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Rockafellar R.T. — Convex analysis

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Название: Convex analysis

Автор: Rockafellar R.T.

Аннотация:

R. Tyrrell Rockafellar's classic study presents readers with a coherent branch of nonlinear mathematical analysis that is especially suited to the study of optimization problems. Rockafellar's theory differs from classical analysis in that differentiability assumptions are replaced by convexity assumptions. The topics treated in this volume include: systems of inequalities, the minimum or maximum of a convex function over a convex set, Lagrange multipliers, minimax theorems and duality, as well as basic results about the structure of convex sets and the continuity and differentiability of convex functions and saddle- functions.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1970

Количество страниц: 472

Добавлена в каталог: 12.09.2007

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

Предметный указатель

Abnormal program      318
Addition of convex cones      22
Addition of convex functions      33 77 145 176 179—180 223 263
Addition of convex processes      415 421
Addition of convex sets      16—17 49 74—75 146 175 183
Addition of epigraphs      34
Addition of saddle-functions      402
Adjoint of a bifunction      309—326 330 353—358 401—412
Adjoint of a convex process      417ff
Adjoint of a linear transformation      3 9 310
aff      see “Affine hull”
Affine functions      23 25 27 102—103
Affine functions, partial      70 107 431
Affine hull      6 45 154
Affine hull of convex cone      15
Affine hull, characterization      113
Affine independence      6—7 154
Affine sets      3—9
Affine sets, closed halves      165—166
Affine sets, representation      4—8
Affine transformations      7—8 44—45
Alternative system of inequalities      201
Asymptotic cone      61
Ball      43
Barrier cone      15 113 123
Barycenter      12
Barycentric coordinates      7
Bi-affine functions      302
Bifunction      291ff (see also “Convex bi-functions”)
Bilinear functions      351—352 411
Boundedness conditions      54 64 68—69 88 123
Caratheodory’s theorem      153—157 427
Chemical equilibrium problem      430
Circulations      204 208 272 337—338
Cl      see “Closure”
Closed bifunction      293
Closed concave function      308
Closed convex function      52 (see also “Closure”)
Closed saddle-function      363 (see also “Closure”)
Closure of a bifunction      293 305—306 310 403 407
Closure of a concave function      307—308
Closure of a convex function      51—59 72—81 102—104 218—219 425
Closure of a convex process      415
Closure of a convex set      43—50 72—81 112 421—422
Closure of a saddle-function      359—369 390
Closure of an epigraph      52
Co-finite      116 259—260 411—412
Complete non-decreasing curves      232 338 428
Composition of a convex function and a linear transformation      38 78
Composition of convex functions      32
Composition of convex processes      416 422—423
Concave bifunctions      308ff
Concave closure of a saddle-function      350—353
Concave functions      25 307—308 426
Concave functions, monotone conjugates      110
Concave programs      308ff
Concave-convex functions      349ff
Cone      13
Conjugacy correspondence      104 123—124
Conjugacy correspondence for saddle-functions      389ff
Conjugate concave functions      111 308
Conjugate convex functions      104—111 113—118 121—124 133—137 140—150 173 179—180 263—264 405 421 425—426
Conjugate convex functions, definition      104
Conjugate convex functions, subgradients      218
Conjugate saddle-functions      390—391 395 432
Consequence      199
Consistency      185 295 309 315
Constancy space      69
Continuity of convex functions      82—89 426
Continuity of derivatives      227—238
Continuity of gradient mappings      246 376—377
Continuity of saddle-functions      370—371
Continuity, joint      89
Continuity, uniform      86—87
Continuous extensions      85
conv      see “Convex hull”
Convergence of convex functions      90—91 426
Convergence of gradients      248—249
Convergence of saddle-functions      372 375—378
Convergence of subgradients      233—236
Convex bifunctions      293—306 309—311 350—358 384—389 401—412 417—418
Convex closure of a saddle-function      350—353
Convex combinations      11—12
Convex combinations of points and directions      154
Convex cones      13—15 22 50
Convex cones, generation      78 122 126 156 178
Convex cones, polar      121—125
Convex cones, polyhedral      170 178
Convex cones, separation      100—101
Convex function      23
Convex function, co-finite      259
Convex function, differential conditions for convexity      26—27
Convex function, finitely generated      172—173
Convex function, interpolation properties      25
Convex function, Legendre type      258
Convex function, partial quadratic      109 431
Convex function, polyhedral      172—177
Convex function, polynomial      268
Convex function, quadratic      27 108
Convex function, separable      270—271 285—290 337—338
Convex function, symmetric      109—110
Convex hull      12 177 427
Convex hull of a set of points and directions      153—155
Convex hull, of a bounded set      158
Convex hull, of a collection of convex functions      37 81 149 156
Convex hull, of a collection of convex sets      18 80 156—157
Convex hull, of a non-convex function      36 103 157—158
Convex hull, of two convex cones      22
Convex hull, relative interior      50
Convex processes      413—423 432
Convex processes, polyhedral      415
Convex programs, generalized      291—326 355—356 385—387
Convex programs, normal      316—319
Convex programs, ordinary      273—291 293—294 296 298 300 320—326 429
Convex programs, polyhedral      301—303
Convex set      10
Convex set as a cross-section of a cone      15
Convex set, finitely generated      170—171
Convex set, polyhedral      11
Convex set, symmetric      16
Convex-concave functions      349ff
Cyclically monotone mappings      238—240
Decomposition principle      285—290 312—313 429
Derivatives, directional      213—221 226 244—245 264 299—301 372—377
Derivatives, partial      241 244 376
Derivatives, right and left      214 216 227—232
Differentiability      241—246 428
Differentiability of saddle-functions      375—376
DIM      see “Dimension”
Dimension of a convex function      23 71
Dimension of a convex set      12—13 45—46 126
Dimension of an affine set      4
Direct sums      19 49
Directed graphs      204 208 272 337—338
Direction      60
Direction of affinity      70
Direction of constancy      69
Direction of linearity      65
Direction of recession      61 69 264—270
Directional derivatives      213—221 226 244—245 264 299—301 372—377
Distance function      28 34
Distributive inequalities      416
DOM      see “Effective domain”
Dual programs      310—338 355—356 429ff
Dual systems of inequalities      201
Effective domain of a bifunction      293
Effective domain of a concave function      307
Effective domain of a convex function      23 25 122
Effective domain of a convex process      413
Effective domain of a saddle-function      362 366 391—392

Effective domain, relative interior      54
Eigensets      423
Elementary vectors      203—208 272 428
Epi      see “Epigraph”
Epigraph      23 307
Epigraph, closure      52
Epigraph, relative interior      54
Epigraph, support function of      119
Equi-Lipschitzian      87—88
Equicontinuity      88
Equilibrium prices      276—277 280 299—300
Equivalent saddle-functions      363—369 383 394
Essentially smooth function      251—258
Essentially strictly convex function      253—260
Euclidean metric      43
Exposed directions      163 168
Exposed faces      162—163
Exposed points      162—163 167—168 243 427
Exposed rays      163 169
Extensions of saddle-functions      349 358 363 366 369
Extreme directions      162—166 172
Extreme points      162—167 172 344—345 427
Extreme points at infinity      162
Extreme rays      162 167
Faces      162—165 171 427
Faces, exposed      162—163
Farkas’ lemma      200—201
Feasible solutions      274 295 308 315
Fenchel’s Duality Theorem      327ft 408 430
Fenchel’s inequality      105 218
Finitely generated convex function      172—173
Finitely generated convex set      170—171
Flat      3
flows      204 208 272 337—338
Fully closed saddle-function      356 365
Gale — Kuhn — Tucker Theorem      317 337 421 430—431
Gauge      28 35 79 124—125 128—131 427
Gauge-like functions      133
Generalized convex programs      291—326 355—356 385—387
Generalized polytope      171
Generalized simplex      154—155
Generators      170
Geometric mean      27 29
Geometric programming      324—326 430
Gradients      213 241—250 300 375—378 396
Graph domain      293
Graph function      292
Half-spaces      10 99 112 160
Half-spaces in $\mathbb{R}^{n+1}$       102
Half-spaces, homogeneous      101
Half-spaces, tangent      169
Half-spaces, upper      102
Half-spaces, vertical      102
Helly’s theorem      191—197 206 267 427—428
Hessian matrix      27
Hyperplanes      5
Hyperplanes, in $\mathbb{R}^{n+1}$       102
Hyperplanes, representation      5
Hyperplanes, supporting      100
Hyperplanes, tangent      169
Hyperplanes, vertical      102
Image of a convex function      38 75 142 175 255 405 409—412 416 421
Image of a convex set      19 48 73 143 174 414—415 421—422
Image-closed bifunction      352—353
Improper convex function      24 34 52—53
Improper saddle-function      366
Incidence matrix      204 208
Inconsistency      185 315
Indicator bifunction      292—293 310 355 417
Indicator function      28 33 425
Indicator function, conjugate      113—114
Inequalities      129—130 425 428
Inequalities, between functions      38 104
Inequalities, between vectors      13
Inequalities, consistent      185
Inequalities, convex      29 55 58 185—197
Inequalities, homogeneous      14
Inequalities, linear      10—11 13—14 62 65 113 122 170 185 198—209
Infimal convolution      34 38 76—77 145 175 179—181 254 425 427
Infimal convolution of bi-functions      401—404
Infimal convolution, partial      39
Inner product equation      355 409—412 419—421
Inner product of a vector and a function      350
Inner product of a vector and a set      417
Inner product of two functions      408—412
Inner product of two sets      422—423
Inner product of two vectors      3
int      see “Interior”
Interior      43—44 47 112
Intersections of convex cones      13 22
Intersections of convex sets      10 64 145
Intersections, relative interiors      47
interval      202
Inverse addition      21
Inverse addition of epigraphs      40
Inverse bifunction      384—385 388—389 401 405—406
Inverse image of a convex function      38 78 141 225
Inverse image of a convex set      19 49 64 143 174
Inverse process      414 418
Kernel of a saddle-function      367—369
Kuhn — Tucker coefficients      274—277 280 429
Kuhn — Tucker conditions      282—284 304 333—338 386—387 429
Kuhn — Tucker theorem      283 387
Kuhn — Tucker vectors      274—290 295—306 309 387
Lagrange multipliers      273—274 280 283 429
Lagrangian function      280—290 296—298 302—305 309 314 385—387
Lattice of convex functions      38
Lattice of convex processes      416
Lattice of convex sets      18
Legendre conjugate      256—260
Legendre transformation      251 256 427 429
Level sets      28—29 55 58—59 70 123 127 222 263—265
Level sets of support functions      118
Line      3—4
Line segment      10 12
Lineality      65 126
Lineality of a convex function      70 117
Lineality space      65 70 117 126
Linear combinations, convex      11
Linear combinations, of convex functions      33;
Linear combinations, of convex sets      17—18
Linear combinations, positive and non-negative      14
Linear programs      301—302 311—312 317 332 334—335 337 425
Linear variety      3
Lipschitz conditions      116 237 370—371
Lipschitzian      86
Locally simplicial sets      84—85 184
Lower boundary      33
Lower closed saddle-function      365
Lower closure      357—359 368
Lower conjugate      389—391
Lower semi-continuity      51—52 72 77
Lower semi-continuous hu      1152 54
Lower simple extension      349 358
Maximum of a convex function      342—346
minimax      379 391—393 397—398 431
Minimum set      263—266
Minimum set of a convex function      263ff
Minkowski metric      132
Monotone conjugacy      111 426
Monotone mappings      240 340 396
Monotonicity      68—69 77 85—86
Moreau’s Theorem      338
Multiplication of bifunctions      409—412
Multiplication of convex processes      422—423
Network programming      272 337—338 431
Non-decreasing curves      232 338
Non-decreasing functions      68—69 77 85—86 232 338
Non-negative orthant      13 122 226
Norm      129—132 136 427

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