12.07.2015
•
Views
818 INDEXinterior, 40map, see transformationmembership relation, 224normal cone, 765set, 38, 61, 62parallel, 38subset, 62, 72hyperplane intersection, 86nullspace basis span, 87projector, 703, 715transformation, see transformationalgebraaffine, 38, 40, 62, 79, 95, 715arg, 338, 653, 654cone, 164face, 95functionlinear, 220support, 240fundamental theorem, 617linear, 85interior, 40, 46, 183linear, 85, 591nullspace, 634optimization, 653, 654projection, 715subspace, 36, 85, 86, 634algebraiccomplement, 96, 97, 156, 203, 708, 712,772projection on, 708, 740multiplicity, 618Alizadeh, 299alternation, 485alternative, see theoremambient, 529, 542vector space, 36anchor, 315, 316, 417, 424angle, 84acute, 72alternating projection, 755brackets, 50, 51constraint, 408, 409, 412, 428dihedral, 29, 84, 429, 491, 649EDM cone, 128halfspace, 73inequality, 400, 489matrix, 433interpoint, 405obtuse, 72positive semidefinite cone, 128relative, 430, 432right, 129, 159, 181, 182torsion, 29, 428Anstreicher, 665antidiagonal, 437antisymmetry, 103, 606arg, 783algebra, 338, 653, 654artificial intelligence, 25asymmetry, 35, 72axioms of the metric, 397axis of revolution, 129, 130, 649− B −, 228ball1-norm, 227, 228nonnegative, 347∞-norm, 227Euclidean, 39, 227, 228, 413, 703, 744smallest, 64nuclear norm, 67packing, 413spectral norm, 70, 745Barvinok, 35, 113, 138, 142, 145, 299, 393,425base, 96station, 425basis, 58, 190, 777discrete cosine transform, 341nonorthogonal, 190, 716projection on, 716nullspace, 87, 91, 613, 710orthogonalprojection, 715orthonormal, 58, 90, 712pursuit, 372standardmatrix, 58, 60, 770vector, 58, 151, 406, 783
INDEX 819bees, 30, 413Bellman, 276bijective, 50, 53, 56, 119, 185Gram form, 439, 442inner-product form, 443, 444invertible, 53, 700isometry, 54, 485linear, 52, 54, 57, 60, 144orthogonal, 54, 485, 563, 647, 665biorthogonaldecomposition, 704, 713expansion, 153, 168, 186, 190, 705EDM, 510projection, 716unique, 188, 190, 193, 511, 638, 717biorthogonality, 188, 191condition, 186, 619, 638Birkhoff, 69Boolean, see problemboundlower, 552polyhedral, 501boundary, 39, 40, 46, 64, 96-point, 94extreme, 111of affine, 38, 46of cone, 106, 178, 183, 185EDM, 499, 507, 514membership, 169, 178, 185, 287PSD, 40, 114, 119, 135–138, 287, 519ray, 107of halfspace, 72of hypercube slice, 383of open set, 41of point, 46of ray, 96relative, 39, 40, 46, 64, 96bounded, 64, 84bowl, 680, 681box, 83, 744bridge, 429, 458, 527, 553, 569Bunt, 137, 734calculus− C −, 36matrix, 669of inequalities, 7of variations, 173Carathéodory’s theorem, 168, 726cardinality, 333, 402, 784, 7861-norm heuristic, 335, 336, 572-one, 227, 347constraint, see constraintconvex envelope, see convexgeometry, 227, 230, 347minimization, 227, 228, 333, 340,351–353, 380Boolean, 292nonnegative, 229, 230, 233, 333, 337,339, 346–350rank connection, 310rank one, 369rank, full, 298quasiconcavity, 233, 334regularization, 345, 370Cartesianaxes, 38, 63, 80, 99, 108, 154, 166, 336,744cone, see orthantcoordinates, 480product, 47, 284, 756, 778cone, 102, 103, 163function, 224subspace, 336, 744Cauchy-Schwarz, 88, 753Cayley-Mengerdeterminant, 491, 493form, 449, 470, 471, 559, 584cellular, 426telephone network, 425centergeometric, 407, 411, 435, 478gravity, 435mass, 435central path, 283certificate, 217, 287, 288, 309, 391null, 170, 179chain rule, 676two argument, 676Chu, 7
-
Page 1 and 2:
DATTORROCONVEXOPTIMIZATION&EUCLIDEA
-
Page 3 and 4:
Convex Optimization&Euclidean Dista
-
Page 5 and 6:
for Jennie Columba♦Antonio♦♦&
-
Page 7 and 8:
PreludeThe constant demands of my d
-
Page 9 and 10:
Convex Optimization&Euclidean Dista
-
Page 11 and 12:
CONVEX OPTIMIZATION & EUCLIDEAN DIS
-
Page 13 and 14:
List of Figures1 Overview 211 Orion
-
Page 15 and 16:
LIST OF FIGURES 1562 Shrouded polyh
-
Page 17:
LIST OF FIGURES 17130 Elliptope E 3
-
Page 21 and 22:
Chapter 1OverviewConvex Optimizatio
-
Page 23 and 24:
ˇx 4ˇx 3ˇx 2Figure 2: Applicatio
-
Page 25 and 26:
25Figure 4: This coarsely discretiz
-
Page 27 and 28:
(biorthogonal expansion) is examine
-
Page 29 and 30:
29cardinality Boolean solution to a
-
Page 31 and 32:
31Figure 8: Robotic vehicles in con
-
Page 33 and 34:
an elaborate exposition offering in
-
Page 35 and 36:
Chapter 2Convex geometryConvexity h
-
Page 37 and 38:
2.1. CONVEX SET 372.1.2 linear inde
-
Page 39 and 40:
2.1. CONVEX SET 392.1.6 empty set v
-
Page 41 and 42:
2.1. CONVEX SET 41(a)R(b)R 2(c)R 3F
-
Page 43 and 44:
2.1. CONVEX SET 43where Q∈ R 3×3
-
Page 45 and 46:
2.1. CONVEX SET 45Now let’s move
-
Page 47 and 48:
2.1. CONVEX SET 47By additive inver
-
Page 49 and 50:
2.1. CONVEX SET 49R nR mR(A T )x px
-
Page 51 and 52:
2.2. VECTORIZED-MATRIX INNER PRODUC
-
Page 53 and 54:
2.2. VECTORIZED-MATRIX INNER PRODUC
-
Page 55 and 56:
2.2. VECTORIZED-MATRIX INNER PRODUC
-
Page 57 and 58:
2.2. VECTORIZED-MATRIX INNER PRODUC
-
Page 59 and 60:
2.2. VECTORIZED-MATRIX INNER PRODUC
-
Page 61 and 62:
2.3. HULLS 61Figure 20: Convex hull
-
Page 63 and 64:
2.3. HULLS 63Aaffine hull (drawn tr
-
Page 65 and 66:
2.3. HULLS 65subset of the affine h
-
Page 67 and 68:
2.3. HULLS 672.3.2.0.2 Example. Nuc
-
Page 69 and 70:
2.3. HULLS 692.3.2.0.3 Exercise. Co
-
Page 71 and 72:
2.3. HULLS 71Figure 24: A simplicia
-
Page 73 and 74:
2.4. HALFSPACE, HYPERPLANE 73H + =
-
Page 75 and 76:
2.4. HALFSPACE, HYPERPLANE 7511−1
-
Page 77 and 78:
2.4. HALFSPACE, HYPERPLANE 772.4.2.
-
Page 79 and 80:
2.4. HALFSPACE, HYPERPLANE 792.4.2.
-
Page 81 and 82:
2.4. HALFSPACE, HYPERPLANE 81tradit
-
Page 83 and 84:
2.4. HALFSPACE, HYPERPLANE 832.4.2.
-
Page 85 and 86:
2.5. SUBSPACE REPRESENTATIONS 852.5
-
Page 87 and 88:
2.5. SUBSPACE REPRESENTATIONS 87(Ex
-
Page 89 and 90:
2.5. SUBSPACE REPRESENTATIONS 89(a)
-
Page 91 and 92:
2.5. SUBSPACE REPRESENTATIONS 91are
-
Page 93 and 94:
2.6. EXTREME, EXPOSED 93A one-dimen
-
Page 95 and 96:
2.6. EXTREME, EXPOSED 952.6.1.1 Den
-
Page 97 and 98:
2.7. CONES 97X(a)00(b)XFigure 33: (
-
Page 99 and 100:
2.7. CONES 99XXFigure 37: Truncated
-
Page 101 and 102:
2.7. CONES 101Figure 39: Not a cone
-
Page 103 and 104:
2.7. CONES 103cone that is a halfli
-
Page 105 and 106:
2.7. CONES 105A pointed closed conv
-
Page 107 and 108:
2.8. CONE BOUNDARY 107That means th
-
Page 109 and 110:
2.8. CONE BOUNDARY 1092.8.1.1 extre
-
Page 111 and 112:
2.8. CONE BOUNDARY 1112.8.2 Exposed
-
Page 113 and 114:
2.9. POSITIVE SEMIDEFINITE (PSD) CO
-
Page 115 and 116:
2.9. POSITIVE SEMIDEFINITE (PSD) CO
-
Page 117 and 118:
2.9. POSITIVE SEMIDEFINITE (PSD) CO
-
Page 119 and 120:
2.9. POSITIVE SEMIDEFINITE (PSD) CO
-
Page 121 and 122:
2.9. POSITIVE SEMIDEFINITE (PSD) CO
-
Page 123 and 124:
2.9. POSITIVE SEMIDEFINITE (PSD) CO
-
Page 125 and 126:
2.9. POSITIVE SEMIDEFINITE (PSD) CO
-
Page 127 and 128:
2.9. POSITIVE SEMIDEFINITE (PSD) CO
-
Page 129 and 130:
2.9. POSITIVE SEMIDEFINITE (PSD) CO
-
Page 131 and 132:
2.9. POSITIVE SEMIDEFINITE (PSD) CO
-
Page 133 and 134:
2.9. POSITIVE SEMIDEFINITE (PSD) CO
-
Page 135 and 136:
2.9. POSITIVE SEMIDEFINITE (PSD) CO
-
Page 137 and 138:
2.9. POSITIVE SEMIDEFINITE (PSD) CO
-
Page 139 and 140:
2.9. POSITIVE SEMIDEFINITE (PSD) CO
-
Page 141 and 142:
2.10. CONIC INDEPENDENCE (C.I.) 141
-
Page 143 and 144:
2.10. CONIC INDEPENDENCE (C.I.) 143
-
Page 145 and 146:
2.10. CONIC INDEPENDENCE (C.I.) 145
-
Page 147 and 148:
2.12. CONVEX POLYHEDRA 147all dimen
-
Page 149 and 150:
2.12. CONVEX POLYHEDRA 149convex po
-
Page 151 and 152:
2.12. CONVEX POLYHEDRA 1512.12.2.2
-
Page 153 and 154:
2.13. DUAL CONE & GENERALIZED INEQU
-
Page 155 and 156:
2.13. DUAL CONE & GENERALIZED INEQU
-
Page 157 and 158:
2.13. DUAL CONE & GENERALIZED INEQU
-
Page 159 and 160:
2.13. DUAL CONE & GENERALIZED INEQU
-
Page 161 and 162:
2.13. DUAL CONE & GENERALIZED INEQU
-
Page 163 and 164:
2.13. DUAL CONE & GENERALIZED INEQU
-
Page 165 and 166:
2.13. DUAL CONE & GENERALIZED INEQU
-
Page 167 and 168:
2.13. DUAL CONE & GENERALIZED INEQU
-
Page 169 and 170:
2.13. DUAL CONE & GENERALIZED INEQU
-
Page 171 and 172:
2.13. DUAL CONE & GENERALIZED INEQU
-
Page 173 and 174:
2.13. DUAL CONE & GENERALIZED INEQU
-
Page 175 and 176:
2.13. DUAL CONE & GENERALIZED INEQU
-
Page 177 and 178:
2.13. DUAL CONE & GENERALIZED INEQU
-
Page 179 and 180:
2.13. DUAL CONE & GENERALIZED INEQU
-
Page 181 and 182:
2.13. DUAL CONE & GENERALIZED INEQU
-
Page 183 and 184:
2.13. DUAL CONE & GENERALIZED INEQU
-
Page 185 and 186:
2.13. DUAL CONE & GENERALIZED INEQU
-
Page 187 and 188:
2.13. DUAL CONE & GENERALIZED INEQU
-
Page 189 and 190:
2.13. DUAL CONE & GENERALIZED INEQU
-
Page 191 and 192:
2.13. DUAL CONE & GENERALIZED INEQU
-
Page 193 and 194:
2.13. DUAL CONE & GENERALIZED INEQU
-
Page 195 and 196:
2.13. DUAL CONE & GENERALIZED INEQU
-
Page 197 and 198:
2.13. DUAL CONE & GENERALIZED INEQU
-
Page 199 and 200:
2.13. DUAL CONE & GENERALIZED INEQU
-
Page 201 and 202:
2.13. DUAL CONE & GENERALIZED INEQU
-
Page 203 and 204:
2.13. DUAL CONE & GENERALIZED INEQU
-
Page 205 and 206:
2.13. DUAL CONE & GENERALIZED INEQU
-
Page 207 and 208:
2.13. DUAL CONE & GENERALIZED INEQU
-
Page 209 and 210:
2.13. DUAL CONE & GENERALIZED INEQU
-
Page 211 and 212:
2.13. DUAL CONE & GENERALIZED INEQU
-
Page 213 and 214:
2.13. DUAL CONE & GENERALIZED INEQU
-
Page 215 and 216:
2.13. DUAL CONE & GENERALIZED INEQU
-
Page 217 and 218:
2.13. DUAL CONE & GENERALIZED INEQU
-
Page 219 and 220:
Chapter 3Geometry of convex functio
-
Page 221 and 222:
3.1. CONVEX FUNCTION 221f 1 (x)f 2
-
Page 223 and 224:
3.1. CONVEX FUNCTION 223Rf(b)f(X
-
Page 225 and 226:
3.2. PRACTICAL NORM FUNCTIONS, ABSO
-
Page 227 and 228:
3.2. PRACTICAL NORM FUNCTIONS, ABSO
-
Page 229 and 230:
3.2. PRACTICAL NORM FUNCTIONS, ABSO
-
Page 231 and 232:
3.2. PRACTICAL NORM FUNCTIONS, ABSO
-
Page 233 and 234:
3.2. PRACTICAL NORM FUNCTIONS, ABSO
-
Page 235 and 236:
3.3. INVERTED FUNCTIONS AND ROOTS 2
-
Page 237 and 238:
3.4. AFFINE FUNCTION 237rather]x >
-
Page 239 and 240:
3.4. AFFINE FUNCTION 239f(z)Az 2z 1
-
Page 241 and 242:
3.5. EPIGRAPH, SUBLEVEL SET 241{a T
-
Page 243 and 244:
3.5. EPIGRAPH, SUBLEVEL SET 243Subl
-
Page 245 and 246:
3.5. EPIGRAPH, SUBLEVEL SET 245wher
-
Page 247 and 248:
3.5. EPIGRAPH, SUBLEVEL SET 247part
-
Page 249 and 250:
3.5. EPIGRAPH, SUBLEVEL SET 249that
-
Page 251 and 252:
3.6. GRADIENT 251respect to its vec
-
Page 253 and 254:
3.6. GRADIENT 253Invertibility is g
-
Page 255 and 256:
3.6. GRADIENT 2553.6.1.0.2 Theorem.
-
Page 257 and 258:
3.6. GRADIENT 257f(Y )[ ∇f(X)−1
-
Page 259 and 260:
3.6. GRADIENT 259αβα ≥ β ≥
-
Page 261 and 262:
3.6. GRADIENT 2613.6.4 second-order
-
Page 263 and 264:
3.7. CONVEX MATRIX-VALUED FUNCTION
-
Page 265 and 266:
3.7. CONVEX MATRIX-VALUED FUNCTION
-
Page 267 and 268:
3.7. CONVEX MATRIX-VALUED FUNCTION
-
Page 269 and 270:
3.8. QUASICONVEX 269exponential alw
-
Page 271 and 272:
3.9. SALIENT PROPERTIES 2713.8.0.0.
-
Page 273 and 274:
Chapter 4Semidefinite programmingPr
-
Page 275 and 276:
4.1. CONIC PROBLEM 275(confer p.162
-
Page 277 and 278:
4.1. CONIC PROBLEM 277PCsemidefinit
-
Page 279 and 280:
4.1. CONIC PROBLEM 279is the affine
-
Page 281 and 282:
4.1. CONIC PROBLEM 281faces of S 3
-
Page 283 and 284:
4.1. CONIC PROBLEM 2834.1.2.3 Previ
-
Page 285 and 286:
4.2. FRAMEWORK 285Semidefinite Fark
-
Page 287 and 288:
4.2. FRAMEWORK 287On the other hand
-
Page 289 and 290:
4.2. FRAMEWORK 2894.2.2.1 Dual prob
-
Page 291 and 292:
4.2. FRAMEWORK 291For symmetric pos
-
Page 293 and 294:
4.2. FRAMEWORK 293has norm ‖x ⋆
-
Page 295 and 296:
4.2. FRAMEWORK 295minimize 1 TˆxX
-
Page 297 and 298:
4.2. FRAMEWORK 297asminimize ‖ỹ
-
Page 299 and 300:
4.3. RANK REDUCTION 2994.3 Rank red
-
Page 301 and 302:
4.3. RANK REDUCTION 301A rank-reduc
-
Page 303 and 304:
4.3. RANK REDUCTION 303(t ⋆ i)
-
Page 305 and 306:
4.3. RANK REDUCTION 3054.3.3.0.1 Ex
-
Page 307 and 308:
4.3. RANK REDUCTION 3074.3.3.0.2 Ex
-
Page 309 and 310:
4.4. RANK-CONSTRAINED SEMIDEFINITE
-
Page 311 and 312:
4.4. RANK-CONSTRAINED SEMIDEFINITE
-
Page 313 and 314:
4.4. RANK-CONSTRAINED SEMIDEFINITE
-
Page 315 and 316:
4.4. RANK-CONSTRAINED SEMIDEFINITE
-
Page 317 and 318:
4.4. RANK-CONSTRAINED SEMIDEFINITE
-
Page 319 and 320:
4.4. RANK-CONSTRAINED SEMIDEFINITE
-
Page 321 and 322:
4.4. RANK-CONSTRAINED SEMIDEFINITE
-
Page 323 and 324:
4.4. RANK-CONSTRAINED SEMIDEFINITE
-
Page 325 and 326:
4.4. RANK-CONSTRAINED SEMIDEFINITE
-
Page 327 and 328:
4.4. RANK-CONSTRAINED SEMIDEFINITE
-
Page 329 and 330:
4.4. RANK-CONSTRAINED SEMIDEFINITE
-
Page 331 and 332:
4.4. RANK-CONSTRAINED SEMIDEFINITE
-
Page 333 and 334:
4.5. CONSTRAINING CARDINALITY 333mi
-
Page 335 and 336:
4.5. CONSTRAINING CARDINALITY 3350R
-
Page 337 and 338:
4.5. CONSTRAINING CARDINALITY 337it
-
Page 339 and 340:
4.5. CONSTRAINING CARDINALITY 339m/
-
Page 341 and 342:
4.5. CONSTRAINING CARDINALITY 341we
-
Page 343 and 344:
4.5. CONSTRAINING CARDINALITY 343fl
-
Page 345 and 346:
4.5. CONSTRAINING CARDINALITY 345We
-
Page 347 and 348:
4.5. CONSTRAINING CARDINALITY 3474.
-
Page 349 and 350:
4.5. CONSTRAINING CARDINALITY 349R
-
Page 351 and 352:
4.5. CONSTRAINING CARDINALITY 351pe
-
Page 353 and 354:
4.6. CARDINALITY AND RANK CONSTRAIN
-
Page 355 and 356:
4.6. CARDINALITY AND RANK CONSTRAIN
-
Page 357 and 358:
4.6. CARDINALITY AND RANK CONSTRAIN
-
Page 359 and 360:
4.6. CARDINALITY AND RANK CONSTRAIN
-
Page 361 and 362:
4.6. CARDINALITY AND RANK CONSTRAIN
-
Page 363 and 364:
4.6. CARDINALITY AND RANK CONSTRAIN
-
Page 365 and 366:
4.6. CARDINALITY AND RANK CONSTRAIN
-
Page 367 and 368:
4.6. CARDINALITY AND RANK CONSTRAIN
-
Page 369 and 370:
4.6. CARDINALITY AND RANK CONSTRAIN
-
Page 371 and 372:
4.6. CARDINALITY AND RANK CONSTRAIN
-
Page 373 and 374:
4.6. CARDINALITY AND RANK CONSTRAIN
-
Page 375 and 376:
4.6. CARDINALITY AND RANK CONSTRAIN
-
Page 377 and 378:
4.6. CARDINALITY AND RANK CONSTRAIN
-
Page 379 and 380:
4.6. CARDINALITY AND RANK CONSTRAIN
-
Page 381 and 382:
4.6. CARDINALITY AND RANK CONSTRAIN
-
Page 383 and 384:
4.6. CARDINALITY AND RANK CONSTRAIN
-
Page 385 and 386:
4.7. CONSTRAINING RANK OF INDEFINIT
-
Page 387 and 388:
4.7. CONSTRAINING RANK OF INDEFINIT
-
Page 389 and 390:
4.7. CONSTRAINING RANK OF INDEFINIT
-
Page 391 and 392:
4.8. CONVEX ITERATION RANK-1 391whi
-
Page 393 and 394:
4.8. CONVEX ITERATION RANK-1 393the
-
Page 395 and 396:
Chapter 5Euclidean Distance MatrixT
-
Page 397 and 398:
5.2. FIRST METRIC PROPERTIES 397to
-
Page 399 and 400:
5.3. ∃ FIFTH EUCLIDEAN METRIC PRO
-
Page 401 and 402:
5.3. ∃ FIFTH EUCLIDEAN METRIC PRO
-
Page 403 and 404:
5.4. EDM DEFINITION 403The collecti
-
Page 405 and 406:
5.4. EDM DEFINITION 4055.4.2 Gram-f
-
Page 407 and 408:
5.4. EDM DEFINITION 407We provide a
-
Page 409 and 410:
5.4. EDM DEFINITION 4095.4.2.3.1 Ex
-
Page 411 and 412:
5.4. EDM DEFINITION 411is the fact:
-
Page 413 and 414:
5.4. EDM DEFINITION 413Figure 120:
-
Page 415 and 416:
5.4. EDM DEFINITION 415is found fro
-
Page 417 and 418:
5.4. EDM DEFINITION 417one less dim
-
Page 419 and 420:
5.4. EDM DEFINITION 419equality con
-
Page 421 and 422:
5.4. EDM DEFINITION 421How much dis
-
Page 423 and 424:
5.4. EDM DEFINITION 423105ˇx 4ˇx
-
Page 425 and 426:
5.4. EDM DEFINITION 425now implicit
-
Page 427 and 428:
5.4. EDM DEFINITION 427by translate
-
Page 429 and 430:
5.4. EDM DEFINITION 429Crippen & Ha
-
Page 431 and 432:
5.4. EDM DEFINITION 431where ([√t
-
Page 433 and 434:
5.4. EDM DEFINITION 433because (A.3
-
Page 435 and 436:
5.5. INVARIANCE 4355.5.1.0.1 Exampl
-
Page 437 and 438:
5.5. INVARIANCE 437x 2 x 2x 3 x 1 x
-
Page 439 and 440:
5.6. INJECTIVITY OF D & UNIQUE RECO
-
Page 441 and 442:
5.6. INJECTIVITY OF D & UNIQUE RECO
-
Page 443 and 444:
5.6. INJECTIVITY OF D & UNIQUE RECO
-
Page 445 and 446:
5.7. EMBEDDING IN AFFINE HULL 4455.
-
Page 447 and 448:
5.7. EMBEDDING IN AFFINE HULL 447Fo
-
Page 449 and 450:
5.7. EMBEDDING IN AFFINE HULL 4495.
-
Page 451 and 452:
5.8. EUCLIDEAN METRIC VERSUS MATRIX
-
Page 453 and 454:
5.8. EUCLIDEAN METRIC VERSUS MATRIX
-
Page 455 and 456:
5.8. EUCLIDEAN METRIC VERSUS MATRIX
-
Page 457 and 458:
5.8. EUCLIDEAN METRIC VERSUS MATRIX
-
Page 459 and 460:
5.9. BRIDGE: CONVEX POLYHEDRA TO ED
-
Page 461 and 462:
5.9. BRIDGE: CONVEX POLYHEDRA TO ED
-
Page 463 and 464:
5.9. BRIDGE: CONVEX POLYHEDRA TO ED
-
Page 465 and 466:
5.10. EDM-ENTRY COMPOSITION 465of a
-
Page 467 and 468:
5.10. EDM-ENTRY COMPOSITION 467Then
-
Page 469 and 470:
5.11. EDM INDEFINITENESS 4695.11.1
-
Page 471 and 472:
5.11. EDM INDEFINITENESS 471we have
-
Page 473 and 474:
5.11. EDM INDEFINITENESS 473So beca
-
Page 475 and 476:
5.11. EDM INDEFINITENESS 475holds o
-
Page 477 and 478:
5.12. LIST RECONSTRUCTION 477where
-
Page 479 and 480:
5.12. LIST RECONSTRUCTION 479(a)(c)
-
Page 481 and 482:
5.13. RECONSTRUCTION EXAMPLES 481Wi
-
Page 483 and 484:
5.13. RECONSTRUCTION EXAMPLES 483d
-
Page 485 and 486:
5.13. RECONSTRUCTION EXAMPLES 485Th
-
Page 487 and 488:
5.14. FIFTH PROPERTY OF EUCLIDEAN M
-
Page 489 and 490:
5.14. FIFTH PROPERTY OF EUCLIDEAN M
-
Page 491 and 492:
5.14. FIFTH PROPERTY OF EUCLIDEAN M
-
Page 493 and 494:
5.14. FIFTH PROPERTY OF EUCLIDEAN M
-
Page 495 and 496:
5.14. FIFTH PROPERTY OF EUCLIDEAN M
-
Page 497 and 498:
Chapter 6Cone of distance matricesF
-
Page 499 and 500:
6.1. DEFINING EDM CONE 4996.1 Defin
-
Page 501 and 502:
6.2. POLYHEDRAL BOUNDS 501This cone
-
Page 503 and 504:
6.4. EDM DEFINITION IN 11 T 503That
-
Page 505 and 506:
6.4. EDM DEFINITION IN 11 T 505N(1
-
Page 507 and 508:
6.4. EDM DEFINITION IN 11 T 507Then
-
Page 509 and 510:
6.4. EDM DEFINITION IN 11 T 5096.4.
-
Page 511 and 512:
6.5. CORRESPONDENCE TO PSD CONE S N
-
Page 513 and 514:
6.5. CORRESPONDENCE TO PSD CONE S N
-
Page 515 and 516:
6.5. CORRESPONDENCE TO PSD CONE S N
-
Page 517 and 518:
6.6. VECTORIZATION & PROJECTION INT
-
Page 519 and 520:
6.6. VECTORIZATION & PROJECTION INT
-
Page 521 and 522:
6.6. VECTORIZATION & PROJECTION INT
-
Page 523 and 524:
6.7. A GEOMETRY OF COMPLETION 523(b
-
Page 525 and 526:
6.7. A GEOMETRY OF COMPLETION 525[3
-
Page 527 and 528:
6.7. A GEOMETRY OF COMPLETION 527wh
-
Page 529 and 530:
6.8. DUAL EDM CONE 529to the geomet
-
Page 531 and 532:
6.8. DUAL EDM CONE 531Proof. First,
-
Page 533 and 534:
6.8. DUAL EDM CONE 533EDM 2 = S 2 h
-
Page 535 and 536:
6.8. DUAL EDM CONE 535therefore the
-
Page 537 and 538:
6.8. DUAL EDM CONE 537Elegance of t
-
Page 539 and 540:
6.8. DUAL EDM CONE 5396.8.1.5 Affin
-
Page 541 and 542:
6.8. DUAL EDM CONE 5416.8.1.7 Schoe
-
Page 543 and 544:
6.8. DUAL EDM CONE 5430dvec rel ∂
-
Page 545 and 546:
6.10. POSTSCRIPT 5456.10 Postscript
-
Page 547 and 548:
Chapter 7Proximity problemsIn the
-
Page 549 and 550:
549project on the subspace, then pr
-
Page 551 and 552:
551HS N h0EDM NK = S N h ∩ R N×N
-
Page 553 and 554:
5537.0.3 Problem approachProblems t
-
Page 555 and 556:
7.1. FIRST PREVALENT PROBLEM: 555fi
-
Page 557 and 558:
7.1. FIRST PREVALENT PROBLEM: 5577.
-
Page 559 and 560:
7.1. FIRST PREVALENT PROBLEM: 559di
-
Page 561 and 562:
7.1. FIRST PREVALENT PROBLEM: 5617.
-
Page 563 and 564:
7.1. FIRST PREVALENT PROBLEM: 563wh
-
Page 565 and 566:
7.1. FIRST PREVALENT PROBLEM: 565Th
-
Page 567 and 568:
7.2. SECOND PREVALENT PROBLEM: 567O
-
Page 569 and 570:
7.2. SECOND PREVALENT PROBLEM: 569S
-
Page 571 and 572:
7.2. SECOND PREVALENT PROBLEM: 571r
-
Page 573 and 574:
7.2. SECOND PREVALENT PROBLEM: 573w
-
Page 575 and 576:
7.2. SECOND PREVALENT PROBLEM: 5757
-
Page 577 and 578:
7.2. SECOND PREVALENT PROBLEM: 577a
-
Page 579 and 580:
7.3. THIRD PREVALENT PROBLEM: 579is
-
Page 581 and 582:
7.3. THIRD PREVALENT PROBLEM: 581We
-
Page 583 and 584:
7.3. THIRD PREVALENT PROBLEM: 583su
-
Page 585 and 586:
7.3. THIRD PREVALENT PROBLEM: 585Gi
-
Page 587 and 588:
7.3. THIRD PREVALENT PROBLEM: 587Op
-
Page 589 and 590:
7.4. CONCLUSION 589filtering, multi
-
Page 591 and 592:
Appendix ALinear algebraA.1 Main-di
-
Page 593 and 594:
A.1. MAIN-DIAGONAL δ OPERATOR, λ
-
Page 595 and 596:
A.1. MAIN-DIAGONAL δ OPERATOR, λ
-
Page 597 and 598:
A.2. SEMIDEFINITENESS: DOMAIN OF TE
-
Page 599 and 600:
A.3. PROPER STATEMENTS 599(AB) T
-
Page 601 and 602:
A.3. PROPER STATEMENTS 601A.3.1Semi
-
Page 603 and 604:
A.3. PROPER STATEMENTS 603For A dia
-
Page 605 and 606:
A.3. PROPER STATEMENTS 605Diagonali
-
Page 607 and 608:
A.3. PROPER STATEMENTS 607For A,B
-
Page 609 and 610:
A.3. PROPER STATEMENTS 609When B is
-
Page 611 and 612:
A.4. SCHUR COMPLEMENT 611A.4 Schur
-
Page 613 and 614:
A.4. SCHUR COMPLEMENT 613A.4.0.0.3
-
Page 615 and 616:
A.4. SCHUR COMPLEMENT 615From Corol
-
Page 617 and 618:
A.5. EIGENVALUE DECOMPOSITION 617wh
-
Page 619 and 620:
A.5. EIGENVALUE DECOMPOSITION 619A.
-
Page 621 and 622:
A.6. SINGULAR VALUE DECOMPOSITION,
-
Page 623 and 624:
A.6. SINGULAR VALUE DECOMPOSITION,
-
Page 625 and 626:
A.7. ZEROS 625A.6.5SVD of symmetric
-
Page 627 and 628:
A.7. ZEROS 627(Transpose.)Likewise,
-
Page 629 and 630:
A.7. ZEROS 629For X,A∈ S M +[34,2
-
Page 631 and 632:
A.7. ZEROS 631A.7.5.0.1 Proposition
-
Page 633 and 634:
Appendix BSimple matricesMathematic
-
Page 635 and 636:
B.1. RANK-ONE MATRIX (DYAD) 635R(v)
-
Page 637 and 638:
B.1. RANK-ONE MATRIX (DYAD) 637B.1.
-
Page 639 and 640:
B.2. DOUBLET 639R([u v ])R(Π)= R([
-
Page 641 and 642:
B.3. ELEMENTARY MATRIX 641has N −
-
Page 643 and 644:
B.4. AUXILIARY V -MATRICES 643is an
-
Page 645 and 646:
B.4. AUXILIARY V -MATRICES 64514. [
-
Page 647 and 648:
B.5. ORTHOGONAL MATRIX 647Given X
-
Page 649 and 650:
B.5. ORTHOGONAL MATRIX 649Figure 15
-
Page 651 and 652:
B.5. ORTHOGONAL MATRIX 651which is
-
Page 653 and 654:
Appendix CSome analytical optimal r
-
Page 655 and 656:
C.2. TRACE, SINGULAR AND EIGEN VALU
-
Page 657 and 658:
C.2. TRACE, SINGULAR AND EIGEN VALU
-
Page 659 and 660:
C.2. TRACE, SINGULAR AND EIGEN VALU
-
Page 661 and 662:
C.3. ORTHOGONAL PROCRUSTES PROBLEM
-
Page 663 and 664:
C.4. TWO-SIDED ORTHOGONAL PROCRUSTE
-
Page 665 and 666:
C.4. TWO-SIDED ORTHOGONAL PROCRUSTE
-
Page 667 and 668:
C.4. TWO-SIDED ORTHOGONAL PROCRUSTE
-
Page 669 and 670:
Appendix DMatrix calculusFrom too m
-
Page 671 and 672:
D.1. DIRECTIONAL DERIVATIVE, TAYLOR
-
Page 673 and 674:
D.1. DIRECTIONAL DERIVATIVE, TAYLOR
-
Page 675 and 676:
D.1. DIRECTIONAL DERIVATIVE, TAYLOR
-
Page 677 and 678:
D.1. DIRECTIONAL DERIVATIVE, TAYLOR
-
Page 679 and 680:
D.1. DIRECTIONAL DERIVATIVE, TAYLOR
-
Page 681 and 682:
D.1. DIRECTIONAL DERIVATIVE, TAYLOR
-
Page 683 and 684:
D.1. DIRECTIONAL DERIVATIVE, TAYLOR
-
Page 685 and 686:
D.1. DIRECTIONAL DERIVATIVE, TAYLOR
-
Page 687 and 688:
D.1. DIRECTIONAL DERIVATIVE, TAYLOR
-
Page 689 and 690:
D.1. DIRECTIONAL DERIVATIVE, TAYLOR
-
Page 691 and 692:
D.2. TABLES OF GRADIENTS AND DERIVA
-
Page 693 and 694:
D.2. TABLES OF GRADIENTS AND DERIVA
-
Page 695 and 696:
D.2. TABLES OF GRADIENTS AND DERIVA
-
Page 697 and 698:
D.2. TABLES OF GRADIENTS AND DERIVA
-
Page 699 and 700:
Appendix EProjectionFor any A∈ R
-
Page 701 and 702:
701U T = U † for orthonormal (inc
-
Page 703 and 704:
E.1. IDEMPOTENT MATRICES 703where A
-
Page 705 and 706:
E.1. IDEMPOTENT MATRICES 705order,
-
Page 707 and 708:
E.1. IDEMPOTENT MATRICES 707are lin
-
Page 709 and 710:
E.3. SYMMETRIC IDEMPOTENT MATRICES
-
Page 711 and 712:
E.3. SYMMETRIC IDEMPOTENT MATRICES
-
Page 713 and 714:
E.3. SYMMETRIC IDEMPOTENT MATRICES
-
Page 715 and 716:
E.4. ALGEBRA OF PROJECTION ON AFFIN
-
Page 717 and 718:
E.5. PROJECTION EXAMPLES 717a ∗ 2
-
Page 719 and 720:
E.5. PROJECTION EXAMPLES 719where Y
-
Page 721 and 722:
E.5. PROJECTION EXAMPLES 721(B.4.2)
-
Page 723 and 724:
E.6. VECTORIZATION INTERPRETATION,
-
Page 725 and 726:
E.6. VECTORIZATION INTERPRETATION,
-
Page 727 and 728:
E.6. VECTORIZATION INTERPRETATION,
-
Page 729 and 730:
E.7. PROJECTION ON MATRIX SUBSPACES
-
Page 731 and 732:
E.7. PROJECTION ON MATRIX SUBSPACES
-
Page 733 and 734:
E.8. RANGE/ROWSPACE INTERPRETATION
-
Page 735 and 736:
E.9. PROJECTION ON CONVEX SET 735As
-
Page 737 and 738:
E.9. PROJECTION ON CONVEX SET 737Wi
-
Page 739 and 740:
E.9. PROJECTION ON CONVEX SET 739R(
-
Page 741 and 742:
E.9. PROJECTION ON CONVEX SET 741E.
-
Page 743 and 744:
E.9. PROJECTION ON CONVEX SET 743E.
-
Page 745 and 746:
E.9. PROJECTION ON CONVEX SET 745Un
-
Page 747 and 748:
E.9. PROJECTION ON CONVEX SET 747ac
-
Page 749 and 750:
E.10. ALTERNATING PROJECTION 749bC
-
Page 751 and 752:
E.10. ALTERNATING PROJECTION 7510
-
Page 753 and 754:
E.10. ALTERNATING PROJECTION 753E.1
-
Page 755 and 756:
E.10. ALTERNATING PROJECTION 755y 2
-
Page 757 and 758:
E.10. ALTERNATING PROJECTION 757Def
-
Page 759 and 760:
E.10. ALTERNATING PROJECTION 759Dis
-
Page 761 and 762:
E.10. ALTERNATING PROJECTION 761mat
-
Page 763 and 764:
E.10. ALTERNATING PROJECTION 763K
-
Page 765 and 766:
E.10. ALTERNATING PROJECTION 765E.1
-
Page 767 and 768:
E.10. ALTERNATING PROJECTION 767E.1
-
Page 769 and 770:
Appendix FNotation and a few defini
-
Page 771 and 772:
771A ij or A(i, j) , ij th entry of
-
Page 773 and 774:
773⊞orthogonal vector sum of sets
-
Page 775 and 776:
775x +vector x whose negative entri
-
Page 777 and 778:
777X point list ((76) having cardin
-
Page 779 and 780:
779SDPSVDSNRdBEDMS n 1S n hS n⊥hS
-
Page 781 and 782:
781vectorentrycubixquartixfeasible
-
Page 783 and 784:
783Oorder of magnitude information
-
Page 785 and 786:
785cofmatrix of cofactors correspon
-
Page 787 and 788:
Bibliography[1] Edwin A. Abbott. Fl
-
Page 789 and 790:
BIBLIOGRAPHY 789[23] Dror Baron, Mi
-
Page 791 and 792:
BIBLIOGRAPHY 791[49] Leonard M. Blu
-
Page 793 and 794:
BIBLIOGRAPHY 793[74] Yves Chabrilla
-
Page 795 and 796:
BIBLIOGRAPHY 795[102] Etienne de Kl
-
Page 797 and 798:
BIBLIOGRAPHY 797[129] Carl Eckart a
-
Page 799 and 800:
BIBLIOGRAPHY 799[154] James Gleik.
-
Page 801 and 802:
BIBLIOGRAPHY 801[182] Johan Håstad
-
Page 803 and 804:
BIBLIOGRAPHY 803[212] Viren Jain an
-
Page 805 and 806:
BIBLIOGRAPHY 805[237] Monique Laure
-
Page 807 and 808:
BIBLIOGRAPHY 807[265] Sunderarajan
-
Page 809 and 810:
BIBLIOGRAPHY 809Notes in Computer S
-
Page 811 and 812:
BIBLIOGRAPHY 811[319] Anthony Man-C
-
Page 813 and 814:
BIBLIOGRAPHY 813[346] Pham Dinh Tao
-
Page 815 and 816:
BIBLIOGRAPHY 815[375] Bernard Widro
-
Page 817:
Index∅, see empty set0-norm, 229,
-
Page 821 and 822:
INDEX 821circular, 130construction,
-
Page 823 and 824:
INDEX 823measure, 295convex, 21, 36
-
Page 825 and 826:
INDEX 825duality, 160, 410gap, 161,
-
Page 827 and 828:
INDEX 827positive semidefinite, 284
-
Page 829 and 830:
INDEX 829point, 39, 62positive semi
-
Page 831 and 832:
INDEX 831isomorphic, 52, 56, 59, 77
-
Page 833 and 834:
INDEX 833nonsymmetric, 703range, 70
-
Page 835 and 836:
INDEX 835Muller, 624multidimensiona
-
Page 837 and 838:
INDEX 837projection on, 739, 745tra
-
Page 839 and 840:
INDEX 839pseudoinverse, 701trace, 5
-
Page 841 and 842:
INDEX 841coordinate system, 156line
-
Page 843 and 844:
INDEX 843nonconvex, 40, 97-101nulls
-
Page 845 and 846:
INDEX 845faces, 106intersection, 10
-
Page 847 and 848:
INDEX 847projection, 517symmetric,
-
Page 849 and 850:
849
-
Page 852:
Convex Optimization & Euclidean Dis
Change language