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Suppose the quadratic function\n\n\[ f\left( {x, y}\right) = {x}^{T}{Ax} + 2{x}^{T}{By} + {y}^{T}{Cy}, \]\n\n(where \( A \) and \( C \) are symmetric) is convex in \( \left( {x, y}\right) \), which means\n\n\[ \left\lbrack \begin{matrix} A & B \\ {B}^{T} & C \end{matrix}\right\rbrack \succcurlyeq 0 \]\n\nWe can express... | By the minimization rule, \( g \) is convex, so we conclude that \( A - B{C}^{ \dagger }{B}^{T} \succcurlyeq 0 \). | Yes |
The distance of a point \( x \) to a set \( S \subseteq {\mathbf{R}}^{n} \), in the norm \( \parallel \cdot \parallel \), is defined as \[ \operatorname{dist}\left( {x, S}\right) = \mathop{\inf }\limits_{{y \in S}}\parallel x - y\parallel . \] | The function \( \parallel x - y\parallel \) is convex in \( \left( {x, y}\right) \), so if the set \( S \) is convex, the distance function \( \operatorname{dist}\left( {x, S}\right) \) is a convex function of \( x \) . | No |
Example 3.17 Suppose \( h \) is convex. Then the function \( g \) defined as\n\n\[ g\left( x\right) = \inf \{ h\left( y\right) \mid {Ay} = x\} \]\n\nis convex. | To see this, we define \( f \) by\n\n\[ f\left( {x, y}\right) = \left\{ \begin{array}{ll} h\left( y\right) & \text{ if }{Ay} = x \\ \infty & \text{ otherwise,} \end{array}\right. \]\n\nwhich is convex in \( \left( {x, y}\right) \) . Then \( g \) is the minimum of \( f \) over \( y \), and hence is convex. (It is not ha... | Yes |
Example 3.18 Euclidean norm squared. The perspective of the convex function \( f\left( x\right) = {x}^{T}x \) on \( {\mathbf{R}}^{n} \) is | \[ g\left( {x, t}\right) = t{\left( x/t\right) }^{T}\left( {x/t}\right) = \frac{{x}^{T}x}{t}, \] which is convex in \( \left( {x, t}\right) \) for \( t > 0 \). We can deduce convexity of \( g \) using several other methods. First, we can express \( g \) as the sum of the quadratic-over-linear functions \( {x}_{i}^{2}/t... | Yes |
We derive the conjugates of some convex functions on \( \mathbf{R} \) . | - Affine function. \( f\left( x\right) = {ax} + b \) . As a function of \( x,{yx} - {ax} - b \) is bounded if and only if \( y = a \), in which case it is constant. Therefore the domain of the conjugate function \( {f}^{ * } \) is the singleton \( \{ a\} \), and \( {f}^{ * }\left( a\right) = - b \) . | Yes |
Consider \( f\left( x\right) = \frac{1}{2}{x}^{T}{Qx} \), with \( Q \in {\mathbf{S}}_{+ + }^{n} \) . The function \( {y}^{T}x - \frac{1}{2}{x}^{T}{Qx} \) is bounded above as a function of \( x \) for all \( y \) . | It attains its maximum at \( x = {Q}^{-1}y \), so\n\n\[ \n{f}^{ * }\left( y\right) = \frac{1}{2}{y}^{T}{Q}^{-1}y.\n\] | Yes |
We consider \( f\left( X\right) = \log \det {X}^{-1} \) on \( {\mathbf{S}}_{+ + }^{n} \) . The conjugate function is defined as \[ {f}^{ * }\left( Y\right) = \mathop{\sup }\limits_{{X \succ 0}}\left( {\operatorname{tr}\left( {YX}\right) + \log \det X}\right) ,\] since \( \operatorname{tr}\left( {YX}\right) \) is the st... | We first show that \( \operatorname{tr}\left( {YX}\right) + \log \det X \) is unbounded above unless \( Y \prec 0 \) . If \( Y \nprec 0 \), then \( Y \) has an eigenvector \( v \) , with \( \parallel v{\parallel }_{2} = 1 \), and eigenvalue \( \lambda \geq 0 \) . Taking \( X = I + {tv}{v}^{T} \) we find that \[ \operat... | Yes |
Let \( {I}_{S} \) be the indicator function of a (not necessarily convex) set \( S \subseteq {\mathbf{R}}^{n} \), i.e., \( {I}_{S}\left( x\right) = 0 \) on \( \operatorname{dom}{I}_{S} = S \). Its conjugate is | \[ {I}_{S}^{ * }\left( y\right) = \mathop{\sup }\limits_{{x \in S}}{y}^{T}x \] which is the support function of the set \( S \). | Yes |
To derive the conjugate of the log-sum-exp function \( f\left( x\right) = \log \left( {\mathop{\sum }\limits_{{i = 1}}^{n}{e}^{{x}_{i}}}\right) \), we first determine the values of \( y \) for which the maximum over \( x \) of \( {y}^{T}x - f\left( x\right) \) is attained. | By setting the gradient with respect to \( x \) equal to zero, we obtain the condition\n\n\[ \n{y}_{i} = \frac{{e}^{{x}_{i}}}{\mathop{\sum }\limits_{{j = 1}}^{n}{e}^{{x}_{j}}},\;i = 1,\ldots, n. \n\]\n\nThese equations are solvable for \( x \) if and only if \( y \succ 0 \) and \( {\mathbf{1}}^{T}y = 1 \) . By substitu... | Yes |
Let \( \parallel \cdot \parallel \) be a norm on \( {\mathbf{R}}^{n} \), with dual norm \( \parallel \cdot {\parallel }_{ * } \). We will show that the conjugate of \( f\left( x\right) = \parallel x\parallel \) is\n\n\[ \n{f}^{ * }\left( y\right) = \left\{ \begin{array}{ll} 0 & \parallel y{\parallel }_{ * } \leq 1 \\ \... | If \( \parallel y{\parallel }_{ * } > 1 \), then by definition of the dual norm, there is a \( z \in {\mathbf{R}}^{n} \) with \( \parallel z\parallel \leq 1 \) and \( {y}^{T}z > 1 \). Taking \( x = {tz} \) and letting \( t \rightarrow \infty \), we have\n\n\[ \n{y}^{T}x - \parallel x\parallel = t\left( {{y}^{T}z - \par... | Yes |
Now consider the function \( f\left( x\right) = \left( {1/2}\right) \parallel x{\parallel }^{2} \), where \( \parallel \cdot \parallel \) is a norm, with dual norm \( \parallel \cdot {\parallel }_{ * } \) . We will show that its conjugate is \( {f}^{ * }\left( y\right) = \left( {1/2}\right) \parallel y{\parallel }_{ * ... | From \( {y}^{T}x \leq \parallel y{\parallel }_{ * }\parallel x\parallel \), we conclude\n\n\[ \n{y}^{T}x - \left( {1/2}\right) \parallel x{\parallel }^{2} \leq \parallel y{\parallel }_{ * }\parallel x\parallel - \left( {1/2}\right) \parallel x{\parallel }^{2} \n\]\n\nfor all \( x \) . The righthand side is a quadratic ... | Yes |
We consider a business or enterprise that consumes \( n \) resources and produces a product that can be sold. We let \( r = \left( {{r}_{1},\ldots ,{r}_{n}}\right) \) denote the vector of resource quantities consumed, and \( S\left( r\right) \) denote the sales revenue derived from the product produced (as a function o... | \[ M\left( p\right) = \mathop{\sup }\limits_{r}\left( {S\left( r\right) - {p}^{T}r}\right) \] The function \( M\left( p\right) \) gives the maximum profit attainable, as a function of the resource prices. In terms of conjugate functions, we can express \( M \) as \[ M\left( p\right) = {\left( -S\right) }^{ * }\left( {-... | Yes |
Example 3.30 Length of a vector. We define the length of \( x \in {\mathbf{R}}^{n} \) as the largest index of a nonzero component, i.e., \[ f\left( x\right) = \max \left\{ {i \mid {x}_{i} \neq 0}\right\} . \] (We define the length of the zero vector to be zero.) | This function is quasiconvex on \( {\mathbf{R}}^{n} \), since its sublevel sets are subspaces: \[ f\left( x\right) \leq \alpha \Leftrightarrow {x}_{i} = 0\text{ for }i = \lfloor \alpha \rfloor + 1,\ldots, n. \] | Yes |
Example 3.31 Consider \( f : {\mathbf{R}}^{2} \rightarrow \mathbf{R} \), with \( \operatorname{dom}f = {\mathbf{R}}_{ + }^{2} \) and \( f\left( {{x}_{1},{x}_{2}}\right) = {x}_{1}{x}_{2} \). This function is neither convex nor concave since its Hessian | \[ {\nabla }^{2}f\left( x\right) = \left\lbrack \begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right\rbrack \] is indefinite; it has one positive and one negative eigenvalue. The function \( f \) is quasiconcave, however, since the superlevel sets \[ \left\{ {x \in {\mathbf{R}}_{ + }^{2} \mid {x}_{1}{x}_{2} \geq \alpha }... | Yes |
Example 3.32 Linear-fractional function. The function\n\n\[ f\left( x\right) = \frac{{a}^{T}x + b}{{c}^{T}x + d} \]\n\nwith \( \operatorname{dom}f = \left\{ {x \mid {c}^{T}x + d > 0}\right\} \), is quasiconvex, and quasiconcave, i.e., quasilinear. Its \( \alpha \) -sublevel set is\n\n\[ {S}_{\alpha } = \left\{ {x \mid ... | which is convex, since it is the intersection of an open halfspace and a closed halfspace. (The same method can be used to show its superlevel sets are convex.) | Yes |
Suppose \( a, b \in {\mathbf{R}}^{n} \), and define\n\n\[ f\left( x\right) = \frac{\parallel x - a{\parallel }_{2}}{\parallel x - b{\parallel }_{2}} \]\n\ni.e., the ratio of the Euclidean distance to \( a \) to the distance to \( b \) . Then \( f \) is quasiconvex on the halfspace \( \left\{ {x \mid \parallel x - a{\pa... | To see this, we consider the \( \alpha \) -sublevel set of \( f \), with \( \alpha \leq 1 \) since \( f\left( x\right) \leq 1 \) on the halfspace \( \left\{ {x \mid \parallel x - a{\parallel }_{2} \leq \parallel x - b{\parallel }_{2}}\right\} \) . This sublevel set is the set of points satisfying\n\n\[ \parallel x - a{... | Yes |
Internal rate of return. Let \( x = \left( {{x}_{0},{x}_{1},\ldots ,{x}_{n}}\right) \) denote a cash flow sequence over \( n \) periods, where \( {x}_{i} > 0 \) means a payment to us in period \( i \), and \( {x}_{i} < 0 \) means a payment by us in period \( i \) . We define the present value of a cash flow, with inter... | Internal rate of return is a quasiconcave function of \( x \) (restricted to \( {x}_{0} < 0,{x}_{1} + \cdots + \) \( \left. {{x}_{n} > 0}\right) \) . To see this, we note that \[ \operatorname{IRR}\left( x\right) \geq R \Leftrightarrow \operatorname{PV}\left( {x, r}\right) > 0\text{ for }0 \leq r < R. \] The lefthand s... | Yes |
Example 3.35 Cardinality of a nonnegative vector. The cardinality or size of a vector \( x \in {\mathbf{R}}^{n} \) is the number of nonzero components, and denoted \( \operatorname{card}\left( x\right) \) . The function card is quasiconcave on \( {\mathbf{R}}_{ + }^{n} \) (but not \( {\mathbf{R}}^{n} \) ). | This follows immediately from the modified Jensen inequality\n\n\[\n\operatorname{card}\left( {x + y}\right) \geq \min \{ \operatorname{card}\left( x\right) ,\operatorname{card}\left( y\right) \}\n\]\n\nwhich holds for \( x, y \succcurlyeq 0 \) . | Yes |
Example 3.36 Rank of positive semidefinite matrix. The function \( \operatorname{rank}X \) is quasi-concave on \( {\mathbf{S}}_{ + }^{n} \) . | This follows from the modified Jensen inequality (3.19), \n\n\[ \operatorname{rank}\left( {X + Y}\right) \geq \min \{ \operatorname{rank}X,\operatorname{rank}Y\} \]\n\nwhich holds for \( X, Y \in {\mathbf{S}}_{ + }^{n} \) . (This can be considered an extension of the previous example, since \( \operatorname{rank}\left(... | Yes |
The maximum generalized eigenvalue of a pair of symmetric matrices \( \left( {X, Y}\right) \), with \( Y \succ 0 \), is defined as\n\n\[ \n{\lambda }_{\max }\left( {X, Y}\right) = \mathop{\sup }\limits_{{u \neq 0}}\frac{{u}^{T}{Xu}}{{u}^{T}{Yu}} = \sup \{ \lambda \mid \det \left( {{\lambda Y} - X}\right) = 0\} .\n\] | To see this we consider the expression\n\n\[ \n{\lambda }_{\max }\left( {X, Y}\right) = \mathop{\sup }\limits_{{u \neq 0}}\frac{{u}^{T}{Xu}}{{u}^{T}{Yu}}.\n\]\n\nFor each \( u \neq 0 \), the function \( {u}^{T}{Xu}/{u}^{T}{Yu} \) is linear-fractional in \( \left( {X, Y}\right) \), hence a quasiconvex function of \( \le... | No |
Example 3.38 Convex over concave function. Suppose \( p \) is a convex function, \( q \) is a concave function, with \( p\left( x\right) \geq 0 \) and \( q\left( x\right) > 0 \) on a convex set \( C \) . Then the function \( f \) defined by \( f\left( x\right) = p\left( x\right) /q\left( x\right) \), on \( C \), is qua... | Here we have\n\n\[ f\left( x\right) \leq t \Leftrightarrow p\left( x\right) - {tq}\left( x\right) \leq 0, \]\n\nso we can take \( {\phi }_{t}\left( x\right) = p\left( x\right) - {tq}\left( x\right) \) for \( t \geq 0 \) . For each \( t,{\phi }_{t} \) is convex and for each \( x,{\phi }_{t}\left( x\right) \) is decreasi... | Yes |
Example 3.40 Log-concave density functions. Many common probability density functions are log-concave. Two examples are the multivariate normal distribution,\n\n\[ f\\left( x\\right) = \\frac{1}{\\sqrt{{\\left( 2\\pi \\right) }^{n}\\det \\sum }}{e}^{-\\frac{1}{2}{\\left( x - \\bar{x}\\right) }^{T}{\\sum }^{-1}\\left( {... | As a more exotic example consider the Wishart distribution, defined as follows. Let \( {x}_{1},\\ldots ,{x}_{p} \\in {\\mathbf{R}}^{n} \) be independent Gaussian random vectors with zero mean and covariance \( \\sum \\in {\\mathbf{S}}^{n} \), with \( p > n \) . The random matrix \( X = \\mathop{\\sum }\\limits_{{i = 1}... | Yes |
Example 3.43 Yield function. Let \( x \in {\mathbf{R}}^{n} \) denote the nominal or target value of a set of parameters of a product that is manufactured. Variation in the manufacturing process causes the parameters of the product, when manufactured, to have the value \( x + w \), where \( w \in {\mathbf{R}}^{n} \) is ... | If the density of the manufacturing error \( w \) is log-concave (for example, Gaussian) and the set \( S \) of product specifications is convex, then the yield function \( Y \) is log-concave. This implies that the \( \alpha \) -yield region, defined as the set of nominal parameters for which the yield exceeds \( \alp... | Yes |
Let \( A \in {\mathbf{R}}^{m \times n} \) . Define\n\n\[ \n{P}_{u} = \left\{ {x \in {\mathbf{R}}^{n} \mid {Ax} \preccurlyeq u}\right\} \n\]\n\nThen its volume vol \( {P}_{u} \) is a log-concave function of \( u \) . | To prove this, note that the function\n\n\[ \n\Psi \left( {x, u}\right) = \left\{ \begin{array}{ll} 1 & {Ax} \preccurlyeq u \\ 0 & \text{ otherwise,} \end{array}\right. \n\]\n\nis log-concave. By the integration result, we conclude that\n\n\[ \n\int \Psi \left( {x, u}\right) {dx} = \operatorname{vol}{P}_{u} \n\]\n\nis ... | Yes |
A differentiable function \( f \), with convex domain, is \( K \) -nondecreasing if and only if\n\n\[ \nabla f\left( x\right) { \succcurlyeq }_{{K}^{ * }}0 \]\n\n(3.24)\n\nfor all \( x \in \operatorname{dom}f \). | First, assume that \( f \) satisfies (3.24) for all \( x \), but is not \( K \) -nondecreasing, i.e., there exist \( x, y \) with \( x{ \preccurlyeq }_{K}y \) and \( f\left( y\right) < f\left( x\right) \) . By differentiability of \( f \) there exists a \( t \in \left\lbrack {0,1}\right\rbrack \) with\n\n\[ \frac{d}{dt... | Yes |
Example 3.47 Convexity with respect to componentwise inequality. A function \( f \) : \( {\mathbf{R}}^{n} \rightarrow {\mathbf{R}}^{m} \) is convex with respect to componentwise inequality (i.e., the generalized inequality induced by \( {\mathbf{R}}_{ + }^{m} \) ) if and only if for all \( x, y \) and \( 0 \leq \theta ... | \[ f\left( {{\theta x} + \left( {1 - \theta }\right) y}\right) \preccurlyeq {\theta f}\left( x\right) + \left( {1 - \theta }\right) f\left( y\right) ,\] | Yes |
Example 3.49 The quadratic matrix function \( g : {\mathbf{R}}^{m \times n} \rightarrow {\mathbf{S}}^{n} \) defined by\n\n\[ g\left( X\right) = {X}^{T}{AX} + {B}^{T}X + {X}^{T}B + C, \]\n\nwhere \( A \in {\mathbf{S}}^{m}, B \in {\mathbf{R}}^{m \times n} \), and \( C \in {\mathbf{S}}^{n} \), is convex when \( A \succcur... | The function \( h : {\mathbf{S}}^{n} \rightarrow \mathbf{R} \) defined by \( h\left( Y\right) = - \log \det \left( {-Y}\right) \) is convex and increasing on \( \operatorname{dom}h = - {\mathbf{S}}_{+ + }^{n} \) .\n\nBy the composition theorem, we conclude that\n\n\[ f\left( X\right) = - \log \det \left( {-\left( {{X}^... | Yes |
Consider the optimization problem\n\n\\[ \n\\text{minimize} \\; {f}_{0}\\left( x\\right) \n\\]\n\n\\[ \n\\text{subject to} \\; {l}_{i} \\leq {x}_{i} \\leq {u}_{i}, \\; i = 1, \\ldots, n \\text{,} \n\\]\n\nwhere \\( x \\in {\\mathbf{R}}^{n} \\) is the variable. The constraints are called variable bounds (since they give... | We can express this problem in standard form as\n\n\\[ \n\\text{minimize} \\; {f}_{0}\\left( x\\right) \n\\]\n\n\\[ \n\\text{subject to} \\; {l}_{i} - {x}_{i} \\leq 0, \\; i = 1, \\ldots, n \n\\]\n\n\\[ \n{x}_{i} - {u}_{i} \\leq 0, \\; i = 1, \\ldots, n. \n\\]\n\nThere are \\( {2n} \\) inequality constraint functions:\... | Yes |
As a simple example consider the unconstrained Euclidean norm minimization problem\n\n\\[ \n\\text{minimize}\\parallel {Ax} - b{\\parallel }_{2}\\text{,}\n\\]\n\n(4.5)\n\nwith variable \\( x \\in {\\mathbf{R}}^{n} \\). | Since the norm is always nonnegative, we can just as well solve the problem\n\n\\[ \n\\text{minimize}\\parallel {Ax} - b{\\parallel }_{2}^{2} = {\\left( Ax - b\\right) }^{T}\\left( {{Ax} - b}\\right) \\text{,}\n\\]\n\n(4.6)\n\nin which we minimize the square of the Euclidean norm. The problems (4.5) and (4.6) are clear... | Yes |
Consider a problem with strictly convex quadratic objective, with some of the variables unconstrained:\n\n\[ \n\\text{minimize}\;{x}_{1}^{T}{P}_{11}{x}_{1} + 2{x}_{1}^{T}{P}_{12}{x}_{2} + {x}_{2}^{T}{P}_{22}{x}_{2} \n\]\n\n\[ \n\\text{subject to}\;{f}_{i}\left( {x}_{1}\right) \leq 0,\;i = 1,\ldots, m\\text{,} \n\]\n\nw... | Here we can analytically minimize over \( {x}_{2} \) :\n\n\[ \n\\mathop{\\inf }\\limits_{{x}_{2}}\\left( {{x}_{1}^{T}{P}_{11}{x}_{1} + 2{x}_{1}^{T}{P}_{12}{x}_{2} + {x}_{2}^{T}{P}_{22}{x}_{2}}\\right) = {x}_{1}^{T}\\left( {{P}_{11} - {P}_{12}{P}_{22}^{-1}{P}_{12}^{T}}\\right) {x}_{1} \n\]\n\n(see §A.5.5). Therefore the... | Yes |
Consider the problem of minimizing the quadratic function\n\n\[ \n{f}_{0}\left( x\right) = \left( {1/2}\right) {x}^{T}{Px} + {q}^{T}x + r, \n\]\n\nwhere \( P \in {\mathbf{S}}_{ + }^{n} \) (which makes \( {f}_{0} \) convex). The necessary and sufficient condition for \( x \) to be a minimizer of \( {f}_{0} \) is\n\n\[ \... | - If \( q \notin \mathcal{R}\left( P\right) \), then there is no solution. In this case \( {f}_{0} \) is unbounded below.\n\n- If \( P \succ 0 \) (which is the condition for \( {f}_{0} \) to be strictly convex), then there is a unique minimizer, \( {x}^{ \star } = - {P}^{-1}q \) . | Yes |
Consider a loss risk constraint of the form\n\n\[ \operatorname{prob}\left( {r \leq \alpha }\right) \leq \beta \]\n\nwhere \( \alpha \) is a given unwanted return level (e.g., a large loss) and \( \beta \) is a given maximum probability. | As in the stochastic interpretation of the robust LP given above, we can express this constraint using the cumulative distribution function \( \Phi \) of a unit Gaussian random variable. The inequality (4.39) is equivalent to\n\n\[ {\bar{p}}^{T}x + {\Phi }^{-1}\left( \beta \right) {\begin{Vmatrix}{\sum }^{1/2}x\end{Vma... | Yes |
Suppose \( y = {Ax} + v \), where \( v \in {\mathbf{R}}^{m} \) is a measurement noise, \( y \in {\mathbf{R}}^{m} \) is a vector of measurements, and \( x \in {\mathbf{R}}^{n} \) is a vector to be estimated, given the measurement \( y \). We assume that \( A \) has rank \( n \), and that the measurement noise satisfies ... | It is a famous result that the problem (4.58) has an optimal solution, the least-squares estimator, or pseudo-inverse,\n\n\[ {F}^{ \star } = {A}^{ \dagger } = {\left( {A}^{T}A\right) }^{-1}{A}^{T}. \]\n\nFor any \( F \) with \( {FA} = I \), we have \( F{F}^{T} \succcurlyeq {F}^{ \star }{F}^{\star T} \). The matrix\n\n\... | Yes |
Minimal upper bound on a set of matrices. We consider the (convex) vector optimization problem, with respect to the positive semidefinite cone,\n\n\\[ \n\\text{minimize (w.r.t.}{\\mathbf{S}}_{ + }^{n}\\text{)}X \n\\]\n\n(4.63)\n\n\\[ \n\\text{subject to}\\;X \\succcurlyeq {A}_{i},\\;i = 1,\\ldots, m\\text{,}\n\\]\n\nwh... | To find a Pareto optimal point, we apply scalarization: we choose any \\( W \\in {\\mathbf{S}}_{+ + }^{n} \\) and form the problem\n\n\\[ \n\\text{minimize}\\;\\operatorname{tr}\\left( {WX}\\right) \n\\]\n\n(4.64)\n\n\\[ \n\\text{subject to}X \\succcurlyeq {A}_{i},\\;i = 1,\\ldots, m\\text{,}\n\\]\n\nwhich is an SDP. D... | Yes |
We consider the problem\n\n\\[ \n\\text{minimize}\\;\\left( {1/2}\\right) {x}^{T}{Px} + {q}^{T}x + r \n\\]\n\n\\( \\left( {5.50}\\right) \\)\n\n\\[ \n\\text{subject to}\\{Ax} = b\\text{,}\n\\]\n\nwhere \\( P \\in {\\mathbf{S}}_{ + }^{n} \\). | The KKT conditions for this problem are\n\n\\[ \nA{x}^{ \\star } = b,\\;P{x}^{ \\star } + q + {A}^{T}{\\nu }^{ \\star } = 0,\n\\]\n\nwhich we can write as\n\n\\[ \n\\left\\lbrack \\begin{matrix} P & {A}^{T} \\\\ A & 0 \\end{matrix}\\right\\rbrack \\left\\lbrack \\begin{matrix} {x}^{ \\star } \\\\ {\\nu }^{ \\star } \\e... | Yes |
We consider the convex optimization problem\n\n\\[ \n\\text{minimize}\; - \\mathop{\\sum }\\limits_{{i = 1}}^{n}\\log \\left( {{\\alpha }_{i} + {x}_{i}}\\right)\n\\]\n\n\\[ \n\\text{subject to}x \\succcurlyeq 0,\\;{\\mathbf{1}}^{T}x = 1\\text{,}\n\\]\n\nwhere \\( {\\alpha }_{i} > 0 \\) . This problem arises in informat... | Introducing Lagrange multipliers \\( {\\lambda }^{ \\star } \\in {\\mathbf{R}}^{n} \\) for the inequality constraints \\( {x}^{ \\star } \\succcurlyeq 0 \\) , and a multiplier \\( {\\nu }^{ \\star } \\in \\mathbf{R} \\) for the equality constraint \\( {\\mathbf{1}}^{T}x = 1 \\), we obtain the KKT conditions\n\n\\[ \n{x... | Yes |
We consider the entropy maximization problem\n\n\\[ \n\\text{ minimize }\\;{f}_{0}\\left( x\\right) = \\mathop{\\sum }\\limits_{{i = 1}}^{n}{x}_{i}\\log {x}_{i}\n\\]\n\n\\[ \n\\text{subject to}\\;{Ax} \\preccurlyeq b\n\\]\n\n\\[ \n{\\mathbf{1}}^{T}x = 1\n\\]\n\nwith domain \\( {\\mathbf{R}}_{+ + }^{n} \\), and its dual... | Suppose we have solved the dual problem. The Lagrangian at \\( \\left( {{\\lambda }^{ \\star },{\\nu }^{ \\star }}\\right) \\) is\n\n\\[ \nL\\left( {x,{\\lambda }^{ \\star },{\\nu }^{ \\star }}\\right) = \\mathop{\\sum }\\limits_{{i = 1}}^{n}{x}_{i}\\log {x}_{i} + {\\lambda }^{\\star T}\\left( {{Ax} - b}\\right) + {\\n... | Yes |
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 {\\mathbf{R}}^{n}, b... | 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 |
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}{Ax}... | 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} \... | Yes |
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 \\preccurlyeq x \\preccurlyeq u \\) are s... | 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 |
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 |
Feasibility of a linear matrix inequality. The following systems are strong alternatives: | \[ F\left( x\right) = {x}_{1}{F}_{1} + \cdots + {x}_{n}{F}_{n} + G \prec 0, \] where \( {F}_{i}, G \in {\mathbf{S}}^{k} \), and \[ Z \succcurlyeq 0,\;Z \neq 0,\;\operatorname{tr}\left( {GZ}\right) \geq 0,\;\operatorname{tr}\left( {{F}_{i}Z}\right) = 0,\;i = 1,\ldots, n, \] where \( Z \in {\mathbf{S}}^{k} \). This follo... | 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\[ \n\\text{minimize}\\parallel A\\left( u\\right) x - b{\\parallel }_{2}^{2}\\text{,}\n\]\n\nwhere \( u \\in \\mathbf{R} \) is an uncertain parameter and \( A\\l... | 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 |
Example 6.6 Comparison of worst-case robust, Tikhonov regularized, and nominal least-squares solutions. 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... | 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}_{... | 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 |
When \( {v}_{i} \) are Gaussian with zero mean and variance \( {\sigma }^{2} \), what is the ML estimate of \( x \) in the log-likelihood function? | The log-likelihood function is\n\n\[ l\left( x\right) = - \left( {m/2}\right) \log \left( {{2\pi }{\sigma }^{2}}\right) - \frac{1}{2{\sigma }^{2}}\parallel {Ax} - y{\parallel }_{2}^{2}, \]\n\nwhere \( A \) is the matrix with rows \( {a}_{1}^{T},\ldots ,{a}_{m}^{T} \). Therefore the ML estimate of \( x \) is \( {x}_{\ma... | Yes |
Example 7.2 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\[ \text{E}{X}^{2} \in \left\lbrack {{0.5},{0.6}}\righ... | 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 the 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},\mathbf... | 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 \( k \) ,\n\n\[ C = \\left\\{ {X \\in {\\mathbf{R}}^{m \\times n} \\mid \\operatorname{rank}X \\leq k}\\right\\} \]\n\nwith \( k \\leq \\min \\{ m, n\\} \), and let \( {X}_{0} \\in {\\mathbf{R}}^{m \\times n} \). We can find a projection of \... | Let\n\n\[ {X}_{0} = \\mathop{\\sum }\\limits_{{i = 1}}^{r}{\\sigma }_{i}{u}_{i}{v}_{i}^{T} \]\n\nbe 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 pro... | 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\\[ \n{\\alpha }_{k} = {M}_{kk}^{2} + \\mathop{\\sum }\\l... | 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 th... | Yes |
Example 9.4 Log barrier for linear inequalities. The function\n\n\[ f\left( x\right) = - \mathop{\sum }\limits_{{i = 1}}^{m}\log \left( {{b}_{i} - {a}_{i}^{T}x}\right) \]\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 |
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\[ \n\\begin{array}{ll} \\text{ minimize } & \\mathop{\\sum }\\limits_{{i = 1}}^{n}{f}_{i}\\left( {x}_{i}\\right) \\\\ \\text{ subject to } & \\mathop{\\sum }\\limits_{{i = 1}}^{n}{x}_{i} = b, \\end{array} \n\]\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 \\\\ - {\\mathbf{1}}^{T} \\end{matrix}\\right\\rbrack \\in {\\mathbf{R}}^{n \\time... | 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 is \(... | Yes |
Example 10.4 Minimum length piecewise-linear curve subject to equality constraints.\n\nWe consider a piecewise-linear curve in \( {\mathbf{R}}^{2} \) with knot points \( \left( {0,0}\right) ,\left( {1,{x}_{1}}\right) ,\ldots ,\left( {n,{x}_{n}}\right) \) . To find the minimum length curve that satisfies the equality co... | In this problem, the objective is a sum of functions of pairs of adjacent variables, so the Hessian \( H \) is tridiagonal. Using block elimination, we can compute the Newton step in around \( {p}^{2}n + \left( {1/3}\right) {p}^{3} \) flops. | 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\n\nis dual feasible, with dual objective value\n\n\\[ \n- {b}^{T}{\\lambda }^{ \\star }\\left( ... | 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 minimize... | 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(11.44) \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\n(11.45) \n\nwith \( \operatorname{dom}\phi = \left\{ {... | Yes |
We consider the SDP with variable \( x \in {\mathbf{R}}^{n} \), minimize \( {c}^{T}x \) subject to \( F\left( x\right) = {x}_{1}{F}_{1} + \cdots + {x}_{n}{F}_{n} + G \preccurlyeq 0 \), where \( G,{F}_{1},\ldots ,{F}_{n} \in {\mathbf{S}}^{p} \). | The dual problem is maximize \( \operatorname{tr}\left( {GZ}\right) \) subject to \( \operatorname{tr}\left( {{F}_{i}Z}\right) + {c}_{i} = 0,\;i = 1,\ldots, n \) and \( Z \succcurlyeq 0 \). Using the generalized logarithm \( \log \det X \) for the positive semidefinite cone \( {\mathbf{S}}_{ + }^{p} \), we have the bar... | 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 |
Proposition 2.1 (Weak Duality) If \( X \) is feasible in (P) and \( \left( {y, S}\right) \) in (D), then\n\n\[ C \bullet X - {b}^{T}y = X \bullet S \geq 0. \] | Proof:\n\nWe find\n\n\[ C \bullet X - {b}^{T}y = (\mathop{\sum }\limits_{{i = 1}}^{m}{y}_{i}{A}_{i} + S) \bullet X - {b}^{T}y = \mathop{\sum }\limits_{{i = 1}}^{m}({A}_{i} \bullet X){y}_{i} + S \bullet X - {b}^{T}y = S \bullet X = X \bullet S. \]\n\nMoreover, since \( X \) is positive semidefinite, it has a square root... | Yes |
Theorem 5.1 \( f\left( X\right) \mathrel{\text{:=}} - \ln \det X \) is an \( n \) -normal barrier for \( S{\mathbb{R}}_{ + }^{n \times n} \) . | Proof:\n\nDefine \( \phi \) as above. Then it is finite on the convex set of \( \alpha \) such that \( X + {\alpha H} \succ 0 \), and on this set\n\n\[ \n{\phi }^{\prime \prime }\left( \alpha \right) = {D}^{2}f\left( \bar{X}\right) \left\lbrack {H, H}\right\rbrack = \left( {{\bar{X}}^{-1}H{\bar{X}}^{-1}}\right) \bullet... | Yes |
Proposition 5.1 For \( \alpha > 0, X \succ 0 \) of order \( n \) ,\n\n\[ \n{f}^{\prime }\left( {\alpha X}\right) = {\alpha }^{-1}{f}^{\prime }\left( X\right) ,\;{f}^{\prime \prime }\left( {\alpha X}\right) = {\alpha }^{-2}{f}^{\prime \prime }\left( X\right) ; \n\]\n\n\[ \n{f}^{\prime }\left( X\right) \bullet X = - n,\;... | The last line also states that the \( X \) -norm of the Newton step for minimizing \( f \) from \( X \) , \( - {\left\lbrack {f}^{\prime \prime }\left( X\right) \right\rbrack }^{-1}{f}^{\prime }\left( X\right) \), is exactly \( \sqrt{n} \) . This shows that \( f \) satisfies also the original definition of Nesterov and... | Yes |
Theorem 5.3 Suppose the operators \( \mathcal{E} \) and \( \mathcal{F} \) map \( {\mathrm{{SIR}}}^{n \times n} \) to itself, and that \( \mathcal{E} \) is nonsingular and \( {\mathcal{E}}^{-1}\mathcal{F} \) is positive definite (but not necessarily self-adjoint). Assume that the linear independence condition holds. The... | Proof:\n\nThe formulae for \( W \) and \( U \) follow directly from the first and third equations. Now substituting for \( W \) in the formula for \( U \), and inserting this in the second equation, we obtain after some manipulation\n\n\[ \n\left( {{\mathcal{{AE}}}^{-1}\mathcal{F}{\mathcal{A}}^{ * }}\right) v = q - {\m... | Yes |
Theorem 5.4 Assume that both \( \left( P\right) \) and \( \left( D\right) \) have strictly feasible solutions and the linear independence condition holds. Then the set of solutions to (7) for all positive \( \nu \) forms a nonempty differentiable path, called the central path. If \( \left( {X\left( \nu \right), y\left(... | \[ C \bullet X\left( \nu \right) - {b}^{T}y\left( \nu \right) = X\left( \nu \right) \bullet S\left( \nu \right) = {n\nu }. \] | Yes |
Theorem 5.5 The existence of strictly feasible solutions to \( \left( P\right) \) and \( \left( D\right) \) and the linear independence condition imply that both have bounded nonempty optimal solution sets, with zero duality gap. | The last part follows from the existence of the central path, since by (12) the duality gap associated to \( X\left( \nu \right) \) and \( \left( {y\left( \nu \right), S\left( \nu \right) }\right) \) is \( {n\nu } \), and this approaches zero as \( \nu \) tends to zero. (In fact, the central path approaches optimal sol... | No |
Theorem 6.1 Suppose \( \left( {{X}_{0},{y}_{0},{S}_{0}}\right) \in {F}^{0}\left( P\right) \times {F}^{0}\left( D\right) \) satisfies\n\n\[ \n{\Psi }_{0}\left( {{X}_{0},{y}_{0},{S}_{0}}\right) \leq \rho \ln \frac{1}{\epsilon } \n\]\n\nfor some \( \epsilon > 0 \) . Then, if we generate a sequence of strictly feasible poi... | Proof:\n\nLet \( K \mathrel{\text{:=}} {2\rho }\ln \left( {1/\epsilon }\right) /\delta \) . Then, using the fact above, we have\n\n\[ \n\rho \ln {X}_{K} \bullet {S}_{K} \leq \rho \ln {X}_{K} \bullet {S}_{K} + {\Psi }_{0}\left( {{X}_{K},{y}_{K},{S}_{K}}\right) \n\]\n\n\[ \n= {\Psi }_{\rho }\left( {{X}_{K},{y}_{K},{S}_{K... | Yes |
Lemma 1 Let \( \\left\\{ {\\alpha }_{k}\\right\\} \) be the stepsizes in the \( {AG} \) method and the sequence \( \\left\\{ {\\theta }_{k}\\right\\} \) satisfies\n\n\[ \n{\\theta }_{k} \\leq \\left( {1 - {\\alpha }_{k}}\\right) {\\theta }_{k - 1} + {\\eta }_{k},\\;k = 1,2,\\ldots ,\n\]\n\n(2.5)\n\nwhere\n\n\[ \n{\\Gam... | Proof. Noting that \( {\\alpha }_{1} = 1 \) and \( {\\alpha }_{k} \\in \\left( {0,1}\\right) \) for any \( k \\geq 2 \) . These observations together with (2.6) then imply that \( {\\Gamma }_{k} > 0 \) for any \( k \\geq 1 \) . Dividing both sides of (2.5) by \( {\\Gamma }_{k} \), we obtain\n\n\[ \n\\frac{{\\theta }_{1... | Yes |
Theorem 1 Let \( {\left\{ {x}_{k}^{md},{x}_{k}^{ag}\right\} }_{k \geq 1} \) be computed by Algorithm 1 and \( {\Gamma }_{k} \) be defined in (2.6).\n\na) If \( \left\{ {\alpha }_{k}\right\} ,\left\{ {\beta }_{k}\right\} \), and \( \left\{ {\lambda }_{k}\right\} \) are chosen such that\n\n\[ \n{C}_{k} \mathrel{\text{:=}... | Proof. We first show part a). Denote \( {\Delta }_{k} \mathrel{\text{:=}} \nabla \Psi \left( {x}_{k - 1}\right) - \nabla \Psi \left( {x}_{k}^{md}\right) \) . By (1.2) and (2.2), we have\n\n\[ \n\begin{Vmatrix}{\Delta }_{k}\end{Vmatrix} = \begin{Vmatrix}{\nabla \Psi \left( {x}_{k - 1}\right) - \nabla \Psi \left( {x}_{k}... | Yes |
a) If \( \left\{ {\lambda }_{k}\right\} \) satisifies\n\n\[{\lambda }_{k} \in \left\lbrack {{\beta }_{k},\left( {1 + \frac{{\alpha }_{k}}{4}}\right) {\beta }_{k}}\right\rbrack \;\forall k \geq 1,\]\n\nthen for any \( N \geq 1 \), we have\n\n\[\mathop{\min }\limits_{{k = 1,\ldots, N}}{\begin{Vmatrix}\nabla \Psi \left( {... | Proof. We first show part a). Note that by (2.6) and (2.27), we have\n\n\[{\Gamma }_{k} = \frac{2}{k\left( {k + 1}\right) }\n\]\nwhich implies that\n\n\[\mathop{\sum }\limits_{{\tau = k}}^{N}{\Gamma }_{\tau } = \mathop{\sum }\limits_{{\tau = k}}^{N}\frac{2}{\tau \left( {\tau + 1}\right) } = 2\mathop{\sum }\limits_{{\ta... | Yes |
Lemma 2 If \( \mathcal{X}\left( \cdot \right) \) is a proper closed convex function with bounded domain, then Assumption 2 is satisfied. | Proof. Denote \( X \equiv \operatorname{dom}\left( \mathcal{X}\right) \mathrel{\text{:=}} \{ u \mid \mathcal{X}\left( u\right) < + \infty \} \) . Note that by assumption, \( X \) is nonempty and bounded. Also observe that (2.37) is equivalent to\n\n\[ \mathcal{P}\left( {x, y, c}\right) = {\operatorname{argmin}}_{u \in ... | Yes |
Lemma 3 Let \( x \in {\mathbb{R}}^{n} \) be given and denote \( g \equiv \nabla \Psi \left( x\right) \) . If \( \parallel \mathcal{G}\left( {x, g, c}\right) \parallel \leq \epsilon \) for some \( c > 0 \), then\n\n\[ - \nabla \Psi \left( {\mathcal{P}\left( {x, g, c}\right) }\right) \in \partial \mathcal{X}\left( {\math... | Proof. By the optimality condition of (2.37), we have \( - \nabla \Psi \left( x\right) - \frac{1}{c}\left( {\mathcal{P}\left( {x, g, c}\right) - x}\right) \in \partial \mathcal{X}\left( {\mathcal{P}\left( {x, g, c}\right) }\right) \), which implies\n\nthat\n\n\[ - \nabla \Psi \left( {\mathcal{P}\left( {x, g, c}\right) ... | Yes |
Lemma 5 Let \( \Psi \left( \cdot \right) \) be defined in (1.3). For any \( x, y \in {\mathbb{R}}^{n} \), we have\n\n\[ - \frac{{L}_{f}}{2}\parallel y - x{\parallel }^{2} \leq \Psi \left( y\right) - \Psi \left( x\right) - \langle \nabla \Psi \left( x\right), y - x\rangle \leq \frac{{L}_{\Psi }}{2}\parallel y - x{\paral... | Proof. We only need to show the first relation since the secone one follows from (2.1). Indeed,\n\n\[ \Psi \left( y\right) - \Psi \left( x\right) = {\int }_{0}^{1}\langle \nabla \Psi \left( {x + t\left( {y - x}\right) }\right), y - x\rangle {dt} \]\n\n\[ = {\int }_{0}^{1}\langle \nabla f\left( {x + t\left( {y - x}\righ... | Yes |
Theorem 2 Suppose that Assumption 2 holds and that \( \left\{ {\alpha }_{k}\right\} ,\left\{ {\beta }_{k}\right\} \), and \( \left\{ {\lambda }_{k}\right\} \) in Algorithm 2 are chosen such that (2.9) and (2.10) hold. Also assume that an optimal solution \( {x}^{ * } \) exists for problem (1.3). Then for any \( N \geq ... | Proof. By the assumption that \( \Psi \in {\mathcal{C}}_{{L}_{\Psi }}^{1,1}\left( {\mathbb{R}}^{n}\right) \), we have\n\n\[ \Psi \left( {x}_{k}^{ag}\right) \leq \Psi \left( {x}_{k}^{md}\right) + \left\langle {\nabla \Psi \left( {x}_{k}^{md}\right) ,{x}_{k}^{ag} - {x}_{k}^{md}}\right\rangle + \frac{{L}_{\Psi }}{2}{\begi... | Yes |
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