v2010.10.26 - Convex Optimization

More documents

Recommendations

Info

66 CHAPTER 2. CONVEX GEOMETRYsvec ∂ S 2 +γI[ α ββ γ]α√2βFigure 22: Two Fantopes. Circle (radius 1/ √ 2), shown here on boundary ofpositive semidefinite cone S 2 + in isometrically isomorphic R 3 from Figure 43,comprises boundary of a Fantope (90) in this dimension (k = 1, N = 2). Lonepoint illustrated is identity matrix I , interior to PSD cone, and is thatFantope corresponding to k = 2, N = 2. (View is from inside PSD conelooking toward origin.)
2.3. HULLS 672.3.2.0.2 Example. Nuclear norm ball: convex hull of rank-1 matrices.From (91), in Example 2.3.2.0.1, we learn that the convex hull of normalizedsymmetric rank-1 matrices is a slice of the positive semidefinite cone.In2.9.2.7 we find the convex hull of all symmetric rank-1 matrices to bethe entire positive semidefinite cone.In the present example we abandon symmetry; instead posing, what isthe convex hull of bounded nonsymmetric rank-1 matrices:conv{uv T | ‖uv T ‖ ≤ 1, u∈ R m , v ∈ R n } = {X ∈ R m×n | ∑ iσ(X) i ≤ 1} (94)where σ(X) is a vector of singular values. (Since ‖uv T ‖= ‖u‖‖v‖ (1610),norm of each vector constituting a dyad uv T (B.1) in the hull is effectivelybounded above by 1.)Proof. (⇐) Suppose ∑ σ(X) i ≤ 1. As inA.6, define a singularvalue decomposition: X = UΣV T where U = [u 1 ... u min{m,n} ]∈ R m×min{m,n} ,V∑= [v 1 ... v min{m,n} ]∈ R n×min{m,n} , and whose sum of singular values isσ(X)i = tr Σ = κ ≤ 1. Then we may write X = ∑ σ i√ √ κuiκ κvTi which is aconvex combination of dyads each of whose norm does not exceed 1. (Srebro)(⇒) Now suppose we are given a convex combination of dyadsX = ∑ ∑α i u i viT such that αi =1, α i ≥ 0 ∀i, and ‖u i vi T ‖≤1 ∀i.Then∑by triangle inequality for singular values [203, cor.3.4.3]σ(X)i ≤ ∑ σ(α i u i vi T )= ∑ α i ‖u i vi T ‖≤ ∑ α i .Given any particular dyad uv T p in the convex hull, because its polar −uv T pand every convex combination of the two belong to that hull, then the uniqueline containing those two points ±uv T p (their affine combination (78)) mustintersect the hull’s boundary at the normalized dyads {±uv T | ‖uv T ‖=1}.Any point formed by convex combination of dyads in the hull must thereforebe expressible as a convex combination of dyads on the boundary: Figure 23.conv{uv T | ‖uv T ‖ ≤ 1, u∈ R m , v ∈ R n } ≡ conv{uv T | ‖uv T ‖ = 1, u∈ R m , v ∈ R n }(95)id est, dyads may be normalized and the hull’s boundary contains them;∂{X ∈ R m×n | ∑ iσ(X) i ≤ 1} ⊇ {uv T | ‖uv T ‖ = 1, u∈ R m , v ∈ R n } (96)Normalized dyads constitute the set of extreme points (2.6.0.0.1) of thisnuclear norm ball which is, therefore, their convex hull.
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 61 and 62: 2.3. HULLS 61Figure 20: Convex hull
Page 63 and 64: 2.3. HULLS 63Aaffine hull (drawn tr
Page 65: 2.3. HULLS 65subset of the affine h
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 81 and 82: 2.4. HALFSPACE, HYPERPLANE 81tradit
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:
Page 121 and 122:
Page 123 and 124:
Page 125 and 126:
Page 127 and 128:
Page 129 and 130:
Page 131 and 132:
Page 133 and 134:
Page 135 and 136:
Page 137 and 138:
Page 139 and 140:
Page 141 and 142:
2.10. CONIC INDEPENDENCE (C.I.) 141
Page 143 and 144:
Page 145 and 146:
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:
Page 157 and 158:
Page 159 and 160:
Page 161 and 162:
Page 163 and 164:
Page 165 and 166:
Page 167 and 168:
Page 169 and 170:
Page 171 and 172:
Page 173 and 174:
Page 175 and 176:
Page 177 and 178:
Page 179 and 180:
Page 181 and 182:
Page 183 and 184:
Page 185 and 186:
Page 187 and 188:
Page 189 and 190:
Page 191 and 192:
Page 193 and 194:
Page 195 and 196:
Page 197 and 198:
Page 199 and 200:
Page 201 and 202:
Page 203 and 204:
Page 205 and 206:
Page 207 and 208:
Page 209 and 210:
Page 211 and 212:
Page 213 and 214:
Page 215 and 216:
Page 217 and 218:
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:
Page 229 and 230:
Page 231 and 232:
Page 233 and 234:
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:
Page 267 and 268:
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:
Page 313 and 314:
Page 315 and 316:
Page 317 and 318:
Page 319 and 320:
Page 321 and 322:
Page 323 and 324:
Page 325 and 326:
Page 327 and 328:
Page 329 and 330:
Page 331 and 332:
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:
Page 357 and 358:
Page 359 and 360:
Page 361 and 362:
Page 363 and 364:
Page 365 and 366:
Page 367 and 368:
Page 369 and 370:
Page 371 and 372:
Page 373 and 374:
Page 375 and 376:
Page 377 and 378:
Page 379 and 380:
Page 381 and 382:
Page 383 and 384:
Page 385 and 386:
4.7. CONSTRAINING RANK OF INDEFINIT
Page 387 and 388:
Page 389 and 390:
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:
Page 443 and 444:
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:
Page 455 and 456:
Page 457 and 458:
Page 459 and 460:
5.9. BRIDGE: CONVEX POLYHEDRA TO ED
Page 461 and 462:
Page 463 and 464:
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:
Page 491 and 492:
Page 493 and 494:
Page 495 and 496:
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:
Page 515 and 516:
Page 517 and 518:
6.6. VECTORIZATION & PROJECTION INT
Page 519 and 520:
Page 521 and 522:
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:
Page 659 and 660:
Page 661 and 662:
C.3. ORTHOGONAL PROCRUSTES PROBLEM
Page 663 and 664:
C.4. TWO-SIDED ORTHOGONAL PROCRUSTE
Page 665 and 666:
Page 667 and 668:
Page 669 and 670:
Appendix DMatrix calculusFrom too m
Page 671 and 672:
D.1. DIRECTIONAL DERIVATIVE, TAYLOR
Page 673 and 674:
Page 675 and 676:
Page 677 and 678:
Page 679 and 680:
Page 681 and 682:
Page 683 and 684:
Page 685 and 686:
Page 687 and 688:
Page 689 and 690:
Page 691 and 692:
D.2. TABLES OF GRADIENTS AND DERIVA
Page 693 and 694:
Page 695 and 696:
Page 697 and 698:
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:
Page 713 and 714:
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:
Page 727 and 728:
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:
Page 767 and 768:
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 and 818:
Index∅, see empty set0-norm, 229,
Page 819 and 820:
INDEX 819bees, 30, 413Bellman, 276b
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
show all

v2010.10.26 - Convex Optimization

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?