0\\text{,}\n\]\n\nwhere we use the fact that the conjugate of \( \\left( {1/2}\\right) \\parallel \\cdot {\\parallel }^{2} \) is... | Yes |
Linear program with box constraints. We consider the linear program\n\n\\[ \n\\text{minimize}\\;{c}^{T}x \n\\]\n\n\\[ \n\\text{subject to}\\;{Ax} = b \n\\]\n\n(5.68)\n\n\\[ \nl \\preccurlyeq x \\preccurlyeq u \n\\]\n\nwhere \\( A \\in {\\mathbf{R}}^{p \\times n} \\) and \\( l \\prec u \\) . The constraints \\( l \\prec... | We can, of course, derive the dual of this linear program. The dual will have a Lagrange multiplier \\( \\nu \\) associated with the equality constraint, \\( {\\lambda }_{1} \\) associated with the inequality constraint \\( x \\preccurlyeq u \\), and \\( {\\lambda }_{2} \\) associated with the inequality constraint \\(... | Yes |
If there is an investment vector \( x \) with \( {p}^{T}x < 0 \), and in all possible scenarios, the final value is nonnegative, i.e., \( {v}^{\left( i\right) T}x \geq 0 \) for \( i = 1,\ldots, m \), then an arbitrage is said to exist. The condition \( {p}^{T}x < 0 \) means you are paid to accept the investment mix, an... | Using Farkas’ lemma, we have no arbitrage if and only if there exists \( y \) such that\n\n\[ \n- {V}^{T}y + p = 0,\;y \succcurlyeq 0. \n\] | Yes |
Example 5.11 Lagrange dual of semidefinite program. We consider a semidefinite program in inequality form,\n\n\[ \n\\text{minimize}\;{c}^{T}x \n\]\n\n\[ \n\\text{subject to}\;{x}_{1}{F}_{1} + \\cdots + {x}_{n}{F}_{n} + G \\preccurlyeq 0 \n\]\n\n\\( \\left( {5.93}\\right) \\)\n\nwhere \\( {F}_{1},\\ldots ,{F}_{n}, G \\i... | We associate with the constraint a dual variable or multiplier \\( Z \\in {\\mathbf{S}}^{k} \\), so the Lagrangian is\n\n\[ \nL\\left( {x, Z}\\right) = {c}^{T}x + \\operatorname{tr}\\left( {\\left( {{x}_{1}{F}_{1} + \\cdots + {x}_{n}{F}_{n} + G}\\right) Z}\\right) \n\]\n\n\[ \n= {x}_{1}\\left( {{c}_{1} + \\operatorname... | Yes |
Example 5.12 Lagrange dual of cone program in standard form. We consider the cone program\n\n\\[ \n\\text{minimize}\\,{c}^{T}x \n\\]\n\n\\[ \n\\text{subject to}\\,{Ax} = b \n\\]\n\n\\[ \nx{ \\succcurlyeq }_{K}0 \n\\]\n\nwhere \\( A \\in {\\mathbf{R}}^{m \\times n}, b \\in {\\mathbf{R}}^{m} \\), and \\( K \\subseteq {\\... | By eliminating \\( \\lambda \\) and defining \\( y = - \\nu \\), this problem can be simplified to\n\n\\[ \n\\begin{array}{ll} \\text{ maximize } & {b}^{T}y \\\\ \\text{ subject to } & {A}^{T}y{ \\preccurlyeq }_{K^{ * }} \\end{array} \n\\]\n\nwhich is a cone program in inequality form, involving the dual generalized in... | Yes |
Example 5.13 Semidefinite program in inequality form. We consider a semidefinite program in inequality form, as in example 5.11. The primal problem is\n\n\[ \n\\text{minimize}\;{c}^{T}x \n\]\n\n\[ \n\\text{subject to}\;F\\left( x\\right) = {x}_{1}{F}_{1} + \\cdots + {x}_{n}{F}_{n} + G \\preccurlyeq 0\\text{,} \n\]\n\nw... | Suppose that \( {x}^{ \\star } \) and \( {Z}^{ \\star } \) are primal and dual optimal, respectively, with zero duality gap. The complementary slackness condition is \( \\operatorname{tr}\\left( {F\\left( {x}^{ \\star }\\right) {Z}^{ \\star }}\\right) = 0 \) . Since \( F\\left( {x}^{ \\star }\\right) \\preccurlyeq 0 \)... | Yes |
The following systems are strong alternatives:\n\n\[ F\left( x\right) = {x}_{1}{F}_{1} + \cdots + {x}_{n}{F}_{n} + G \prec 0, \] \n\nwhere \( {F}_{i}, G \in {\mathbf{S}}^{k} \), and\n\n\[ Z \succcurlyeq 0,\;Z \neq 0,\;\operatorname{tr}\left( {GZ}\right) \geq 0,\;\operatorname{tr}\left( {{F}_{i}Z}\right) = 0,\;i = 1,\ld... | This follows from the general result, if we take for \( K \) the positive semidefinite cone \( {\mathbf{S}}_{ + }^{k} \), and\n\n\[ g\left( Z\right) = \mathop{\inf }\limits_{x}\left( {\operatorname{tr}\left( {F\left( x\right) Z}\right) }\right) = \left\{ \begin{array}{ll} \operatorname{tr}\left( {GZ}\right) & \operator... | Yes |
Example 6.2 Robust regression. Figure 6.5 shows 42 points \( \left( {{t}_{i},{y}_{i}}\right) \) in a plane, with two obvious outliers (one at the upper left, and one at lower right). The dashed line shows the least-squares approximation of the points by a straight line \( f\left( t\right) = \alpha + {\beta t} \) . The ... | The solid line shows the robust least-squares approximation, obtained by minimizing the Huber penalty function\n\n\[ \n\text{ minimize }\;\mathop{\sum }\limits_{{i = 1}}^{{42}}{\phi }_{\mathrm{{hub}}}\left( {{y}_{i} - \alpha - \beta {t}_{i}}\right) , \n\]\n\nwith \( M = 1 \) . This approximation is far less affected by... | Yes |
Example 6.3 Optimal input design. We consider a dynamical system with scalar input sequence \( u\left( 0\right), u\left( 1\right) ,\ldots, u\left( N\right) \), and scalar output sequence \( y\left( 0\right), y\left( 1\right) ,\ldots, y\left( N\right) \) , related by convolution:\n\n\[ y\left( t\right) = \mathop{\sum }\... | Now we consider a specific example, with \( N = {200} \), and impulse response\n\n\[ h\left( t\right) = \frac{1}{9}{\left( {0.9}\right) }^{t}\left( {1 - {0.4}\cos \left( {2t}\right) }\right) . \]\n\nFigure 6.6 shows the optimal input, and corresponding output (along with the desired trajectory \( {y}_{\text{des }} \) )... | Yes |
Example 6.4 Regressor selection problem. We are given a matrix \( A \in {\mathbf{R}}^{m \times n} \) , whose columns are potential regressors, and a vector \( b \in {\mathbf{R}}^{m} \) that is to be fit by a linear combination of \( k < n \) columns of \( A \) . The problem is to choose the subset of \( k \) regressors... | One straightforward approach is to check every possible sparsity pattern in \( x \) with \( k \) nonzero entries. For a fixed sparsity pattern, we can find the optimal \( x \) by solving a least-squares problem, i.e., minimizing \( \parallel \widetilde{A}\widetilde{x} - b{\parallel }_{2} \), where \( \widetilde{A} \) d... | Yes |
To illustrate the difference between the stochastic and worst-case formulations of the robust approximation problem, we consider the least-squares problem\n\n\[ \n\\text{ minimize }\\parallel A\\left( u\\right) x - b{\\parallel }_{2}^{2}, \n\]\n\nwhere \( u \\in \\mathbf{R} \) is an uncertain parameter and \( A\\left( ... | We find three approximate solutions:\n\n- Nominal optimal. The optimal solution \( {x}_{\\text{nom }} \) is found, assuming \( A\\left( u\\right) \) has its nominal value \( {A}_{0} \).\n\n- Stochastic robust approximation. We find \( {x}_{\\text{stoch }} \), which minimizes \( \\mathbf{E}\\parallel A\\left( u\\right) ... | Yes |
We consider an instance of the robust approximation problem\n\n\\[ \n\\text{minimize}\\mathop{\\sup }\\limits_{{\\parallel u{\\parallel }_{2} \\leq 1}}{\\begin{Vmatrix}\\left( \\bar{A} + {u}_{1}{A}_{1} + {u}_{2}{A}_{2}\\right) x - b\\end{Vmatrix}}_{2}\\text{,}\n\\]\n\n\\( \\left( {6.16}\\right) \\)\n\nwith dimensions \... | We compute the optimal solution of the robust least-squares problem (6.16) \\( {x}_{\\mathrm{{rls}}} \\), as well as the solution of the nominal least-squares problem \\( {x}_{\\mathrm{{ls}}} \\) (i.e., assuming \\( u = 0 \\) ), and also the Tikhonov regularized solution \\( {x}_{\\text{tik }} \\), with \\( \\delta = 1... | Yes |
In another variation of function fitting, we have fewer data points than the dimension of the subspace of functions. In the simplest case, we require that the function we choose must satisfy the interpolation conditions\n\n\[ f\left( {u}_{i}\right) = {y}_{i},\;i = 1,\ldots, m, \] | which are linear equality constraints on \( x \) . Among the functions that satisfy these interpolation conditions, we might seek one that is smoothest, or smallest. These lead to least-norm problems. | No |
We consider baskets of two goods (so we can easily plot the goods baskets). To generate the consumer preference data \( \mathcal{P} \), we compute 40 random points in \( {\left\lbrack 0,1\right\rbrack }^{2} \), and then compare them using the utility function\n\n\[ u\left( {{x}_{1},{x}_{2}}\right) = \left( {{1.1}{x}_{1... | We find that 21 of the baskets are definitely rejected in favor of \( \left( {{0.5},{0.5}}\right) \), and 14 of the baskets are definitely preferred. We cannot make any conclusion, from the consumer preference data, about the remaining 5 baskets. These results are shown in figure 6.26. Note that goods baskets below and... | Yes |
We consider a probability distribution on 100 equidistant points \( {\alpha }_{i} \) in the interval \( \left\lbrack {-1,1}\right\rbrack \) . We impose the following prior assumptions:\n\n\[ \mathbf{E}X \in \left\lbrack {-{0.1},{0.1}}\right\rbrack \]\n\n\[ \mathbf{E}{X}^{2} \in \left\lbrack {{0.5},{0.6}}\right\rbrack \... | Figure 7.2 shows the maximum entropy distribution that satisfies these constraints. The maximum entropy distribution satisfies\n\n\[ \mathbf{E}X = {0.056} \]\n\n\[ \text{E}{X}^{2} = {0.5} \]\n\n\[ \mathbf{E}\left( {3{X}^{3} - {2X}}\right) = - {0.2} \]\n\n\[ \operatorname{prob}\left( {X < 0}\right) = {0.4}\text{.} \] | Yes |
Bounding risk probability with known marginal distributions. Suppose \( X \) and \( Y \) are two random variables that give the return on two investments. We assume that \( X \) takes values in \( \left\{ {{\alpha }_{1},\ldots ,{\alpha }_{n}}\right\} \subseteq \mathbf{R} \) and \( Y \) takes values in \( \left\{ {{\bet... | \[ \mathop{\sum }\limits_{{j = 1}}^{m}{p}_{ij} = {r}_{i},\;i = 1,\ldots, n,\;\mathop{\sum }\limits_{{i = 1}}^{n}{p}_{ij} = {q}_{j},\;j = 1,\ldots, m, \] (7.9) but otherwise nothing is known about the joint distribution \( p \) . This defines a polyhedron of joint distributions consistent with the given marginals. Now s... | Yes |
Example 7.4 We consider a binary hypothesis testing example with \( n = 4 \), and\n\n\[ P = \left\lbrack \begin{array}{ll} {0.70} & {0.10} \\ {0.20} & {0.10} \\ {0.05} & {0.70} \\ {0.05} & {0.10} \end{array}\right\rbrack \]\n\nThe optimal trade-off curve between \( {P}_{\mathrm{{fn}}} \) and \( {P}_{\mathrm{{fp}}} \), ... | This minimax detector yields equal probability of a false positive and false negative, which in this case is \( 1/6 \). Every deterministic detector has either a false positive or false negative probability that exceeds \( 1/6 \) , so this is an example where a randomized detector outperforms every deterministic detect... | Yes |
Designing a robust minimax detector for this set \( \mathcal{P} \) can be interpreted as a binary hypothesis testing problem: based on an observation of a random variable \( X \in \) \( \left\{ {{a}_{1},\ldots ,{a}_{n}}\right\} \), choose between the following two hypotheses:\n\n1. \( \mathbf{E}X = {\alpha }_{1},\mathb... | Let \( {\widetilde{t}}^{T} \) denote the first row of \( T \) (and so, \( {\left( \mathbf{1} - \widetilde{t}\right) }^{T} \) is the second row). For given \( \widetilde{t} \), the worst-case probabilities of correct detection are\n\n\[ {D}_{11}^{\mathrm{{wc}}} = \inf \left\{ {{\widetilde{t}}^{T}p\left| {\;\mathop{\sum ... | Yes |
Example 8.1 Projection on the unit square in \( {\mathbf{R}}^{2} \) . Consider the (boundary of the) unit square in \( {\mathbf{R}}^{2} \), i.e., \( C = \left\{ {x \in {\mathbf{R}}^{2} \mid \parallel x{\parallel }_{\infty } = 1}\right\} \) . We take \( {x}_{0} = 0 \) . | In the \( {\ell }_{1} \) -norm, the four points \( \left( {1,0}\right) ,\left( {0, - 1}\right) ,\left( {-1,0}\right) \), and \( \left( {0,1}\right) \) are closest to \( {x}_{0} = 0 \) , with distance 1, so we have \( \operatorname{dist}\left( {{x}_{0}, C}\right) = 1 \) in the \( {\ell }_{1} \) -norm. The same statement... | Yes |
Consider the set of \( m \times n \) matrices with rank less than or equal to \( k \), \[ C = \left\{ {X \in {\mathbf{R}}^{m \times n} \mid \operatorname{rank}X \leq k}\right\} \] with \( k \leq \min \{ m, n\} \), and let \( {X}_{0} \in {\mathbf{R}}^{m \times n} \). We can find a projection of \( {X}_{0} \) on \( C \),... | Let \[ {X}_{0} = \mathop{\sum }\limits_{{i = 1}}^{r}{\sigma }_{i}{u}_{i}{v}_{i}^{T} \] be the singular value decomposition of \( {X}_{0} \), where \( r = \operatorname{rank}{X}_{0} \). Then the matrix \( Y = \mathop{\sum }\limits_{{i = 1}}^{{\min \{ k, r\} }}{\sigma }_{i}{u}_{i}{v}_{i}^{T} \) is a projection of \( {X}_... | Yes |
Bounding correlation coefficients. We consider an example in \( {\mathbf{R}}^{4} \), where we are given\n\n\[ \n{0.6} \leq {\rho }_{12} \leq {0.9},\;{0.8} \leq {\rho }_{13} \leq {0.9}\n\]\n\n\[ \n{0.5} \leq {\rho }_{24} \leq {0.7},\; - {0.8} \leq {\rho }_{34} \leq - {0.4}.\n\]\n\nTo find the minimum and maximum possibl... | The minimum and maximum values (to two significant digits) are -0.39 and 0.23 , with corresponding correlation matrices\n\n\[ \n\left\lbrack \begin{array}{rrrr} {1.00} & {0.60} & {0.87} & - {0.39} \\ {0.60} & {1.00} & {0.33} & {0.50} \\ {0.87} & {0.33} & {1.00} & - {0.55} \\ - {0.39} & {0.50} & - {0.55} & {1.00} \end{a... | Yes |
Consider the case where only one point \( \left( {u, v}\right) \in {\mathbf{R}}^{2} \) is free, and we minimize the sum of the distances to fixed points \( \left( {{u}_{1},{v}_{1}}\right) ,\ldots ,\left( {{u}_{K},{v}_{K}}\right) \) . | - \( {\ell }_{1} \) -norm. We can find a point that minimizes\n\n\[\n\mathop{\sum }\limits_{{i = 1}}^{K}\left( {\left| {u - {u}_{i}}\right| + \left| {v - {v}_{i}}\right| }\right)\n\]\n\nanalytically. An optimal point is any median of the fixed points. In other words, \( u \) can be taken to be any median of the points ... | Yes |
Consider the case where only one point \( x \) is free, and we minimize the sum of the squares of the Euclidean distances to fixed points \( {x}_{1},\ldots ,{x}_{K} \), \[ {\begin{Vmatrix}x - {x}_{1}\end{Vmatrix}}_{2}^{2} + {\begin{Vmatrix}x - {x}_{2}\end{Vmatrix}}_{2}^{2} + \cdots + {\begin{Vmatrix}x - {x}_{K}\end{Vma... | Taking derivatives, we see that the optimal \( x \) is given by \[ \frac{1}{K}\left( {{x}_{1} + {x}_{2} + \cdots + {x}_{K}}\right) \] i.e., the average of the fixed points. | Yes |
Example 8.6 We consider a placement problem in \( {\mathbf{R}}^{2} \) with 6 free points,8 fixed points, and 27 links. Figures 8.15-8.17 show the optimal solutions for the criteria\n\n\[ \mathop{\sum }\limits_{{\left( {i, j}\right) \in \mathcal{A}}}{\begin{Vmatrix}{x}_{i} - {x}_{j}\end{Vmatrix}}_{2},\;\mathop{\sum }\li... | i.e., using the penalty functions \( h\left( z\right) = z, h\left( z\right) = {z}^{2} \), and \( h\left( z\right) = {z}^{4} \) . The figures also show the resulting distributions of the link lengths.\n\nComparing the results, we see that the linear placement concentrates the free points in a small area, while the quadr... | Yes |
Condition number of an ellipsoid. Let \( \mathcal{E} \) be the ellipsoid\n\n\[ \mathcal{E} = \left\{ {x \mid {\left( x - {x}_{0}\right) }^{T}{A}^{-1}\left( {x - {x}_{0}}\right) \leq 1}\right\} \]\n\nwhere \( A \in {\mathbf{S}}_{+ + }^{n} \) . The width of \( \mathcal{E} \) in the direction \( q \) is\n\n\[ \mathop{\sup... | It follows that its minimum and maximum width are\n\n\[ {W}_{\min } = 2{\lambda }_{\min }{\left( A\right) }^{1/2},\;{W}_{\max } = 2{\lambda }_{\max }{\left( A\right) }^{1/2}, \]\n\nand its condition number is\n\n\[ \operatorname{cond}\left( \mathcal{E}\right) = \frac{{\lambda }_{\max }\left( A\right) }{{\lambda }_{\min... | Yes |
In \( §{4.5.4} \) we encountered the unconstrained geometric program\n\n\[\n\\text{minimize}\\mathop{\\sum }\\limits_{{i, j = 1}}^{n}{M}_{ij}^{2}{d}_{i}^{2}/{d}_{j}^{2}\\text{,}\n\]\n\nwhere \( M \\in {\\mathbf{R}}^{n \\times n} \) is given, and the variable is \( d \\in {\\mathbf{R}}^{n} \) . Using the change of varia... | It is easy to minimize \( f \) one component at a time. Keeping all components except the \( k \) th fixed, we can write \( f\\left( x\\right) = \\log \\left( {{\\alpha }_{k} + {\\beta }_{k}{e}^{-{x}_{k}} + {\\gamma }_{k}{e}^{{x}_{k}}}\\right) \), where\n\n\[{\\alpha }_{k} = {M}_{kk}^{2} + \\mathop{\\sum }\\limits_{{i,... | Yes |
The function \( f\left( x\right) = - \log x \) is self-concordant. | Using \( {f}^{\prime \prime }\left( x\right) = 1/{x}^{2},{f}^{\prime \prime \prime }\left( x\right) = - 2/{x}^{3} \), we find that\n\n\[ \frac{\left| {f}^{\prime \prime \prime }\left( x\right) \right| }{2{f}^{\prime \prime }{\left( x\right) }^{3/2}} = \frac{2/{x}^{3}}{2{\left( 1/{x}^{2}\right) }^{3/2}} = 1 \]\n\nso the... | Yes |
Example 9.4 Log barrier for linear inequalities. The function\n\n\[ \nf\left( x\right) = - \mathop{\sum }\limits_{{i = 1}}^{m}\log \left( {{b}_{i} - {a}_{i}^{T}x}\right) \]\n\nwith \( \operatorname{dom}f = \left\{ {x \mid {a}_{i}^{T}x < {b}_{i}, i = 1,\ldots, m}\right\} \), is self-concordant. | Each term \( - \log \left( {{b}_{i} - }\right. \) \( \left. {{a}_{i}^{T}x}\right) \) is the composition of \( - \log y \) with the affine transformation \( y = {b}_{i} - {a}_{i}^{T}x \), and hence self-concordant. Therefore the sum is also self-concordant. | Yes |
The function \( f\left( X\right) = - \log \det X \) is self-concordant on \( \operatorname{dom}f = {\mathbf{S}}_{+ + }^{n} \) | To show this, we consider the function \( \widetilde{f}\left( t\right) = f\left( {X + {tV}}\right) \), where \( X \succ 0 \) and \( V \in {\mathbf{S}}^{n} \) . It can be expressed as\n\n\[ \widetilde{f}\left( t\right) = - \log \det \left( {{X}^{1/2}\left( {I + t{X}^{-1/2}V{X}^{-1/2}}\right) {X}^{1/2}}\right) \]\n\n\[ =... | Yes |
Example 9.6 Log of concave quadratic. The function\n\n\[ f\left( x\right) = - \log \left( {{x}^{T}{Px} + {q}^{T}x + r}\right) ,\]\n\nwhere \( P \in - {\mathbf{S}}_{ + }^{n} \), is self-concordant on\n\n\[ \operatorname{dom}f = \left\{ {x \mid {x}^{T}{Px} + {q}^{T}x + r > 0}\right\} . \] | To show this, it suffices to consider the case \( n = 1 \) (since by restricting \( f \) to a line, the general case reduces to the \( n = 1 \) case). We can then express \( f \) as\n\n\[ f\left( x\right) = - \log \left( {p{x}^{2} + {qx} + r}\right) = - \log \left( {-p\left( {x - a}\right) \left( {b - x}\right) }\right... | Yes |
The following functions \( g \) satisfy the condition (9.43). - \( g\left( x\right) = - {x}^{p} \) for \( 0 < p \leq 1 \) . - \( g\left( x\right) = - \log x \) . - \( g\left( x\right) = x\log x \) . - \( g\left( x\right) = {x}^{p} \) for \( - 1 \leq p \leq 0 \) . - \( g\left( x\right) = {\left( ax + b\right) }^{2}/x \)... | It follows that in each case, the function \( f\left( x\right) = - \log \left( {-g\left( x\right) }\right) - \log x \) is self-concordant. More generally, the function \( f\left( x\right) = - \log \left( {-g\left( x\right) - a{x}^{2} - {bx} - c}\right) - \log x \) is self-concordant on its domain, \[ \left\{ {x \mid x ... | Yes |
Consider the problem of minimizing \( f : {\mathbf{R}}^{n} \rightarrow \mathbf{R} \), which has the form\n\n\[ f\left( x\right) = {\psi }_{1}\left( {{x}_{1},{x}_{2}}\right) + {\psi }_{2}\left( {{x}_{2},{x}_{3}}\right) + \cdots + {\psi }_{n - 1}\left( {{x}_{n - 1},{x}_{n}}\right) ,\]\n\nwhere \( {\psi }_{i} : {\mathbf{R... | Using Cholesky factorization and forward and back substitution algorithms for tridiagonal matrices, we can solve the Newton system for this problem in order \( n \) flops. This should be compared to order \( {n}^{3} \) flops, if the special form of \( f \) were not exploited. | Yes |
Example 10.1 Optimal allocation with resource constraint. We consider the problem\n\n\[ \n\\begin{array}{ll} \\text{ minimize } & \\mathop{\\sum }\\limits_{{i = 1}}^{n}{f}_{i}\\left( {x}_{i}\\right) \\\\\n\\text{ subject to } & \\mathop{\\sum }\\limits_{{i = 1}}^{n}{x}_{i} = b, \\end{array}\n\]\n\nwhere the functions \... | We can eliminate \( {x}_{n} \) (for example) using the parametrization\n\n\[ \n{x}_{n} = b - {x}_{1} - \\cdots - {x}_{n - 1}\n\]\n\nwhich corresponds to the choices\n\n\[ \n\\widehat{x} = b{e}_{n},\\;F = \\left\\lbrack \\begin{matrix} I \\\\\n- {\\mathbf{1}}^{T} \\end{matrix}\\right\\rbrack \\in {\\mathbf{R}}^{n \\time... | Yes |
We consider the problem\n\n\[ \n\\text{ minimize }\\;f\\left( x\\right) = - \\mathop{\\sum }\\limits_{{i = 1}}^{n}\\log {x}_{i}\n\]\n\n\\( \\left( {10.7}\\right) \\)\n\n\[ \n\\text{subject to}{Ax} = b\\text{,}\n\]\n\nwhere \\( A \\in {\\mathbf{R}}^{p \\times n} \\), with implicit constraint \\( x \\succ 0 \\) . | Using\n\n\[ \n{f}^{ * }\\left( y\\right) = \\mathop{\\sum }\\limits_{{i = 1}}^{n}\\left( {-1 - \\log \\left( {-{y}_{i}}\\right) }\\right) = - n - \\mathop{\\sum }\\limits_{{i = 1}}^{n}\\log \\left( {-{y}_{i}}\\right)\n\]\n\n(with \\( \\operatorname{dom}{f}^{ * } = - {\\mathbf{R}}_{+ + }^{n} \\) ), the dual problem is\n... | Yes |
We consider the problem\n\n\\[ \n\\text{ minimize }\\; - \\mathop{\\sum }\\limits_{{i = 1}}^{n}\\log {x}_{i}\n\\]\n\n\\[\n\\text{subject to}{Ax} = b.\\text{.}\n\\] | Here the objective is separable, so the Hessian at \\( x \\) is diagonal:\n\n\\[ \nH = \\operatorname{diag}\\left( {{x}_{1}^{-2},\\ldots ,{x}_{n}^{-2}}\\right) .\n\\]\n\nIf we compute the Newton direction using a generic method such as an \\( {\\mathrm{{LDL}}}^{\\mathrm{T}} \\) factorization of the KKT matrix, the cost... | Yes |
The logarithmic barrier function for an LP in inequality form, \[ \begin{array}{ll} \text{ minimize } & {c}^{T}x \\ \text{ subject to } & {Ax} \preccurlyeq b, \end{array} \] is given by \[ \phi \left( x\right) = - \mathop{\sum }\limits_{{i = 1}}^{m}\log \left( {{b}_{i} - {a}_{i}^{T}x}\right) ,\;\operatorname{dom}\phi =... | The gradient and Hessian of the barrier function are \[ \nabla \phi \left( x\right) = \mathop{\sum }\limits_{{i = 1}}^{m}\frac{1}{{b}_{i} - {a}_{i}^{T}x}{a}_{i},\;{\nabla }^{2}\phi \left( x\right) = \mathop{\sum }\limits_{{i = 1}}^{m}\frac{1}{{\left( {b}_{i} - {a}_{i}^{T}x\right) }^{2}}{a}_{i}{a}_{i}^{T}, \] or, more c... | Yes |
The dual of the inequality form LP (11.8) is\n\n\\[ \n\\text{maximize}\\; - {b}^{T}\\lambda \n\\]\n\n\\[ \n\\text{subject to}\\{A}^{T}\\lambda + c = 0 \n\\]\n\n\\[ \n\\lambda \\succcurlyeq 0\\text{.} \n\\] | From the optimality conditions (11.9), it is clear that\n\n\\[ \n{\\lambda }_{i}^{ \\star }\\left( t\\right) = \\frac{1}{t\\left( {{b}_{i} - {a}_{i}^{T}{x}^{ \\star }\\left( t\\right) }\\right) },\\;i = 1,\\ldots, m, \n\\]\n\nis dual feasible, with dual objective value\n\n\\[ \n- {b}^{T}{\\lambda }^{ \\star }\\left( t\... | Yes |
The force field associated with the \( i \) th constraint of the LP (11.8) is\n\n\[ \n{F}_{i}\left( x\right) = \frac{-{a}_{i}}{{b}_{i} - {a}_{i}^{T}x}.\n\] | This force is in the direction of the inward pointing normal to the constraint plane \( {\mathcal{H}}_{i} = \left\{ {x \mid {a}_{i}^{T}x = {b}_{i}}\right\} \), and has magnitude inversely proportional to the distance to \( {\mathcal{H}}_{i} \), i.e.,\n\n\[ \n{\begin{Vmatrix}{F}_{i}\left( x\right) \end{Vmatrix}}_{2} = \... | Yes |
Example 11.4 Comparison of phase I methods. We apply two phase I methods to an infeasible set of inequalities \( {Ax} \preccurlyeq b \) with dimensions \( m = {100}, n = {50} \) . The first method is the basic phase I method\n\nminimize \( \;s \)\n\nsubject to \( \;{Ax} \preccurlyeq b + \mathbf{1}s \) ,\n\nwhich minimi... | Figure 11.9 shows the distributions of the infeasibilities \( {b}_{i} - {a}_{i}^{T}x \) for these two values of \( x \), denoted \( {x}_{\max } \) and \( {x}_{\text{sum }} \), respectively. The point \( {x}_{\max } \) satisfies 39 of the 100 inequalities, whereas the point \( {x}_{\text{sum }} \) satisfies 79 of the in... | Yes |
The function \( \psi \left( x\right) = \mathop{\sum }\limits_{{i = 1}}^{n}\log {x}_{i} \) is a generalized logarithm for \( K = {\mathbf{R}}_{ + }^{n} \), with degree \( n \). | For \( x \succ 0 \), \[ \nabla \psi \left( x\right) = \left( {1/{x}_{1},\ldots ,1/{x}_{n}}\right) \] so \( \nabla \psi \left( x\right) \succ 0 \), and \( {x}^{T}\nabla \psi \left( x\right) = n \). | Yes |
The function\n\n\[ \psi \left( x\right) = \log \left( {{x}_{n + 1}^{2} - \mathop{\sum }\limits_{{i = 1}}^{n}{x}_{i}^{2}}\right) \]\n\nis a generalized logarithm for the second-order cone\n\n\[ K = \left\{ {x \in {\mathbf{R}}^{n + 1}\left| {\;{\left( \mathop{\sum }\limits_{{i = 1}}^{n}{x}_{i}^{2}\right) }^{1/2} \leq {x}... | The gradient of \( \psi \) at a point \( x \in \operatorname{int}K \) is given by\n\n\[ \frac{\partial \psi \left( x\right) }{\partial {x}_{j}} = \frac{-2{x}_{j}}{\left( {x}_{n + 1}^{2} - \mathop{\sum }\limits_{{i = 1}}^{n}{x}_{i}^{2}\right) },\;j = 1,\ldots, n \]\n\n\[ \frac{\partial \psi \left( x\right) }{\partial {x... | Yes |
The function \( \psi \left( X\right) = \log \det X \) is a generalized logarithm for the cone \( {\mathbf{S}}_{ + }^{p} \) . The degree is \( p \), since\n\n\[ \log \det \left( {sX}\right) = \log \det X + p\log s \] \n\nfor \( s > 0 \) . | The gradient of \( \psi \) at a point \( X \in {\mathbf{S}}_{+ + }^{p} \) is equal to \n\n\[ \nabla \psi \left( X\right) = {X}^{-1} \] \n\nThus, we have \( \nabla \psi \left( X\right) = {X}^{-1} \succ 0 \), and the inner product of \( X \) and \( \nabla \psi \left( X\right) \) is equal to \( \operatorname{tr}\left( {X{... | Yes |
We consider an SOCP with variable \( x \in {\mathbf{R}}^{n} \): \n\n\[ \n\text{minimize}\;{f}^{T}x \n\] \n\n\[ \n\text{subject to}{\begin{Vmatrix}{A}_{i}x + {b}_{i}\end{Vmatrix}}_{2} \leq {c}_{i}^{T}x + {d}_{i},\;i = 1,\ldots, m\text{,} \n\] \n\nwhere \( {A}_{i} \in {\mathbf{R}}^{{n}_{i} \times n} \). | The corresponding logarithmic barrier function for (11.44) is \n\n\[ \n\phi \left( x\right) = - \mathop{\sum }\limits_{{i = 1}}^{m}\log \left( {{\left( {c}_{i}^{T}x + {d}_{i}\right) }^{2} - {\begin{Vmatrix}{A}_{i}x + {b}_{i}\end{Vmatrix}}_{2}^{2}}\right) , \n\] \n\nwith \( \operatorname{dom}\phi = \left\{ {x \mid {\beg... | Yes |
Example 11.9 Semidefinite programming in inequality form. We consider the SDP with variable \( x \in {\mathbf{R}}^{n} \), \[ \text{minimize}\;{c}^{T}x \] \[ \text{subject to}\;F\left( x\right) = {x}_{1}{F}_{1} + \cdots + {x}_{n}{F}_{n} + G \preccurlyeq 0\text{,} \] where \( G,{F}_{1},\ldots ,{F}_{n} \in {\mathbf{S}}^{p... | Using the generalized logarithm \( \log \det X \) for the positive semidefinite cone \( {\mathbf{S}}_{ + }^{p} \), we have the barrier function (for the primal problem) \[ \phi \left( x\right) = \log \det \left( {-F{\left( x\right) }^{-1}}\right) \] with \( \operatorname{dom}\phi = \{ x \mid F\left( x\right) \prec 0\} ... | Yes |
The second-order cone has generalized logarithm \( \psi \left( x\right) = \log \left( {{x}_{p + 1}^{2} - \mathop{\sum }\limits_{{i = 1}}^{p}{x}_{i}^{2}}\right) \), with \( \operatorname{dom}\psi = \left\{ {x \in {\mathbf{R}}^{p + 1} \mid {x}_{p + 1} > {\left( \mathop{\sum }\limits_{{i = 1}}^{p}{x}_{i}^{2}\right) }^{1/2... | The associated dual logarithm is \[ \bar{\psi }\left( y\right) = \log \left( {{y}_{p + 1}^{2} - \mathop{\sum }\limits_{{i = 1}}^{p}{y}_{i}^{2}}\right) + 2 - \log 4, \] with \( \operatorname{dom}\psi = \left\{ {y \in {\mathbf{R}}^{p + 1} \mid {y}_{p + 1} > {\left( \mathop{\sum }\limits_{{i = 1}}^{p}{y}_{i}^{2}\right) }^... | No |
The dual logarithm associated with \( \psi \left( X\right) = \log \det X \), with \( \operatorname{dom}\psi = {\mathbf{S}}_{+ + }^{p} \), is | \[ \bar{\psi }\left( Y\right) = \log \det Y + p \] with domain \( \operatorname{dom}{\psi }^{ * } = {\mathbf{S}}_{+ + }^{p} \) (see example 3.23). Again, it is the same generalized logarithm, except for a constant. | Yes |
That weakly holomorphic functions (with values in a complex Banach space) are strongly holomorphic | was proved by N. Dunford in Trans. Amer. Math. Soc., vol. 44, pp. 304-356, 1938. | Yes |
Theorem 11.9. When \( A \) has no unit, then \( \Delta \) is locally compact (but not compact) and \( \widehat{A} \subset {C}_{0}\left( \Delta \right) \) ; the origin of \( {A}^{ * } \) is then in the closure of \( \Delta \) . | See [16], pp. 52–53. | No |
Theorem 11.18 was proved by Gelfand and Naimark in Mat. Sbornik, vol. 12, pp. 197-213, 1943. In the same paper they also proved that every \( {B}^{ * } \) -algebra \( A \) (commutative or not) is isometrically *-isomorphic to an algebra of bounded operators on some Hilbert space (Theorem 12.41), if \( e + {x}^{ * }x \)... | That this additional hypothesis is redundant was proved 15 years later by I. Kaplansky \( \lbrack \left( f\right) \) of Theorem 11.28]. See [21], p. 248, for references to the rather tangled history of this theorem. | No |
Theorem 13.6 was first proved by A. Wintner, Phys. Rev., vol. 71, pp. 738-739, 1947. The more algebraic proof of the text is H. Wielandt's, Math. Ann., vol. 121, p. 21, 1949. | It was generalized by D. C. Kleinecke (Proc. Amer. Math. Soc., vol. 8, pp. 535-536, 1957), to yield the following theorem about derivations: If \( D \) is a continuous linear operator in a Banach algebra \( A \) such that \( D\left( {xy}\right) = {xDy} + \left( {Dx}\right) y \) for all \( x, y \in A \), then the spectr... | Yes |
Lemma 1. C: closed, convex, y & C.\n\nThen \( \exists !{xec} \), s.t. \( \parallel {11y} - x\parallel = \inf \{ {11y} - {z11} \mid {zec}\} = d\left( {y, c}\right) \) | 证明: \n\n\( \forall {\varepsilon }_{n} = \frac{1}{n}\exists {x}_{n} \in C, \)\n\n\[ \begin{array}{ll} \text{ s.t. } & \text{ 令 } \\ \text{ IIy-x_nll } \leq d\left( {y, c}\right) + {\varepsilon }_{n} & \text{ (i) } \end{ar... | Yes |
Lemma 2. C: closed, convex, y & C.\n\nThen \( \exists a \neq 0, b \), s.t. \( {a}^{T}y < b \) ,*) \( {a}^{T}x \geq b,\forall x \in C\left( {x * }\right) \) | Proof: By Lemma 1， \( \exists x \in C \) , \( \left( {y \neq x}\right) \)\ns.t. \( \parallel y - x\parallel = d\left( {y, c}\right) \)\n\nLet \( a = x - y \neq 0, b = \frac{||x{|}^{2} - ||y|{|}^{2}}{2} \)\n\nLet \( a = x - y \neq 0, b = \frac{||x{|}^{2} - ||y|{|}^{2}}{2} \)\n\n\( \Rightarrow {a}^{T}y = {x}^{T}y - {y}^{... | Yes |
Lemma 3. If C is convex, y k C, | Then \( \exists a \neq 0, b \) s.t. \( {a}^{\top }y \leq b,{a}^{\top }x \geq b,\forall x \in C\left( {* * * }\right) \) Proof: \( \widetilde{c} = c - y \) \( \xi = 1 - {0EC}\left( {\widetilde{c}{i}_{s}\operatorname{convex}}\right) \) | No |
Theorem 3. Let \( \mathbf{x} \) have the spectral decomposition \( \mathbf{x} = {\lambda }_{1}{\mathbf{c}}_{\mathbf{1}} + {\lambda }_{2}{\mathbf{c}}_{\mathbf{2}} \). Then,\n\n1. \( \operatorname{Arw}\left( \mathbf{x}\right) \) and \( {Q}_{\mathbf{x}} \) commute and thus share a system of eigenvectors.\n\n2. \( {\lambda... | Proof. The statement that \( \operatorname{Arw}\left( \mathbf{x}\right) \) and \( {Q}_{\mathbf{x}} \) commute is a direct consequence of definition of \( {Q}_{\mathbf{x}} \) as \( 2{\operatorname{Arw}}^{2}\left( \mathbf{x}\right) - \operatorname{Arw}\left( {\mathbf{x}}^{2}\right) \), and the fact that \( \operatorname{... | Yes |
Theorem 6. The following statements are equivalent:\n\n1. \( \mathbf{x} \) and \( \mathbf{y} \) operator commute.\n\n2. \( \operatorname{Arw}\left( \mathbf{x}\right) \) and \( \operatorname{Arw}\left( \mathbf{y}\right) \) commute. Therefore, for all \( \mathbf{z} \in {\Re }^{n},\mathbf{x} \circ \left( {\mathbf{y} \circ... | Proof. Immediate from the eigenvalue structure of \( \operatorname{Arw}\left( \mathbf{x}\right) \) and \( {Q}_{\mathbf{x}} \) stated in Theorem 3. | No |
Corollary 7. The vectors \( \mathbf{x} \) and \( \mathbf{y} \) operator commute if and only if \( \overline{\mathbf{x}} = \mathbf{0} \) or \( \overline{\mathbf{y}} = \mathbf{0} \) or \( \overline{\mathbf{x}} \) and \( \overline{\mathbf{y}} \) are proportional, that is \( \overline{\mathbf{x}} = \alpha \overline{\mathbf... | Therefore, \( \mathbf{x} \) and \( \mathbf{y} \) commute if either one of them is a multiple of the identity, or else their projections on the hyperplane \( {x}_{0} = 0 \) are collinear. | Yes |
Theorem 8. For each \( \mathbf{x},\mathbf{y},\mathbf{u} \in {\Re }^{n} \) , \( \mathbf{x} \) nonsingular, \( \alpha \in \Re \), and integer \( t \) we have\n\n1. \( {Q}_{\alpha \mathbf{y}} = {\alpha }^{2}{Q}_{\mathbf{y}} \) .\n\n2. \( {Q}_{\mathbf{x}}{\mathbf{x}}^{-1} = \mathbf{x} \) and thus \( {Q}_{\mathbf{x}}^{-1}\m... | Proof. (1) is trivial. Note that \( {\mathbf{x}}^{-1} = R\mathbf{x}/\det \left( \mathbf{x}\right) \) and\n\n\[ \n{Q}_{{\mathbf{x}}^{-1}} = \frac{1}{\mathop{\det }\limits^{2}\left( \mathbf{x}\right) }\left( \begin{matrix} \parallel \mathbf{x}{\parallel }^{2} & - 2{x}_{0}{\overline{\mathbf{x}}}^{\top } \\ - 2{x}_{0}\over... | Yes |
Theorem 9. Let \( \mathbf{p} \in {\Re }^{n} \) be nonsingular. Then \( {Q}_{\mathbf{p}}\left( \mathcal{Q}\right) = \mathcal{Q} \) ; likewise, \( {Q}_{\mathbf{p}}\left( {\operatorname{int}\mathcal{Q}}\right) = \) int \( \mathcal{Q} \) . | Proof. Let \( \mathbf{x} \in \mathcal{Q} \) (respectively, \( \mathbf{x} \in \) int \( \mathcal{Q} \) ) and \( \mathbf{y} = {Q}_{\mathbf{p}}\mathbf{x} \) . By part 8 of Theorem 8 \( \det \left( \mathbf{y}\right) \geq 0 \) (respectively, \( \det \left( \mathbf{y}\right) > 0 \) ). Therefore, either \( \mathbf{y} \in \mat... | Yes |
Theorem 10. Let \( \mathbf{x},\mathbf{z},\mathbf{p} \) be all positive definite. Define \( \widetilde{\mathbf{x}} = {Q}_{\mathbf{p}}\mathbf{x} \) and \( \mathbf{z} = {Q}_{{\mathbf{p}}^{-1}}\mathbf{z} \) . Then\n\n1. the vectors \( \mathbf{a} = {Q}_{{\mathbf{x}}^{1/2}}\mathbf{z} \) and \( \mathbf{b} = {Q}_{{\widetilde{\... | Proof. By part 3 of Theorem 3 the eigenvalues of \( {Q}_{\mathbf{a}} \) and \( {Q}_{\mathbf{b}} \) are pairwise products of eigenvalues of \( \mathbf{a} \) and \( \mathbf{b} \), respectively. Since by Theorem 9 the eigenvalues of \( \mathbf{a} \) and \( \mathbf{b} \) are all positive, \( \mathbf{a} \) and \( \mathbf{b}... | Yes |
Theorem 13 (Strong Duality Theorem). If the primal and dual problems in (2) have strictly feasible solutions \( \left( {{\mathbf{x}}_{\mathbf{i}} \succ \mathcal{Q}\mathbf{0},{\mathbf{z}}_{\mathbf{i}}{ \succ }_{\mathcal{Q}}\mathbf{0},\forall i}\right) \), then both have optimal solutions \( {\mathbf{x}}^{ * } \) , \( \l... | This theorem and Theorem 14 below are in fact true for all cone-LPs over any closed, pointed, convex and full-dimensional cone \( \mathcal{K} \), and can be proved by using a general form of Farkas' lemma (e.g. see [NN94]). | Yes |
Lemma 15 (Complementarity Conditions). Suppose that \( \mathbf{x} \in \mathcal{Q} \) and \( \mathbf{z} \in \mathcal{Q} \) ,(i.e., \( {\mathbf{x}}_{\mathbf{i}}{ \succcurlyeq }_{\mathcal{Q}}\mathbf{0} \) and \( {\mathbf{z}}_{\mathbf{i}}{ \succcurlyeq }_{\mathcal{Q}}\mathbf{0} \), for all \( i = 1,\ldots, r \) ). Then \( ... | Proof. If \( {x}_{i0} = 0 \) or \( {z}_{i0} = 0 \) the result is trivial, since \( {\mathbf{x}}_{\mathbf{i}} = \mathbf{0} \) in the first case and \( {\mathbf{z}}_{\mathbf{i}} = \mathbf{0} \) in the second. Therefore, we need only consider the case where \( \left( {{x}_{i0} > 0}\right. \) and \( \left. {{z}_{i0} > 0}\r... | Yes |
Theorem 20. For each boundary block \( {\mathbf{x}}_{\mathbf{i}} = {\alpha }_{i}{\mathbf{c}}_{\mathbf{i}}^{\prime } \), let \( {Q}_{i} = \left( {\sqrt{2}{\mathbf{c}}_{\mathbf{i}},{\widehat{Q}}_{i},\sqrt{2}{\mathbf{c}}_{\mathbf{i}}^{\prime }}\right) = \) \( \left( {\sqrt{2}{\mathbf{c}}_{\mathbf{i}},\overline{{Q}_{i}}}\r... | Proof. Set\n\n\[ {G}_{\mathcal{Q}} = {Q}_{1} \oplus \cdots \oplus {Q}_{p} \oplus I \oplus I. \]\n\n(51)\n\nSince \( {Q}_{i}^{\top }{\mathbf{c}}_{\mathbf{i}} = \frac{\mathbf{e}}{\sqrt{2}}, i = 1,\ldots, p \), multiplying \( {H}_{\mathcal{Q}} \) from the right by \( {G}_{\mathcal{Q}} \) yields\n\n\[ {H}_{\mathcal{Q}}{G}_... | Yes |
i. If a primal optimal solution is nondegenerate, then the optimal dual solution is unique. | Proof. We shall prove the contrapositive of both statements.\n\ni. Let \( \left( {{\mathbf{y}}^{\prime };{\mathbf{z}}^{\prime }}\right) \) and \( \left( {{\mathbf{y}}^{\prime \prime };{\mathbf{z}}^{\prime \prime }}\right) \) be two distinct optimal dual solutions and \( \mathbf{x} \) be an optimal primal solution. Defi... | Yes |
Corollary 24. \( \left( {\mathbf{x},\mathbf{y},\mathbf{z}}\right) \) satisfies strict complementarity if for each block \( i \), either both \( {\mathbf{x}}_{\mathbf{i}} \) and \( {\mathbf{z}}_{\mathbf{i}} \) are nonzero and in \( \mathrm{{bd}}{\mathcal{Q}}_{{n}_{i}} \), or if one is zero, the other is in the interior ... | It can be easily verified that at the optimum \( {z}_{i0}\det \left( {\mathbf{x}}_{\mathbf{i}}\right) = {x}_{i0}\det \left( {\mathbf{z}}_{\mathbf{i}}\right) = 0 \) . Thus strict complementarity holds if and only if for all blocks exactly one of \( {x}_{i0} \) or \( \det \left( {\mathbf{z}}_{\mathbf{i}}\right) \) is zer... | No |
Lemma 27. Every primal-dual block canonical matrix is nonsingular. | Proof. (This proof is essentially the same as the proof of Theorem 1 of [AHO98].) Interchange first and third rows and second and last columns. Then, since \( {B}_{1} \) and \( {V}_{2} \) are nonsingular, one can apply block Gaussian elimination to get\n\n\[ \left( \begin{matrix} {B}_{1} & 0 & 0 & 0 & {B}_{2} \\ 0 & I ... | Yes |
Theorem 28. For the SOCP problem, primal and dual nondegeneracy together with strict complementarity imply that the Jacobian matrix \( {J}_{\mathcal{Q}} \) is nonsingular at the optimum. | Proof. Define \( {P}_{\mathcal{Q}} \) as the block diagonal matrix\n\n\[ \n{P}_{\mathcal{Q}}\overset{\text{ def }}{ = }\left( {{Q}_{1} \oplus \cdots \oplus {Q}_{p} \oplus I \oplus I}\right) \oplus I \oplus \left( {{Q}_{1} \oplus \cdots \oplus {Q}_{p} \oplus I \oplus I}\right) \]\n\n(55)\n\nwith \( {Q}_{i} \) correspond... | Yes |
Lemma 30. For a given nonsingular \( \mathbf{p} \) we have:\n\ni. \( {\mathbf{x}}^{\top }\mathbf{z} = {\widetilde{\mathbf{x}}}^{\top }\mathbf{z} \).\n\nii. For each vector \( \mathbf{u}, A\mathbf{u} = \mathbf{0} \) if and only if \( A\widetilde{\mathbf{u}} = \mathbf{0} \).\n\niii. \( {d}_{ \bullet }\left( {\mathbf{x},\... | Proof. Parts i. and ii. follow from part 4 of Theorem 8. Part iv. is a consequence of part iii. We now prove part iii. Let \( {\widetilde{\mathbf{w}}}_{\mathbf{i}} = {Q}_{{\widetilde{\mathbf{x}}}^{1/2}}\mathbf{z}.{d}_{ \bullet }\left( {\mathbf{x},\mathbf{z}}\right) \) depends only on the spectrum of \( {\mathbf{w}}_{\m... | Yes |
Lemma 33. Let \( \mathbf{x} \in {\Re }^{n} \) and \( \mathbf{z} \in {\Re }^{n} \) be such that \( \mathbf{z} \) is nonsingular. Then\n\n\[ G\left( {\mathbf{x},\mathbf{z}}\right) = {\operatorname{Arw}}^{-1}\left( \mathbf{z}\right) \operatorname{Arw}\left( \mathbf{x}\right) = {\beta I} - {\gamma R}\mathbf{x}{\mathbf{e}}^... | Proof. Noting that \( {\mathbf{z}}^{-1} = \frac{1}{\det \left( \mathbf{z}\right) }R\mathbf{z} \), this lemma can be proved by inspection. | No |
Theorem 34. Let \( \mathbf{x} \in {\Re }^{n} \) and \( \mathbf{z} \in {\Re }^{n} \) be such that \( \mathbf{z} \) is nonsingular. For all nonsingular p we have\n\n\[ F = {Q}_{{\mathbf{p}}^{-1}}\left( {{\operatorname{Arw}}^{-1}\left( \mathbf{z}\right) \operatorname{Arw}\left( \widetilde{\mathbf{x}}\right) }\right) {Q}_{... | Proof. First we prove the theorem for the single block case where \( r = 1 \) . Multiplying \( G\left( {\widetilde{\mathbf{x}},\mathbf{z}}\right) \) in Lemma 33 from the left and the right by \( {Q}_{{\mathbf{p}}^{-1}} \), setting \( \mathbf{q} = {\mathbf{p}}^{-1},\theta = \beta \det {\left( \mathbf{q}\right) }^{2} \) ... | Yes |
Example 2.3 The second-order cone is the norm cone for the Euclidean norm, i.e., | \[ C = \left\{ {\left( {x, t}\right) \in {\mathbf{R}}^{n + 1} \mid \parallel x{\parallel }_{2} \leq t}\right\} \] \[ = \left\{ {\left. \left\lbrack \begin{array}{l} x \\ t \end{array}\right\rbrack \right| \;{\left\lbrack \begin{array}{l} x \\ t \end{array}\right\rbrack }^{T}\left\lbrack \begin{matrix} I & 0 \\ 0 & - 1 ... | Yes |
To describe the simplex (2.7) as a polyhedron, i.e., in the form (2.6), we proceed as follows. By definition, \( x \in C \) if and only if \( x = {\theta }_{0}{v}_{0} + {\theta }_{1}{v}_{1} + \cdots + {\theta }_{k}{v}_{k} \) for some \( \theta \succcurlyeq 0 \) with \( {\mathbf{1}}^{T}\theta = 1 \) . Equivalently, if w... | \[ B = \left\lbrack \begin{array}{lll} {v}_{1} - {v}_{0} & \cdots & {v}_{k} - {v}_{0} \end{array}\right\rbrack \in {\mathbf{R}}^{n \times k}, \] we can say that \( x \in C \) if and only if \[ x = {v}_{0} + {By} \] (2.8) for some \( y \succcurlyeq 0 \) with \( {\mathbf{1}}^{T}y \leq 1 \) . Now we note that affine indep... | Yes |
The positive semidefinite cone \( {\mathbf{S}}_{ + }^{n} \) can be expressed as\n\n\[ \mathop{\bigcap }\limits_{{z \neq 0}}\left\{ {X \in {\mathbf{S}}^{n} \mid {z}^{T}{Xz} \geq 0}\right\} \] | For each \( z \neq 0,{z}^{T}{Xz} \) is a (not identically zero) linear function of \( X \), so the sets\n\n\[ \left\{ {X \in {\mathbf{S}}^{n} \mid {z}^{T}{Xz} \geq 0}\right\} \]\n\nare, in fact, halfspaces in \( {\mathbf{S}}^{n} \). Thus the positive semidefinite cone is the intersection of an infinite number of halfsp... | Yes |
Example 2.8 We consider the set\n\n\\[ \nS = \\left\\{ {x \\in {\\mathbf{R}}^{m}\\left| \\right| p\\left( t\\right) \\mid \\leq 1\\text{ for }\\left| t\\right| \\leq \\pi /3}\\right\\} \n\\]\n\nwhere \\( p\\left( t\\right) = \\mathop{\\sum }\\limits_{{k = 1}}^{m}{x}_{k}\\cos {kt} \\) . The set \\( S \\) can be expresse... | and so is convex. The definition and the set are illustrated in figures 2.13 and 2.14, for \\( m = 2 \\) . | No |
Example 2.9 Polyhedron. The polyhedron \( \{ x \mid {Ax} \preccurlyeq b,{Cx} = d\} \) can be expressed as the inverse image of the Cartesian product of the nonnegative orthant and the origin under the affine function \( f\left( x\right) = \left( {b - {Ax}, d - {Cx}}\right) \) : | \[ \{ x \mid {Ax} \preccurlyeq b,{Cx} = d\} = \left\{ {x \mid f\left( x\right) \in {\mathbf{R}}_{ + }^{m}\times \{ 0\} }\right\} . \] | Yes |
The solution set of a linear matrix inequality, \( \{ x \mid A\left( x\right) \preccurlyeq B\} \), is convex. | Indeed, it is the inverse image of the positive semidefinite cone under the affine function \( f : {\mathbf{R}}^{n} \rightarrow {\mathbf{S}}^{m} \) given by \( f\left( x\right) = B - A\left( x\right) \). | Yes |
Example 2.13 Conditional probabilities. Suppose \( u \) and \( v \) are random variables that take on values in \( \{ 1,\ldots, n\} \) and \( \{ 1,\ldots, m\} \), respectively, and let \( {p}_{ij} \) denote \( \operatorname{prob}\left( {u = i, v = j}\right) \) . Then the conditional probability \( {f}_{ij} = \operatorn... | \[ {f}_{ij} = \frac{{p}_{ij}}{\mathop{\sum }\limits_{{k = 1}}^{n}{p}_{kj}}. \] | Yes |
Example 2.19 Separation of an affine and a convex set. Suppose \( C \) is convex and \( D \) is affine, i.e., \( D = \left\{ {{Fu} + g \mid u \in {\mathbf{R}}^{m}}\right\} \), where \( F \in {\mathbf{R}}^{n \times m} \). Suppose \( C \) and \( D \) are disjoint, so by the separating hyperplane theorem there are \( a \n... | Now \( {a}^{T}x \geq b \) for all \( x \in D \) means \( {a}^{T}{Fu} \geq b - {a}^{T}g \) for all \( u \in {\mathbf{R}}^{m} \). But a linear function is bounded below on \( {\mathbf{R}}^{m} \) only when it is zero, so we conclude \( {a}^{T}F = 0 \) (and hence, \( b \leq {a}^{T}g \) ).\n\nThus we conclude that there exi... | Yes |
Strict separation of a point and a closed convex set. Let \( C \) be a closed convex set and \( {x}_{0} \notin C \) . Then there exists a hyperplane that strictly separates \( {x}_{0} \) from \( C \) . | To see this, note that the two sets \( C \) and \( B\left( {{x}_{0},\epsilon }\right) \) do not intersect for some \( \epsilon > 0 \) . By the separating hyperplane theorem, there exist \( a \neq 0 \) and \( b \) such that \( {a}^{T}x \leq b \) for \( x \in C \) and \( {a}^{T}x \geq b \) for \( x \in B\left( {{x}_{0},\... | Yes |
We derive the necessary and sufficient conditions for solvability of a system of strict linear inequalities\n\n\[ \n{Ax} \prec b.\n\]\n\n\( \left( {2.17}\right) \) | These inequalities are infeasible if and only if the (convex) sets\n\n\[ \nC = \left\{ {b - {Ax} \mid x \in {\mathbf{R}}^{n}}\right\} ,\;D = {\mathbf{R}}_{+ + }^{m} = \left\{ {y \in {\mathbf{R}}^{m} \mid y \succ 0}\right\}\n\]\n\ndo not intersect. The set \( D \) is open; \( C \) is an affine set. Hence by the result a... | Yes |
Example 2.23 Nonnegative orthant. The cone \( {\mathbf{R}}_{ + }^{n} \) is its own dual: | \[ {x}^{T}y \geq 0\text{ for all }x \succcurlyeq 0 \Leftrightarrow y \succcurlyeq 0. \] | Yes |
Positive semidefinite cone. On the set of symmetric \( n \times n \) matrices \( {\mathbf{S}}^{n} \), we use the standard inner product \( \operatorname{tr}\left( {XY}\right) = \mathop{\sum }\limits_{{i, j = 1}}^{n}{X}_{ij}{Y}_{ij} \) (see §A.1.1). The positive semidefinite cone \( {\mathbf{S}}_{ + }^{n} \) is self-dua... | We will establish this fact.\n\nSuppose \( Y \notin {\mathbf{S}}_{ + }^{n} \) . Then there exists \( q \in {\mathbf{R}}^{n} \) with\n\n\[ {q}^{T}{Yq} = \operatorname{tr}\left( {q{q}^{T}Y}\right) < 0. \]\n\nHence the positive semidefinite matrix \( X = q{q}^{T} \) satisfies \( \operatorname{tr}\left( {XY}\right) < 0 \) ... | Yes |
Example 2.25 Dual of a norm cone. Let \( \parallel \cdot \parallel \) be a norm on \( {\mathbf{R}}^{n} \) . The dual of the associated cone \( K = \left\{ {\left( {x, t}\right) \in {\mathbf{R}}^{n + 1} \mid \parallel x\parallel \leq t}\right\} \) is the cone defined by the dual norm, i.e., \[ {K}^{ * } = \left\{ {\left... | To prove the result we have to show that \[ {x}^{T}u + {tv} \geq 0\text{ whenever }\parallel x\parallel \leq t \Leftrightarrow \parallel u{\parallel }_{ * } \leq v. \] Let us start by showing that the righthand condition on \( \left( {u, v}\right) \) implies the lefthand condition. Suppose \( \parallel u{\parallel }_{ ... | Yes |
Suppose \( K \subseteq {\mathbf{R}}^{m} \) is a proper cone. Consider the strict generalized inequality\n\n\[ \n{Ax}{ \prec }_{K}b \n\]\n\nwhere \( x \in {\mathbf{R}}^{n} \). We will derive a theorem of alternatives for this inequality. | Suppose it is infeasible, i.e., the affine set \( \left\{ {b - {Ax} \mid x \in {\mathbf{R}}^{n}}\right\} \) does not intersect the open convex set int \( K \) . Then there is a separating hyperplane, i.e., a nonzero \( \lambda \in {\mathbf{R}}^{m} \) and \( \mu \in \mathbf{R} \) such that \( {\lambda }^{T}\left( {b - {... | Yes |
Example 2.27 Pareto optimal production frontier. We consider a product which requires \( n \) resources (such as labor, electricity, natural gas, water) to manufacture. The product can be manufactured or produced in many ways. With each production method, we associate a resource vector \( x \in {\mathbf{R}}^{n} \), whe... | We can find Pareto optimal production methods (i.e., minimal resource vectors) by minimizing\n\n\[ \n{\lambda }^{T}x = {\lambda }_{1}{x}_{1} + \cdots + {\lambda }_{n}{x}_{n} \n\]\n\nover the set \( P \) of production vectors, using any \( \lambda \) that satisfies \( \lambda \succ 0 \) .\n\nHere the vector \( \lambda \... | Yes |
Example 3.1 Indicator function of a convex set. Let \( C \subseteq {\mathbf{R}}^{n} \) be a convex set, and consider the (convex) function \( {I}_{C} \) with domain \( C \) and \( {I}_{C}\left( x\right) = 0 \) for all \( x \in C \) . In other words, the function is identically zero on the set \( C \) . Its extended-val... | is given by\n\n\[ \n{\widetilde{I}}_{C}\left( x\right) = \left\{ \begin{array}{ll} 0 & x \in C \\ \infty & x \notin C. \end{array}\right. \]\n\nThe convex function \( {\widetilde{I}}_{C} \) is called the indicator function of the set \( C \) . | Yes |
Consider the quadratic function \( f : {\mathbf{R}}^{n} \rightarrow \mathbf{R} \), with dom \( f = {\mathbf{R}}^{n} \), given by \[ f\left( x\right) = \left( {1/2}\right) {x}^{T}{Px} + {q}^{T}x + r, \] with \( P \in {\mathbf{S}}^{n}, q \in {\mathbf{R}}^{n} \), and \( r \in \mathbf{R} \). | Since \( {\nabla }^{2}f\left( x\right) = P \) for all \( x, f \) is convex if and only if \( P \succcurlyeq 0 \) (and concave if and only if \( P \preccurlyeq 0 \). For quadratic functions, strict convexity is easily characterized: \( f \) is strictly convex if and only if \( P \succ 0 \) (and strictly concave if and o... | Yes |
The function \( f : {\mathbf{R}}^{n} \times {\mathbf{S}}^{n} \rightarrow \mathbf{R} \), defined as\n\n\[ f\left( {x, Y}\right) = {x}^{T}{Y}^{-1}x \]\n\nis convex on \( \operatorname{dom}f = {\mathbf{R}}^{n} \times {\mathbf{S}}_{+ + }^{n} \) . | One easy way to establish convexity of \( f \) is via its epigraph:\n\n\[ \text{ epi }f = \left\{ {\left( {x, Y, t}\right) \mid Y \succ 0,{x}^{T}{Y}^{-1}x \leq t}\right\} \]\n\n\[ = \left\{ {\left( {x, Y, t}\right) \left| {\;\left\lbrack \begin{matrix} Y & x \\ {x}^{T} & t \end{matrix}\right\rbrack \succcurlyeq 0}\righ... | Yes |
The function\n\n\[ f\left( x\right) = \max \left\{ {{a}_{1}^{T}x + {b}_{1},\ldots ,{a}_{L}^{T}x + {b}_{L}}\right\} \]\n\ndefines a piecewise-linear (or really, affine) function (with \( L \) or fewer regions). It is convex since it is the pointwise maximum of affine functions. | The converse can also be shown: any piecewise-linear convex function with \( L \) or fewer regions can be expressed in this form. (See exercise 3.29.) | No |
Example 3.6 Sum of \( r \) largest components. For \( x \in {\mathbf{R}}^{n} \) we denote by \( {x}_{\left\lbrack i\right\rbrack } \) the \( i \) th largest component of \( x \), i.e., \[ {x}_{\left\lbrack 1\right\rbrack } \geq {x}_{\left\lbrack 2\right\rbrack } \geq \cdots \geq {x}_{\left\lbrack n\right\rbrack } \] ar... | This can be seen by writing it as \[ f\left( x\right) = \mathop{\sum }\limits_{{i = 1}}^{r}{x}_{\left\lbrack i\right\rbrack } = \max \left\{ {{x}_{{i}_{1}} + \cdots + {x}_{{i}_{r}} \mid 1 \leq {i}_{1} < {i}_{2} < \cdots < {i}_{r} \leq n}\right\} , \] i.e., the maximum of all possible sums of \( r \) different component... | Yes |
Example 3.7 Support function of a set. Let \( C \subseteq {\mathbf{R}}^{n} \), with \( C \neq \varnothing \) . The support function \( {S}_{C} \) associated with the set \( C \) is defined as \[ {S}_{C}\left( x\right) = \sup \left\{ {{x}^{T}y \mid y \in C}\right\} \] (and, naturally, \( \left. {\operatorname{dom}{S}_{C... | For each \( y \in C,{x}^{T}y \) is a linear function of \( x \), so \( {S}_{C} \) is the pointwise supremum of a family of linear functions, hence convex. | Yes |
Example 3.8 Distance to farthest point of a set. Let \( C \subseteq {\mathbf{R}}^{n} \) . The distance (in any norm) to the farthest point of \( C \) , \[ f\left( x\right) = \mathop{\sup }\limits_{{y \in C}}\parallel x - y\parallel \] | is convex. To see this, note that for any \( y \), the function \( \parallel x - y\parallel \) is convex in \( x \) . Since \( f \) is the pointwise supremum of a family of convex functions (indexed by \( y \in C \) ), it is a convex function of \( x \) . | Yes |
Example 3.9 Least-squares cost as a function of weights. Let \( {a}_{1},\ldots ,{a}_{n} \in {\mathbf{R}}^{m} \) . In a weighted least-squares problem we minimize the objective function \( \mathop{\sum }\limits_{{i = 1}}^{n}{w}_{i}\left( {{a}_{i}^{T}x - }\right. \) \( {\left. {b}_{i}\right) }^{2} \) over \( x \in {\math... | We define the (optimal) weighted least-squares cost as\n\n\[ g\left( w\right) = \mathop{\inf }\limits_{x}\mathop{\sum }\limits_{{i = 1}}^{n}{w}_{i}{\left( {a}_{i}^{T}x - {b}_{i}\right) }^{2}, \]\n\nwith domain\n\n\[ \operatorname{dom}g = \left\{ {w\left| {\;\mathop{\inf }\limits_{x}\mathop{\sum }\limits_{{i = 1}}^{n}{w... | Yes |
Example 3.10 Maximum eigenvalue of a symmetric matrix. The function \( f\left( X\right) = \) \( {\lambda }_{\max }\left( X\right) \), with \( \operatorname{dom}f = {\mathbf{S}}^{m} \), is convex. | To see this, we express \( f \) as\n\n\[ f\left( X\right) = \sup \left\{ {{y}^{T}{Xy} \mid \parallel y{\parallel }_{2} = 1}\right\} \]\n\ni.e., as the pointwise supremum of a family of linear functions of \( X \) (i.e., \( {y}^{T}{Xy} \) ) indexed by \( y \in {\mathbf{R}}^{m} \). | Yes |
Consider \( f\left( X\right) = \parallel X{\parallel }_{2} \) with \( \operatorname{dom}f = {\mathbf{R}}^{p \times q} \) , where \( \parallel \cdot {\parallel }_{2} \) denotes the spectral norm or maximum singular value. Convexity of \( f \) follows from | \[ f\left( X\right) = \sup \left\{ {{u}^{T}{Xv} \mid \parallel u{\parallel }_{2} = 1,\parallel v{\parallel }_{2} = 1}\right\} ,\] which shows it is the pointwise supremum of a family of linear functions of \( X \) . | Yes |
if \( g \) is convex, \( h \) is convex, and \( \widetilde{h} \) is nondecreasing, then \( f = h \circ g \) is convex. | Assume that \( x, y \in \operatorname{dom}f \), and \( 0 \leq \theta \leq 1 \) . Since \( x, y \in \operatorname{dom}f \), we have that \( x, y \in \operatorname{dom}g \) and \( g\left( x\right), g\left( y\right) \in \operatorname{dom}h \) . Since \( \operatorname{dom}g \) is convex, we conclude that \( {\theta x} + \l... | Yes |
Suppose \( p \geq 1 \), and \( {g}_{1},\ldots ,{g}_{k} \) are convex and nonnegative. Then the function \( {\left( \mathop{\sum }\limits_{{i = 1}}^{k}{g}_{i}{\left( x\right) }^{p}\right) }^{1/p} \) is convex. | To show this, we consider the function \( h : {\mathbf{R}}^{k} \rightarrow \mathbf{R} \) defined as\n\n\[ h\left( z\right) = {\left( \mathop{\sum }\limits_{{i = 1}}^{k}\max {\left\{ {z}_{i},0\right\} }^{p}\right) }^{1/p}, \]\n\nwith \( \operatorname{dom}h = {\mathbf{R}}^{k} \), so \( h = \widetilde{h} \). This function... | Yes |
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