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| Ïîèñê ïî óêàçàòåëÿì |
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| Lange K. — Optimization |
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| Ïðåäìåòíûé óêàçàòåëü |
ABO genetic locus 123
Active constraint 69
Adaptive barrier methods 208—216
Adaptive barrier methods, linear programming 211
Adaptive barrier methods, logarithmic 208—210
Admixtures see “EM algorithm mixture
Allele frequency estimation 123—125 140—141
Arfine function 70
Arithmetic-geometric mean inequality 2—3 8
Armijo rule 200
Attenuation coefficient 130
Backward algorithm, Baum’s 148
Ball 27
Baum’s algorithms 147—149
Bernstein polynomial 116
Binomial distribution 110
Bivariate normal distribution, missing data 151
Blood type genes 123 134 140
Boundary point 28
Bounded set 26
Bradley — Terry model 127
Bregman distance 209
Broyden — Fletcher — Goldfarb — Shanno update 183
Cauchy sequence 23
Cauchy — Schwarz inequality 7—8 109
Censored variable 151
Chain rule 54
Chain rule for second differential 84
Chebyshev’s inequality 110
Chernoff bound 115
Cholesky decomposition 114 236
Closed set 26
Closure 29
Coercive function 196 205
Coloring 129
Compact set 29
Completeness 23
Completeness and existence of suprema 24
Concave function 9 95
Conjugate gradient algorithm 177—180
Conjugate vectors 177
Connected set 35
Connected set, arcwise 35
Continuous function 30
Continuously differentiable function 56
Convergent sequence 22
Convex cone 27
Convex function 9 95
Convex function, minimization of 104—109
Convex programming 207—228
Convex programming, convergence of MM algorithm 212—215
Convex programming, dual programs see “Dual programs”
Convex programming, Dykstra’s algorithm 216—219
Convex programming, for a geometric program 210
Convex programming, linear classification 223—226
Convex regression 217
Convex set 94
Critical point 3
Davidon — Fletcher — Powell update 185
Davidon’s formula 182
Derivative, directional 50
Derivative, equality of mixed partials 51—52
Derivative, partial 50
Derivative, second order partial see “Second differential”
Derivative, univariate 43
Descent direction 159
Differentiable function, Caratheodory’s definition 52—53
Differentiable function, Frechet’s definition 52
Differential 52 53
Differential, of a matrix-valued function 61—65
Differential, second see “Second differential”
Directional derivative 50
Distance 31
Dual programs 219—223
Dual programs, Duffin’s counterexample 222
Dual programs, Fenchel conjugate 220—222
Dual programs, linear programming 221
Dual programs, quadratic programming 221—222
Dykstra’s algorithm 216—219
Dykstra’s algorithm, hybrid MM- 225—226
EM algorithm 137—154
EM algorithm, allele frequency estimation 140
EM algorithm, ascent property 138—140
EM algorithm, bivariate normal parameters 151
EM algorithm, E step 138
EM algorithm, estimating multinomial parameters 153
EM algorithm, exponential family 150
EM algorithm, factor analysis 144—147
EM algorithm, linear regression with right censoring 151
EM algorithm, M step 138
EM algorithm, mixture parameter 152
EM algorithm, transmission tomography 141—143
entropy 149
Epigraph 97
Equality constraint 69
Euclidean matrix norm 20
Euclidean norm 19—20
Exponential family 166—167
Exponential family, EM algorithm 150
Exponential family, generalized linear models 167
Extremal value 3
Extremal value, distinguishing from a saddle point 82
Factor analysis 143
Factor loading matrix 145
Feasible point 69
Feature space 226
Fenchel biconjugate 222
Fenchel conjugate 6—7 222 230
Fermat’s principle 9
Fletcher — Reeves update 179
Forward algorithm, Baum’s 148
Free variable 70
Function, affine 70
Function, coercive 197 205
Function, concave 9 95
Function, continuous 30
Function, continuously differentiable 56
Function, convex 9 95
Function, differentiable see “Differentiable function”
Function, Gamma 104
Function, Huber’s 171
Function, Lagrangian 11
Function, link 167
Function, log-convex 103
Function, loglikelihood 12 107 134
Function, logposterior 132
Function, majorizing 120
Function, matrix exponential 25—26
Function, objective 69
Function, potential 132
Function, Riemann’s zeta 113
Function, slope see “Slope function”
Function, square-integrable 228
Function, twice continuously differentiable 79
Function, twice differentiable 79
Function, uniformly continuous 34
Gamma function 104
Gauge function 57
Gauge integral 46—47 57
Gauss — Newton algorithm 162
Gauss — Newton algorithm, scoring, and 164—166
Generalized linear model 167—168
Geometric programming 108 210
Gibbs prior 132
Gibbs’ lemma 90
Golden search 181
Gradient direction 10
Gradient vector 8
Hadamard product 231
Hadamard’s inequality 91
| Halfspace 27
Hardy — Weinberg law 123
Hermite interpolation 180
Hessian matrix 8
Hestenes — Stiefel update 179
Hidden trials, EM algorithm for 153
Hidden trials, multinomial 153
Hidden trials, Poisson or exponential 153
Holder’s Inequality 90 112
Huber’s function 171
Hyperplane 11 27
Implicit function theorem 60—61
Inactive constraint 69
Induced matrix norm 21
Inequality constraint 69
Inequality, arithmetic-geometric mean 2—3 8
Inequality, Cauchy — Schwarz 7—8 109
Inequality, Chebyshev’s 110
Inequality, Hadamard’s 91
Inequality, Holder’s 90 112
Inequality, information 138
Inequality, Jensen’s 111
Inequality, Lipschitz 98
Inequality, Markov’s 109
Inequality, Minkowski’s triangle 116
Inequality, Schlomilch’s 111—112
Information Inequality 138
Interior 28
Intermediate Value Theorem 36
Inverse function theorem 58—59
Isotone regression 217
Jensen’s Inequality 111
Karush — Kuhn — Tucker theory, Kuhn — Tucker constraint qualification 75—76
Karush — Kuhn — Tucker theory, multiplier rule see “Lagrange multiplier rule”
Karush — Kuhn — Tucker theory, sufficient condition for a minimum 85—88
Kernel 227
Kronecker product 62 227
Lagrange multiplier rule 71—73
Lagrangian function 11 219—221
Least squares estimation 9—10 217
Least squares estimation, nonlinear regression functions 161—162
Least squares estimation, right-censored data 151
Leibnitz’s formula 65
Limit inferior 24
Limit superior 24
Line search methods 180—182
Linear classification 223—226
Linear convergence 192
Linear logistic regression 127—128
Linear programming 70 74 211
Linear programming, dual for 221
Link function 167
Lipschitz inequality 98
Log-convex function 103
Logarithmic barrier method 208—210
Loglikelihood function 12 107 134
Logposterior function 132
L’Hopital’s Rule 65
Majorizing function 120
Mangasarian — Promovitz constraint qualification 70 77
Markov chain, hidden 147—149
Markov’s inequality 109
Marquardt’s method 172
Matrix exponential function 25—26
Matrix exponential function and differential equations 48
Matrix logarithm 49
Matrix, eigenvalues of a symmetric 13
Matrix, factor loading 145
Matrix, Hessian 9
Matrix, nilpotent 39
Matrix, observed information 13
Matrix, skew-symmetric 39
Matrix, square root 172
Maximum likelihood estimation, allele frequency 123
Maximum likelihood estimation, Dirichlet distribution 160—161
Maximum likelihood estimation, exponential distribution 162—163
Maximum likelihood estimation, hidden Markov chains see “Markov chain”
Maximum likelihood estimation, multinomial distribution 12—13 148—149 163 215
Maximum likelihood estimation, multivariate normal distribution 107
Maximum likelihood estimation, Poisson distribution 162
Maximum likelihood estimation, power series family, for a 171
Maxwell — Boltzmann distribution 150
Mean value theorem, failure of 57
Mean value theorem, multivariate 56
Mean value theorem, univariate 45
Method of false position 180
Minkowski’s triangle inequality 116
Missing data, EM algorithm 138 147
Mixtures see “EM algorithm mixture
MM algorithm 119—136
MM algorithm, allele frequency estimation see “Allele frequency estimation”
MM algorithm, Bradley — Terry model 127
MM algorithm, convergence for convex program 211—215
MM algorithm, descent property 120
MM algorithm, global convergence of 196—199
MM algorithm, hybrid Dykstra 225—226
MM algorithm, linear logistic regression 127—128
MM algorithm, linear regression 125—126
MM algorithm, majorization 121—122
MM algorithm, transmission tomography see “Transmission tomography”
MM gradient algorithm 160—161
MM gradient algorithm, convergence of 194—196
MM gradient algorithm, Dirichlet distribution, estimation with 160—161
Multivariate normal distribution, maximum entropy property 149
Multivariate normal distribution, maximum likelihood for 107
Neighborhood 28
Newton’s method 155—168
Newton’s method, convergence of 193—194
Newton’s method, least squares estimation 161—162
Newton’s method, MM gradient algorithm see “MM gradient algorithm”
Newton’s method, root finding 156—158
Newton’s method, scoring see “Scoring”
Newton’s method, transmission tomography 160
Nilpotent matrix 39
Norm, equivalence of 33
Norm, Euclidean 19—20
Norm, Euclidean matrix 20
Norm, induced matrix 21
Normal distribution 233—236
Normal distribution, mixtures 152
Normal distribution, multivariate 235—236
Normal distribution, univariate 233—234
Normal equation 9
Objective function 69
Observed information 155
Observed information matrix 13
Open set 28
Partial derivative 50
Pattern space 226
Pixel 131
Poisson admixture model 150
Poisson process 128
Polak — Ribiere update 179
Population genetics see “Allele frequency estimation”
Population genetics, inference of maternal/paternal alleles in offspring 13—15
Posterior mode 132
Posynomial 108
Potential function 132
Power series family 171
Primal program, convex 220
Projection operators 216
Proposition, Bolzano — Weierstrass 29
Proposition, Ekeland 76
Proposition, Gordon 77 104
Proposition, Heine 34
Proposition, Liapunov 198
Proposition, Ostrowski 192
Proposition, Weierstrass 33
q quantile 134
QR decomposition 235
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