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Lange K. Optimization
Lange K.  Optimization









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: Optimization

: Lange K.

:

This introduction to optimization attempts to strike a balance between presentation of mathematical theory and development of numerical algorithms. Building on students' skills in calculus and linear algebra, the text provides a rigorous exposition without undue abstraction. Its stress on convexity serves as bridge between linear and nonlinear programming and makes it possible to give a modern exposition of linear programming based on the interior point method rather than the simplex method. The emphasis on statistical applications will be especially appealing to graduate students of statistics and biostatistics. The intended audience also includes graduate students in applied mathematics, computational biology, computer science, economics, and physics as well as upper division undergraduate majors in mathematics who want to see rigorous mathematics combined with real applications.


: en

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ed2k: ed2k stats

: 2004

: 252

: 23.04.2005

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ABO genetic locus      123
Active constraint      69
Adaptive barrier methods      208216
Adaptive barrier methods, linear programming      211
Adaptive barrier methods, logarithmic      208210
Admixtures      see EM algorithm mixture
Allele frequency estimation      123125 140141
Arfine function      70
Arithmetic-geometric mean inequality      23 8
Armijo rule      200
Attenuation coefficient      130
Backward algorithm, Baums      148
Ball      27
Baums algorithms      147149
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      78 109
Censored variable      151
Chain rule      54
Chain rule for second differential      84
Chebyshevs 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      177180
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      104109
Convex programming      207228
Convex programming, convergence of MM algorithm      212215
Convex programming, dual programs      see Dual programs
Convex programming, Dykstras algorithm      216219
Convex programming, for a geometric program      210
Convex programming, linear classification      223226
Convex regression      217
Convex set      94
Critical point      3
Davidon Fletcher Powell update      185
Davidons formula      182
Derivative, directional      50
Derivative, equality of mixed partials      5152
Derivative, partial      50
Derivative, second order partial      see Second differential
Derivative, univariate      43
Descent direction      159
Differentiable function, Caratheodorys definition      5253
Differentiable function, Frechets definition      52
Differential      52 53
Differential, of a matrix-valued function      6165
Differential, second      see Second differential
Directional derivative      50
Distance      31
Dual programs      219223
Dual programs, Duffins counterexample      222
Dual programs, Fenchel conjugate      220222
Dual programs, linear programming      221
Dual programs, quadratic programming      221222
Dykstras algorithm      216219
Dykstras algorithm, hybrid MM-      225226
EM algorithm      137154
EM algorithm, allele frequency estimation      140
EM algorithm, ascent property      138140
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      144147
EM algorithm, linear regression with right censoring      151
EM algorithm, M step      138
EM algorithm, mixture parameter      152
EM algorithm, transmission tomography      141143
entropy      149
Epigraph      97
Equality constraint      69
Euclidean matrix norm      20
Euclidean norm      1920
Exponential family      166167
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      67 222 230
Fermats principle      9
Fletcher Reeves update      179
Forward algorithm, Baums      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, Hubers      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      2526
Function, objective      69
Function, potential      132
Function, Riemanns 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      4647 57
Gauss Newton algorithm      162
Gauss Newton algorithm, scoring, and      164166
Generalized linear model      167168
Geometric programming      108 210
Gibbs prior      132
Gibbs lemma      90
Golden search      181
Gradient direction      10
Gradient vector      8
Hadamard product      231
Hadamards 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
Holders Inequality      90 112
Hubers function      171
Hyperplane      11 27
Implicit function theorem      6061
Inactive constraint      69
Induced matrix norm      21
Inequality constraint      69
Inequality, arithmetic-geometric mean      23 8
Inequality, Cauchy Schwarz      78 109
Inequality, Chebyshevs      110
Inequality, Hadamards      91
Inequality, Holders      90 112
Inequality, information      138
Inequality, Jensens      111
Inequality, Lipschitz      98
Inequality, Markovs      109
Inequality, Minkowskis triangle      116
Inequality, Schlomilchs      111112
Information Inequality      138
Interior      28
Intermediate Value Theorem      36
Inverse function theorem      5859
Isotone regression      217
Jensens Inequality      111
Karush Kuhn Tucker theory, Kuhn Tucker constraint qualification      7576
Karush Kuhn Tucker theory, multiplier rule      see Lagrange multiplier rule
Karush Kuhn Tucker theory, sufficient condition for a minimum      8588
Kernel      227
Kronecker product      62 227
Lagrange multiplier rule      7173
Lagrangian function      11 219221
Least squares estimation      910 217
Least squares estimation, nonlinear regression functions      161162
Least squares estimation, right-censored data      151
Leibnitzs formula      65
Limit inferior      24
Limit superior      24
Line search methods      180182
Linear classification      223226
Linear convergence      192
Linear logistic regression      127128
Linear programming      70 74 211
Linear programming, dual for      221
Link function      167
Lipschitz inequality      98
Log-convex function      103
Logarithmic barrier method      208210
Loglikelihood function      12 107 134
Logposterior function      132
LHopitals Rule      65
Majorizing function      120
Mangasarian Promovitz constraint qualification      70 77
Markov chain, hidden      147149
Markovs inequality      109
Marquardts method      172
Matrix exponential function      2526
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      160161
Maximum likelihood estimation, exponential distribution      162163
Maximum likelihood estimation, hidden Markov chains      see Markov chain
Maximum likelihood estimation, multinomial distribution      1213 148149 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
Minkowskis triangle inequality      116
Missing data, EM algorithm      138 147
Mixtures      see EM algorithm mixture
MM algorithm      119136
MM algorithm, allele frequency estimation      see Allele frequency estimation
MM algorithm, Bradley Terry model      127
MM algorithm, convergence for convex program      211215
MM algorithm, descent property      120
MM algorithm, global convergence of      196199
MM algorithm, hybrid Dykstra      225226
MM algorithm, linear logistic regression      127128
MM algorithm, linear regression      125126
MM algorithm, majorization      121122
MM algorithm, transmission tomography      see Transmission tomography
MM gradient algorithm      160161
MM gradient algorithm, convergence of      194196
MM gradient algorithm, Dirichlet distribution, estimation with      160161
Multivariate normal distribution, maximum entropy property      149
Multivariate normal distribution, maximum likelihood for      107
Neighborhood      28
Newtons method      155168
Newtons method, convergence of      193194
Newtons method, least squares estimation      161162
Newtons method, MM gradient algorithm      see MM gradient algorithm
Newtons method, root finding      156158
Newtons method, scoring      see Scoring
Newtons method, transmission tomography      160
Nilpotent matrix      39
Norm, equivalence of      33
Norm, Euclidean      1920
Norm, Euclidean matrix      20
Norm, induced matrix      21
Normal distribution      233236
Normal distribution, mixtures      152
Normal distribution, multivariate      235236
Normal distribution, univariate      233234
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      1315
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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