Berkovitz L.D. — Convexity and Optimization in Rn :: Электронная библиотека попечительского совета мехмата МГУ

Главная Ex Libris Книги Журналы Статьи Серии Каталог Wanted Загрузка ХудЛит Справка Поиск по индексам Поиск Форум

Авторизация

Поиск по указателям

Berkovitz L.D. — Convexity and Optimization in Rn

Обсудите книгу на научном форуме

Нашли опечатку?
Выделите ее мышкой и нажмите Ctrl+Enter

Название: Convexity and Optimization in Rn

Автор: Berkovitz L.D.

Аннотация:

A textbook for a one-semester beginning graduate course for students of engineering, economics, operations research, and mathematics. Students are expected to have a good grounding in basic real analysis and linear algebra.

Язык:

Рубрика: Математика/Оптимизация и управление/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2002

Количество страниц: 268

Добавлена в каталог: 22.04.2005

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID

Предметный указатель

Affine hull      74
Affine transformations      78—114
Algebra of sets      4—5
Alternative theorems, convex functions      113—117
Alternative theorems, convex sets, linear inequalities      61—68
Alternative theorems, convex sets, linear inequalities, Farkas’s lemma      63—65 67—68
Alternative theorems, convex sets, linear inequalities, Gale’s theorem      68
Alternative theorems, convex sets, linear inequalities, Gordan’s theorem      65—67
Alternative theorems, convex sets, linear inequalities, Motzkin’s theorem      66—67
Anticycling routine, simplex method, phase I procedure      254
Anticycling routine, termination and cycling      250 254
Backward transformation, revised simplex method      259
Barycentric coordinates      74
Basic solution, simplex method      231
Basic variables, phase II procedure      235—245
Basic variables, simplex method      230
Bland’s rule, simplex method, termination and cycling      248—250
Bolzano—Weierstrass property      15
Boundary points, convex function optimization      137
Boundary points, supporting hyperplanes      56—57
Canonical linear programming problem      234—260
Canonical linear programming problem, optimization      144
Canonical linear programming problem, simplex method      224—225
Canonical linear programming problem, simplex method, extreme points, feasible sets      225—229
Canonical linear programming problem, simplex method, phase I      251—255
Canonical linear programming problem, simplex method, revised simplex method      258—260
Caratheodory theorem      41—43
Cartesian products      16—18
Cauchy—Schwarz inequality      3—4
Closed half spaces      35—36
Closed line segments      31
Closed sets      6—8
Closest point characterization      50—51
Coercive functions      20
Colonel Blotto game      122—127
Compactness      14—15
Compactness, convex sets, separation theorem      54—55
Compactness, supporting hyperplanes, extreme points      58—61
Complementary slackness condition, linear programming      141—145
Completeness property, real numbers      12—13
Concave functions, defined      88
Concave functions, game theory      117
Consistency, convex programming, perturbation theory      192—200
Consistency, convex programming, problem II (CPII)      180—181
Consistency, strong in (CPII)      183—187
Constraint qualifications, convex programming, necessary conditions      183—188
Constraint qualifications, convex programming, quadratic programming      212
Constraint qualifications, differentiable nonlinear programming      154—163
Constraint qualifications, differentiable nonlinear programming, Kuhn—Tucker constraint qualification      162—163
Constraint qualifications, differentiable nonlinear programming, tangential constraints      165—178
Continuity      8—11
Convex cones      44 62
Convex functions, alternative theorems      113—117
Convex functions, definitions and properties      87—101
Convex functions, differentiable functions      106—113
Convex functions, differentiable nonlinear programming      159—163
Convex functions, game theory applications      117—121
Convex functions, Jensen’s inequality      90—91
Convex functions, optimization problems      137—139
Convex functions, partial derivatives      99 106—109
Convex functions, subgradients      102—106
Convex hulls      41—43
Convex hulls, extreme points      85—86
Convex programming problem I (CPI), problem statement      179—180
Convex programming problem II (CPII), problem dual to (DCPII)      200—207
Convex programming problem II (CPII), problem statement      180—181
Convex programming, duality, geometric interpretation      207—210
Convex programming, duality, Lagrangian duality      200—207
Convex programming, duality, linear programming      215—221
Convex programming, duality, quadratic programming      212
Convex sets, affine geometry      69—80
Convex sets, affine geometry, barycentric coordinates      74
Convex sets, affine geometry, dependence      71—72
Convex sets, affine geometry, hull dimensions      74
Convex sets, affine geometry, independence      72—73
Convex sets, affine geometry, linear manifold      69—71
Convex sets, affine geometry, linear transformations      78
Convex sets, affine geometry, nonvoid interiors      76—77
Convex sets, affine geometry, parallel subspace      70
Convex sets, affine geometry, simplex dimension      76
Convex sets, extreme points      56—61
Convex sets, hyperplane support      56—61
Convex sets, linear inequalities, systems of      61—68
Convex sets, linear inequalities, systems of, Farkas’s lemma      63—65 67
Convex sets, linear inequalities, systems of, Gale’s lemma      68
Convex sets, linear inequalities, systems of, Gordan’s lemma      65—67
Convex sets, linear inequalities, systems of, Motzkin’s lemma      66—67
Convex sets, nonempty relative interiors      81—82
Convex sets, properties of      35—45
Convex sets, properties of, Caratheodory theorem      41—42
Convex sets, properties of, convex cone      44
Convex sets, properties of, convex hull      41 43
Convex sets, properties of, half spaces      35—36
Convex sets, properties of, points, convex combinations      40—41
Convex sets, properties of, scalar combinations      37
Convex sets, relative interior points      80—81
Convex sets, separation theorems      45—56
Convex sets, separation theorems, disjoint convex sets      53—54 81
Convex sets, separation theorems, proper separation      46 53
Convex sets, separation theorems, strict separation      46
Convex sets, separation theorems, strong separation      47—49 54
Current basis, simplex method      234—245
Current value, simplex method      234—245
Cycling, simplex method      245—250
Degenerate basic solution, simplex method      231 245—247
Derivatives, convex functions      94—96 99
Derivatives, Euclidean n-space $(\mathbb{R}^n)$ , differentiation      22—29
Derivatives, Euclidean n-space $(\mathbb{R}^n)$ , linear transformation      22—25
Derivatives, second derivative test, differentiable, unconstrained problems      130—136
Differentiability, convex functions      99 106—113
Differentiability, Euclidean n-space $(\mathbb{R}^n)$ , linear transformation      24
Differentiable nonlinear programming, first-order conditions      145—163
Differentiable nonlinear programming, second-order conditions      163—178
Differentiable unconstrained problems      129—136
Differentiable unconstrained problems, linear regression      134—135
Differentiable unconstrained problems, local minimizer      131
Differentiable unconstrained problems, orthogonal mean-square approximations      135—136
Differentiable unconstrained problems, positive semidefinite functions      131—132
Differentiable unconstrained problems, second derivative test      131
Differentiable unconstrained problems, strict minimizers      130—136
Differentiation, Euclidean n-space $(\mathbb{R}^n)$       25—29
Directional derivatives, convex functions      95—96
Disjoint convex sets, hyperplane separation      53
Duality theorems, convex programming, dual quadratic programming (DQP)      214—215
Duality theorems, convex programming, gaps, convex programming      202—207
Duality theorems, convex programming, gaps, quadratic programming      212
Duality theorems, convex programming, geometric interpretation      207—210
Duality theorems, convex programming, Lagrangian duality      200—207
Duality theorems, convex programming, linear programming      215—221
Elementary operations, simplex method      232—234
Empty set      4—5
Entering variable, revised simplex method      256—260
Entering variable, simplex method      239—245
Epigraphs, convex functions      88—89
Equality constraints, convex programming problem (CPI)      180
Equality constraints, differentiable nonlinear programming, necessary conditions      146—159 168—171
Equality constraints, differentiable nonlinear programming, sufficient conditions      159—160 172—177
Equilibrium prices      196
Equivalent norms, Euclidean n-space $(\mathbb{R}^n)$       16—18
Eta file, revised simplex method      258—260
Eta matrix, revised simplex method      257—260
Euclidean n-space $(\mathbb{R}^n)$ , defined      1
Euclidean n-space $(\mathbb{R}^n)$ , metric topology      5—8
Euclidean n-space $(\mathbb{R}^n)$ , vectors      1—4
Euclidean norm      2—3
Extreme points, convex sets      56—61 85
Extreme points, simplex method, feasible sets      225—230
Farkas’s lemma, differentiable nonlinear programming      162—163
Farkas’s lemma, linear inequalities      63—64 67—68
Farkas’s lemma, linear programming      142

Farkas’s lemma, linear programming duality      220
Feasible sets, optimization problems      129
Feasible sets, simplex method, auxiliary problem      251—255
Feasible sets, simplex method, extreme points      225—229
Finite-intersection property      15
First-order conditions, differentiable nonlinear programming      145—163
Forward transformation, revised simplex method      258—260
Frechet differentiable      24
Free variables, simplex method      230
Fritz — John theorem, differentiable nonlinear programming      146—163
Fritz — John theorem, differentiable nonlinear programming, second-order conditions      172—174
Fundamental existence theorem      18—20
Gale’s theorem, linear inequalities      68
Game theory      117—127
Game theory, optimal mixed strategies      126—127
Game theory, optimal pure strategies      125—126
Gauge Functions      101
Gauss elimination      232—234
Gauss — Jordan elimination      233—234
Geometric interpretation, convex programming duality      207—210
Gordan’s theorem, generalization to convex functions      215
Gordan’s theorem, linear inequalities      65—67
Gordan’s theorem, quadratic programming      211—215
Half spaces      35—36
Half spaces, intersection theorems      55—56
Half-open line segments      31
Hessian matrix, differentiable convex functions      110—113
Hessian matrix, differentiable unconstrained problems      131—132
Hyperplanes, convex functions, subgradients      102—106
Hyperplanes, convex programming duality, geometric interpretation      208—210
Hyperplanes, convex set properties      35—36
Hyperplanes, convex set properties, Farkas’s lemma      65
Hyperplanes, definition      33—34
Hyperplanes, supporting hyperplanes      56—59 61 85
Implicit function theorem      163—164
Infimum      12—14
Inner product      2
Interior points, continuity of convex functions      96—98
Interior points, convex sets, n-dimensional      76
Interior points, convex sets, relative interior points      80—86
Jacobian matrix      25 165
Jensen’s inequality, convex functions      90—91
Karush — Kuhn — Tucker theorem, differentiable nonlinear programming      155—163
Karush — Kuhn — Tucker theorem, differentiable nonlinear programming, constraint qualifications      154—163
Karush — Kuhn — Tucker theorem, differentiable nonlinear programming, second-order conditions      168—172
Karush — Kuhn — Tucker theorem, differentiable nonlinear programming, sufficient conditions      159—160
Krein — Milman theorem      60
Kuhn — Tucker vectors as multipliers      184—185
Kuhn — Tucker vectors as subgradients of value function      194—196
Kuhn — Tucker vectors, definition      184
Kuhn — Tucker vectors, examples      197—200
Lagrange multipliers, convex programming as Kuhn—Tucker vectors      184—815
Lagrange multipliers, convex programming as subgradients of value function      194—196
Lagrange multipliers, convex programming, necessary conditions      181—184
Lagrange multipliers, convex programming, saddle point conditions      187—188
Lagrange multipliers, convex programming, sufficient conditions      186—187
Lagrange multipliers, differentiable nonlinear programming, first-order conditions      146—163
Lagrangian duality, convex programming      200—207
Lagrangian duality, convex programming, quadratic programming      211—215
Leaving variables, revised simplex method      256—260
Leaving variables, simplex method      239—245
Left-hand limit, real numbers      14
Limit points, continuity      9—11
Limit points, metric toplogy      6
Limits      9—11
Line segments      31
Linear functionals, Euclidean n-space $(\mathbb{R}^n)$       21—24
Linear functionals, hyperplane definition      33—34
Linear inequalities, convex sets, Farkas’s lemma      63—65 67
Linear inequalities, convex sets, Gale’s theorem      68
Linear inequalities, convex sets, Gordan’s theorem      65—67
Linear inequalities, convex sets, Motzkin’s theorem      66—67
Linear inequalities, convex sets, theorems of the alternative      61—68
Linear manifolds      69—71
Linear manifolds, dimension      70—71
Linear programming      see also “Nonlinear programming”
Linear programming, canonical linear programming problem      144
Linear programming, complementary slackness condition      141—145
Linear programming, duality in      215—221
Linear programming, formulation      139—140
Linear programming, necessary and sufficient conditions      141—145
Linear programming, simplex method, auxiliary problem      251—255
Linear programming, simplex method, examples      222—225
Linear programming, simplex method, extreme points, feasible set      225—229
Linear programming, simplex method, phase I procedure      251—255
Linear programming, simplex method, phase II procedure      234—245
Linear programming, simplex method, preliminaries      230—234
Linear programming, simplex method, revised method      255—260
Linear programming, simplex method, termination and cycling      245—250
Linear programming, standard vs. canonical linear forms      144—145
Linear regression      134—135
Linear transformation as derivative      22—25
Linear transformation, Euclidean n-space $(\mathbb{R}^n)$       21
Lipschitz continuous functions      98
Local minimizers, convex functions      137
Local minimizers, differentiable nonlinear programming, differentiable unconstrained problems      129—136
Local minimizers, differentiable nonlinear programming, second-order sufficiency theorems      172—178
Logarithmic convexity      113
Lower bounds, real numbers      12—13
Matrix games      122—127
Maximization of convex functions      137—139
Mean-square approximation by orthogonal functions      135
Metric toplogy, Euclidean n-space $(\mathbb{R}^n)$       5—8
Minimizers, convex function optimization      137—139
Minimizers, convex programming, necessary and sufficient conditions      183—188
Minimizers, convex programming, quadratic programming      213—215
Minimizers, differentiable nonlinear programming      145—163
Minimizers, fundamental existence theorem      19—20
Minimizers, linear programming      140—145
Minkowski distance function      101
Mixed strategies      125—127
Morgan’s law      4—5
Motzkin’s theorem, linear inequalities      66—67
Multipliers      see “Kuhn — Tucker vectors; Lagrange multipliers”
Necessary conditions, convex programming      181—188
Necessary conditions, differentiable nonlinear programming, first-order conditions      145—163
Necessary conditions, differentiable nonlinear programming, second-order conditions      163—178
Necessary conditions, linear programming duality      215—221
Necessary conditions, quadratic programming      211—215
Negative half space, convex set properties      35—36
Nonbasic variables, simplex method      230
Nonlinear programming, differentiable problems, first-order conditions      145—163
Nonlinear programming, differentiable problems, second-order conditions      163—178
Nontrivial supporting hyperplane      56—57 85
Null set      4—5
Open cover      14—15
Open line segment      31
Open sets, metric toplogy      6—8
Optimal solution, simplex method      231 238—245
Optimal strategies      see “Game theory”
Orthogonality      4
Parallel hyperplanes      34
Parallel subspace      70
Payoff matrix      123—127
Perturbation theory      188—200
Pivot position, simplex method      244—245
Pivot position, simplex method, auxiliary problems      252—255
Positive definite matrix      111—112 135—136
Positive half space      35—36
Primal convex programming problem (CPII)      180—188
Primal convex programming problem (CPII), Lagrangian duality      200—207
Proper separation, convex sets      46
Quadratic programming      210—215
Quadratic programming, dual quadratic program (DQP)      214—215
Real numbers, boundedness properties      11—14
Relative boundaries, convex sets, support and separation theorems      81—86
Relative interior, convex sets, support and separation theorems      80—86
Revised simplex method      255—260
Right-hand limit      13—14
Saddle points, convex programming, Lagrangian duality      206—207

1 2

О проекте