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Rockafellar R.T. Ч Convex analysis
Rockafellar R.T. Ч Convex analysis



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Ќазвание: Convex analysis

јвтор: Rockafellar R.T.

јннотаци€:

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: en

–убрика: ћатематика/

—татус предметного указател€: √отов указатель с номерами страниц

ed2k: ed2k stats

√од издани€: 1970

 оличество страниц: 472

ƒобавлена в каталог: 12.09.2007

ќперации: ѕоложить на полку | —копировать ссылку дл€ форума | —копировать ID
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ѕредметный указатель
Abnormal program      318
Addition of convex cones      22
Addition of convex functions      33 77 145 176 179Ч180 223 263
Addition of convex processes      415 421
Addition of convex sets      16Ч17 49 74Ч75 146 175 183
Addition of epigraphs      34
Addition of saddle-functions      402
Adjoint of a bifunction      309Ч326 330 353Ч358 401Ч412
Adjoint of a convex process      417ff
Adjoint of a linear transformation      3 9 310
aff      see УAffine hullФ
Affine functions      23 25 27 102Ч103
Affine functions, partial      70 107 431
Affine hull      6 45 154
Affine hull of convex cone      15
Affine hull, characterization      113
Affine independence      6Ч7 154
Affine sets      3Ч9
Affine sets, closed halves      165Ч166
Affine sets, representation      4Ч8
Affine transformations      7Ч8 44Ч45
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 УConvex bi-functionsФ)
Bilinear functions      351Ч352 411
Boundedness conditions      54 64 68Ч69 88 123
CaratheodoryТs theorem      153Ч157 427
Chemical equilibrium problem      430
Circulations      204 208 272 337Ч338
Cl      see УClosureФ
Closed bifunction      293
Closed concave function      308
Closed convex function      52 (see also УClosureФ)
Closed saddle-function      363 (see also УClosureФ)
Closure of a bifunction      293 305Ч306 310 403 407
Closure of a concave function      307Ч308
Closure of a convex function      51Ч59 72Ч81 102Ч104 218Ч219 425
Closure of a convex process      415
Closure of a convex set      43Ч50 72Ч81 112 421Ч422
Closure of a saddle-function      359Ч369 390
Closure of an epigraph      52
Co-finite      116 259Ч260 411Ч412
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 422Ч423
Concave bifunctions      308ff
Concave closure of a saddle-function      350Ч353
Concave functions      25 307Ч308 426
Concave functions, monotone conjugates      110
Concave programs      308ff
Concave-convex functions      349ff
Cone      13
Conjugacy correspondence      104 123Ч124
Conjugacy correspondence for saddle-functions      389ff
Conjugate concave functions      111 308
Conjugate convex functions      104Ч111 113Ч118 121Ч124 133Ч137 140Ч150 173 179Ч180 263Ч264 405 421 425Ч426
Conjugate convex functions, definition      104
Conjugate convex functions, subgradients      218
Conjugate saddle-functions      390Ч391 395 432
Consequence      199
Consistency      185 295 309 315
Constancy space      69
Continuity of convex functions      82Ч89 426
Continuity of derivatives      227Ч238
Continuity of gradient mappings      246 376Ч377
Continuity of saddle-functions      370Ч371
Continuity, joint      89
Continuity, uniform      86Ч87
Continuous extensions      85
conv      see УConvex hullФ
Convergence of convex functions      90Ч91 426
Convergence of gradients      248Ч249
Convergence of saddle-functions      372 375Ч378
Convergence of subgradients      233Ч236
Convex bifunctions      293Ч306 309Ч311 350Ч358 384Ч389 401Ч412 417Ч418
Convex closure of a saddle-function      350Ч353
Convex combinations      11Ч12
Convex combinations of points and directions      154
Convex cones      13Ч15 22 50
Convex cones, generation      78 122 126 156 178
Convex cones, polar      121Ч125
Convex cones, polyhedral      170 178
Convex cones, separation      100Ч101
Convex function      23
Convex function, co-finite      259
Convex function, differential conditions for convexity      26Ч27
Convex function, finitely generated      172Ч173
Convex function, interpolation properties      25
Convex function, Legendre type      258
Convex function, partial quadratic      109 431
Convex function, polyhedral      172Ч177
Convex function, polynomial      268
Convex function, quadratic      27 108
Convex function, separable      270Ч271 285Ч290 337Ч338
Convex function, symmetric      109Ч110
Convex hull      12 177 427
Convex hull of a set of points and directions      153Ч155
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 156Ч157
Convex hull, of a non-convex function      36 103 157Ч158
Convex hull, of two convex cones      22
Convex hull, relative interior      50
Convex processes      413Ч423 432
Convex processes, polyhedral      415
Convex programs, generalized      291Ч326 355Ч356 385Ч387
Convex programs, normal      316Ч319
Convex programs, ordinary      273Ч291 293Ч294 296 298 300 320Ч326 429
Convex programs, polyhedral      301Ч303
Convex set      10
Convex set as a cross-section of a cone      15
Convex set, finitely generated      170Ч171
Convex set, polyhedral      11
Convex set, symmetric      16
Convex-concave functions      349ff
Cyclically monotone mappings      238Ч240
Decomposition principle      285Ч290 312Ч313 429
Derivatives, directional      213Ч221 226 244Ч245 264 299Ч301 372Ч377
Derivatives, partial      241 244 376
Derivatives, right and left      214 216 227Ч232
Differentiability      241Ч246 428
Differentiability of saddle-functions      375Ч376
DIM      see УDimensionФ
Dimension of a convex function      23 71
Dimension of a convex set      12Ч13 45Ч46 126
Dimension of an affine set      4
Direct sums      19 49
Directed graphs      204 208 272 337Ч338
Direction      60
Direction of affinity      70
Direction of constancy      69
Direction of linearity      65
Direction of recession      61 69 264Ч270
Directional derivatives      213Ч221 226 244Ч245 264 299Ч301 372Ч377
Distance function      28 34
Distributive inequalities      416
DOM      see УEffective domainФ
Dual programs      310Ч338 355Ч356 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 391Ч392
Effective domain, relative interior      54
Eigensets      423
Elementary vectors      203Ч208 272 428
Epi      see УEpigraphФ
Epigraph      23 307
Epigraph, closure      52
Epigraph, relative interior      54
Epigraph, support function of      119
Equi-Lipschitzian      87Ч88
Equicontinuity      88
Equilibrium prices      276Ч277 280 299Ч300
Equivalent saddle-functions      363Ч369 383 394
Essentially smooth function      251Ч258
Essentially strictly convex function      253Ч260
Euclidean metric      43
Exposed directions      163 168
Exposed faces      162Ч163
Exposed points      162Ч163 167Ч168 243 427
Exposed rays      163 169
Extensions of saddle-functions      349 358 363 366 369
Extreme directions      162Ч166 172
Extreme points      162Ч167 172 344Ч345 427
Extreme points at infinity      162
Extreme rays      162 167
Faces      162Ч165 171 427
Faces, exposed      162Ч163
FarkasТ lemma      200Ч201
Feasible solutions      274 295 308 315
FenchelТs Duality Theorem      327ft 408 430
FenchelТs inequality      105 218
Finitely generated convex function      172Ч173
Finitely generated convex set      170Ч171
Flat      3
flows      204 208 272 337Ч338
Fully closed saddle-function      356 365
Gale Ч Kuhn Ч Tucker Theorem      317 337 421 430Ч431
Gauge      28 35 79 124Ч125 128Ч131 427
Gauge-like functions      133
Generalized convex programs      291Ч326 355Ч356 385Ч387
Generalized polytope      171
Generalized simplex      154Ч155
Generators      170
Geometric mean      27 29
Geometric programming      324Ч326 430
Gradients      213 241Ч250 300 375Ч378 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
HellyТs theorem      191Ч197 206 267 427Ч428
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 409Ч412 416 421
Image of a convex set      19 48 73 143 174 414Ч415 421Ч422
Image-closed bifunction      352Ч353
Improper convex function      24 34 52Ч53
Improper saddle-function      366
Incidence matrix      204 208
Inconsistency      185 315
Indicator bifunction      292Ч293 310 355 417
Indicator function      28 33 425
Indicator function, conjugate      113Ч114
Inequalities      129Ч130 425 428
Inequalities, between functions      38 104
Inequalities, between vectors      13
Inequalities, consistent      185
Inequalities, convex      29 55 58 185Ч197
Inequalities, homogeneous      14
Inequalities, linear      10Ч11 13Ч14 62 65 113 122 170 185 198Ч209
Infimal convolution      34 38 76Ч77 145 175 179Ч181 254 425 427
Infimal convolution of bi-functions      401Ч404
Infimal convolution, partial      39
Inner product equation      355 409Ч412 419Ч421
Inner product of a vector and a function      350
Inner product of a vector and a set      417
Inner product of two functions      408Ч412
Inner product of two sets      422Ч423
Inner product of two vectors      3
int      see УInteriorФ
Interior      43Ч44 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      384Ч385 388Ч389 401 405Ч406
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      367Ч369
Kuhn Ч Tucker coefficients      274Ч277 280 429
Kuhn Ч Tucker conditions      282Ч284 304 333Ч338 386Ч387 429
Kuhn Ч Tucker theorem      283 387
Kuhn Ч Tucker vectors      274Ч290 295Ч306 309 387
Lagrange multipliers      273Ч274 280 283 429
Lagrangian function      280Ч290 296Ч298 302Ч305 309 314 385Ч387
Lattice of convex functions      38
Lattice of convex processes      416
Lattice of convex sets      18
Legendre conjugate      256Ч260
Legendre transformation      251 256 427 429
Level sets      28Ч29 55 58Ч59 70 123 127 222 263Ч265
Level sets of support functions      118
Line      3Ч4
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      17Ч18
Linear combinations, positive and non-negative      14
Linear programs      301Ч302 311Ч312 317 332 334Ч335 337 425
Linear variety      3
Lipschitz conditions      116 237 370Ч371
Lipschitzian      86
Locally simplicial sets      84Ч85 184
Lower boundary      33
Lower closed saddle-function      365
Lower closure      357Ч359 368
Lower conjugate      389Ч391
Lower semi-continuity      51Ч52 72 77
Lower semi-continuous hu      1152 54
Lower simple extension      349 358
Maximum of a convex function      342Ч346
minimax      379 391Ч393 397Ч398 431
Minimum set      263Ч266
Minimum set of a convex function      263ff
Minkowski metric      132
Monotone conjugacy      111 426
Monotone mappings      240 340 396
Monotonicity      68Ч69 77 85Ч86
MoreauТs Theorem      338
Multiplication of bifunctions      409Ч412
Multiplication of convex processes      422Ч423
Network programming      272 337Ч338 431
Non-decreasing curves      232 338
Non-decreasing functions      68Ч69 77 85Ч86 232 338
Non-negative orthant      13 122 226
Norm      129Ч132 136 427
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