v2009.01.01 - Convex Optimization

More documents

Recommendations

Info

366 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX (a) (c) x 3 x 4 x 5 x 5 x 6 √ d13√d14 x 3 ˇx 3 ˇx 4 √ d12 ˇx 2 x 1 x 1 x 2 x 4 x 1 x 1 Figure 97: (a) Given three distances indicated with absolute point positions ˇx 2 , ˇx 3 , ˇx 4 known and noncollinear, absolute position of x 1 in R 2 can be precisely and uniquely determined by trilateration; solution to a system of nonlinear equations. Dimensionless EDM graphs (b) (c) (d) represent EDMs in various states of completion. Line segments represent known absolute distances and may cross without vertex at intersection. (b) Four-point list must always be embeddable in affine subset having dimension rankVN TDV N not exceeding 3. To determine relative position of x 2 ,x 3 ,x 4 , triangle inequality is necessary and sufficient (5.14.1). Knowing all distance information, then (by injectivity of D (5.6)) point position x 1 is uniquely determined to within an isometry in any dimension. (c) When fifth point is introduced, only distances to x 3 ,x 4 ,x 5 are required to determine relative position of x 2 in R 2 . Graph represents first instance of missing distance information; √ d 12 . (d) Three distances are absent ( √ d 12 , √ d 13 , √ d 23 ) from complete set of interpoint distances, yet unique isometric reconstruction (5.4.2.2.6) of six points in R 2 is certain. (b) (d)
5.4. EDM DEFINITION 367 where Φ ij = (e i − e j )(e i − e j ) T ∈ S N + (796) and where the constraint on distance-square ďi1 is equivalently written as a constraint on the Gram matrix via (809). There are 9 independent affine equality constraints on that Gram matrix while the sensor is constrained only by dimensioning to lie in R 2 . Although trG the objective of minimization 5.12 insures a solution on the boundary of positive semidefinite cone S 4 + , for this problem, we claim that the set of feasible Gram matrices forms a line (2.5.1.1) in isomorphic R 10 tangent (2.1.7.2) to the positive semidefinite cone boundary. (5.4.2.2.5, confer4.2.1.3) By Schur complement (A.4,2.9.1.0.1), any feasible G and X provide G ≽ X T X (842) which is a convex relaxation of the desired (nonconvex) equality constraint [ ] [ ] I X I [I X ] X T = G X T (843) expected positive semidefinite rank-2 under noiseless conditions. by (1391), the relaxation admits But, (3 ≥) rankG ≥ rankX (844) (a third dimension corresponding to an intersection of three spheres, not circles, were there noise). If rank of an optimal solution equals 2, [ ] I X rank ⋆ X ⋆T G ⋆ = 2 (845) then G ⋆ = X ⋆T X ⋆ by Theorem A.4.0.0.4. As posed, this localization problem does not require affinely independent (Figure 23, three noncollinear) anchors. Assuming the anchors exhibit no rotational or reflective symmetry in their affine hull (5.5.2) and assuming the sensor x 1 lies in that affine hull, then sensor position solution x ⋆ 1= X ⋆ (:, 1) is unique under noiseless measurement. [278] 5.12 Trace (trG = 〈I , G〉) minimization is a heuristic for rank minimization. (7.2.2.1) It may be interpreted as squashing G which is bounded below by X T X as in (842); id est, G−X T X ≽ 0 ⇒ trG ≥ tr X T X (1389). δ(G−X T X)=0 ⇔ G=X T X (A.7.2) ⇒ tr G = tr X T X which is a condition necessary for equality.
Page 1 and 2:
DATTORRO CONVEX OPTIMIZATION & EUCL
Page 3 and 4:
Convex Optimization & Euclidean Dis
Page 5 and 6:
for Jennie Columba ♦ Antonio ♦
Page 7 and 8:
Prelude The constant demands of my
Page 9 and 10:
Page 11 and 12:
CONVEX OPTIMIZATION & EUCLIDEAN DIS
Page 13 and 14:
List of Figures 1 Overview 19 1 Ori
Page 15 and 16:
LIST OF FIGURES 15 3 Geometry of co
Page 17 and 18:
LIST OF FIGURES 17 126 Decomposing
Page 19 and 20:
Chapter 1 Overview Convex Optimizat
Page 21 and 22:
ˇx 4 ˇx 3 ˇx 2 Figure 2: Applica
Page 23 and 24:
23 Figure 4: This coarsely discreti
Page 25 and 26:
ases (biorthogonal expansion). We e
Page 27 and 28:
27 Figure 7: These bees construct a
Page 29 and 30:
that establish its membership to th
Page 31 and 32:
31 appendices Provided so as to be
Page 33 and 34:
Chapter 2 Convex geometry Convexity
Page 35 and 36:
2.1. CONVEX SET 35 2.1.2 linear ind
Page 37 and 38:
2.1. CONVEX SET 37 2.1.6 empty set
Page 39 and 40:
2.1. CONVEX SET 39 2.1.7.1 Line int
Page 41 and 42:
2.1. CONVEX SET 41 (a) R 2 (b) R 3
Page 43 and 44:
2.1. CONVEX SET 43 This theorem in
Page 45 and 46:
2.2. VECTORIZED-MATRIX INNER PRODUC
Page 47 and 48:
Page 49 and 50:
Page 51 and 52:
Page 53 and 54:
Page 55 and 56:
2.3. HULLS 55 Figure 16: Convex hul
Page 57 and 58:
2.3. HULLS 57 The affine hull of tw
Page 59 and 60:
2.3. HULLS 59 2.3.2 Convex hull The
Page 61 and 62:
2.3. HULLS 61 In case k = N , the F
Page 63 and 64:
2.3. HULLS 63 2.3.2.0.3 Exercise. C
Page 65 and 66:
2.3. HULLS 65 Figure 20: A simplici
Page 67 and 68:
2.4. HALFSPACE, HYPERPLANE 67 H + a
Page 69 and 70:
2.4. HALFSPACE, HYPERPLANE 69 1 1
Page 71 and 72:
2.4. HALFSPACE, HYPERPLANE 71 Recal
Page 73 and 74:
2.4. HALFSPACE, HYPERPLANE 73 C H
Page 75 and 76:
2.4. HALFSPACE, HYPERPLANE 75 2.4.2
Page 77 and 78:
2.4. HALFSPACE, HYPERPLANE 77 (conf
Page 79 and 80:
2.5. SUBSPACE REPRESENTATIONS 79 Ra
Page 81 and 82:
2.5. SUBSPACE REPRESENTATIONS 81 If
Page 83 and 84:
2.6. EXTREME, EXPOSED 83 2.6 Extrem
Page 85 and 86:
2.6. EXTREME, EXPOSED 85 A B C D Fi
Page 87 and 88:
2.7. CONES 87 2.6.1.3.1 Definition.
Page 89 and 90:
2.7. CONES 89 0 Figure 30: Boundary
Page 91 and 92:
2.7. CONES 91 2.7.2 Convex cone We
Page 93 and 94:
2.7. CONES 93 Then a pointed closed
Page 95 and 96:
2.7. CONES 95 A pointed closed conv
Page 97 and 98:
2.8. CONE BOUNDARY 97 So the ray th
Page 99 and 100:
2.8. CONE BOUNDARY 99 2.8.1.1 extre
Page 101 and 102:
2.8. CONE BOUNDARY 101 2.8.2 Expose
Page 103 and 104:
2.8. CONE BOUNDARY 103 From Theorem
Page 105 and 106:
2.9. POSITIVE SEMIDEFINITE (PSD) CO
Page 107 and 108:
Page 109 and 110:
Page 111 and 112:
Page 113 and 114:
Page 115 and 116:
Page 117 and 118:
Page 119 and 120:
Page 121 and 122:
Page 123 and 124:
Page 125 and 126:
Page 127 and 128:
2.10. CONIC INDEPENDENCE (C.I.) 127
Page 129 and 130:
Page 131 and 132:
Page 133 and 134:
2.12. CONVEX POLYHEDRA 133 It follo
Page 135 and 136:
2.12. CONVEX POLYHEDRA 135 Coeffici
Page 137 and 138:
2.12. CONVEX POLYHEDRA 137 2.12.3 U
Page 139 and 140:
2.12. CONVEX POLYHEDRA 139
Page 141 and 142:
2.13. DUAL CONE & GENERALIZED INEQU
Page 143 and 144:
Page 145 and 146:
Page 147 and 148:
Page 149 and 150:
Page 151 and 152:
Page 153 and 154:
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:
Chapter 3 Geometry of convex functi
Page 197 and 198:
3.1. CONVEX FUNCTION 197 f 1 (x) f
Page 199 and 200:
3.1. CONVEX FUNCTION 199 3.1.3 norm
Page 201 and 202:
3.1. CONVEX FUNCTION 201 A B 1 Figu
Page 203 and 204:
3.1. CONVEX FUNCTION 203 k/m 1 0.9
Page 205 and 206:
k∑ i=1 3.1. CONVEX FUNCTION 205 S
Page 207 and 208:
3.1. CONVEX FUNCTION 207 rather x >
Page 209 and 210:
3.1. CONVEX FUNCTION 209 rather ] x
Page 211 and 212:
3.1. CONVEX FUNCTION 211 3.1.6.0.2
Page 213 and 214:
3.1. CONVEX FUNCTION 213 q(x) f(x)
Page 215 and 216:
3.1. CONVEX FUNCTION 215 3.1.7.0.2
Page 217 and 218:
3.1. CONVEX FUNCTION 217 We learned
Page 219 and 220:
3.1. CONVEX FUNCTION 219 Since opti
Page 221 and 222:
3.1. CONVEX FUNCTION 221 2 1.5 1 0.
Page 223 and 224:
3.1. CONVEX FUNCTION 223 Setting th
Page 225 and 226:
3.1. CONVEX FUNCTION 225 Similarly,
Page 227 and 228:
3.1. CONVEX FUNCTION 227 For vector
Page 229 and 230:
3.1. CONVEX FUNCTION 229 This means
Page 231 and 232:
3.1. CONVEX FUNCTION 231 f(Y ) −
Page 233 and 234:
3.2. MATRIX-VALUED CONVEX FUNCTION
Page 235 and 236:
Page 237 and 238:
Page 239 and 240:
3.3. QUASICONVEX 239 exponential al
Page 241 and 242:
3.3. QUASICONVEX 241 Unlike convex
Page 243 and 244:
3.4. SALIENT PROPERTIES 243 6. (af
Page 245 and 246:
Chapter 4 Semidefinite programming
Page 247 and 248:
4.1. CONIC PROBLEM 247 where K is a
Page 249 and 250:
4.1. CONIC PROBLEM 249 4.1.1.2 Redu
Page 251 and 252:
4.1. CONIC PROBLEM 251 In any SDP f
Page 253 and 254:
4.1. CONIC PROBLEM 253 Proposition
Page 255 and 256:
4.2. FRAMEWORK 255 sets are closed
Page 257 and 258:
4.2. FRAMEWORK 257 4.2.1.1.3 Exampl
Page 259 and 260:
4.2. FRAMEWORK 259 4.2.2 Duals The
Page 261 and 262:
4.2. FRAMEWORK 261 When equality is
Page 263 and 264:
4.2. FRAMEWORK 263 The pseudoinvers
Page 265 and 266:
4.2. FRAMEWORK 265 For the data giv
Page 267 and 268:
4.2. FRAMEWORK 267 minimizes an aff
Page 269 and 270:
4.3. RANK REDUCTION 269 whose rank
Page 271 and 272:
4.3. RANK REDUCTION 271 and where m
Page 273 and 274:
4.3. RANK REDUCTION 273 4.3.3 Optim
Page 275 and 276:
4.3. RANK REDUCTION 275 Initialize:
Page 277 and 278:
4.4. RANK-CONSTRAINED SEMIDEFINITE
Page 279 and 280:
Page 281 and 282:
Page 283 and 284:
Page 285 and 286:
Page 287 and 288:
Page 289 and 290:
Page 291 and 292:
Page 293 and 294:
Page 295 and 296:
4.5. CONSTRAINING CARDINALITY 295 m
Page 297 and 298:
4.5. CONSTRAINING CARDINALITY 297 m
Page 299 and 300:
4.5. CONSTRAINING CARDINALITY 299 a
Page 301 and 302:
4.5. CONSTRAINING CARDINALITY 301 f
Page 303 and 304:
4.5. CONSTRAINING CARDINALITY 303 n
Page 305 and 306:
4.5. CONSTRAINING CARDINALITY 305 W
Page 307 and 308:
4.5. CONSTRAINING CARDINALITY 307 t
Page 309 and 310:
4.6. CARDINALITY AND RANK CONSTRAIN
Page 311 and 312:
Page 313 and 314:
Page 315 and 316: 4.6. CARDINALITY AND RANK CONSTRAIN
Page 341 and 342: 4.7. CONVEX ITERATION RANK-1 341 fi
Page 343 and 344: 4.7. CONVEX ITERATION RANK-1 343 Gi
Page 345 and 346: Chapter 5 Euclidean Distance Matrix
Page 347 and 348: 5.2. FIRST METRIC PROPERTIES 347 co
Page 349 and 350: 5.3. ∃ FIFTH EUCLIDEAN METRIC PRO
Page 351 and 352: 5.3. ∃ FIFTH EUCLIDEAN METRIC PRO
Page 353 and 354: 5.4. EDM DEFINITION 353 The collect
Page 355 and 356: 5.4. EDM DEFINITION 355 5.4.2 Gram-
Page 357 and 358: 5.4. EDM DEFINITION 357 D ∈ EDM N
Page 359 and 360: 5.4. EDM DEFINITION 359 5.4.2.2.1 E
Page 361 and 362: 5.4. EDM DEFINITION 361 ten affine
Page 363 and 364: 5.4. EDM DEFINITION 363 spheres: Th
Page 365: 5.4. EDM DEFINITION 365 By eliminat
Page 369 and 370: 5.4. EDM DEFINITION 369 5.4.2.2.6 D
Page 371 and 372: 5.4. EDM DEFINITION 371 10 5 ˇx 4
Page 373 and 374: 5.4. EDM DEFINITION 373 corrected b
Page 375 and 376: 5.4. EDM DEFINITION 375 by translat
Page 377 and 378: 5.4. EDM DEFINITION 377 Crippen & H
Page 379 and 380: 5.4. EDM DEFINITION 379 where ([√
Page 381 and 382: 5.4. EDM DEFINITION 381 because (A.
Page 383 and 384: 5.5. INVARIANCE 383 5.5.1.0.1 Examp
Page 385 and 386: 5.5. INVARIANCE 385 x 2 x 2 x 3 x 1
Page 387 and 388: 5.6. INJECTIVITY OF D & UNIQUE RECO
Page 393 and 394: 5.7. EMBEDDING IN AFFINE HULL 393 5
Page 395 and 396: 5.7. EMBEDDING IN AFFINE HULL 395 F
Page 397 and 398: 5.7. EMBEDDING IN AFFINE HULL 397 5
Page 399 and 400: 5.8. EUCLIDEAN METRIC VERSUS MATRIX
Page 407 and 408: 5.9. BRIDGE: CONVEX POLYHEDRA TO ED
Page 413 and 414: 5.10. EDM-ENTRY COMPOSITION 413 of
Page 415 and 416: 5.10. EDM-ENTRY COMPOSITION 415 The
Page 417 and 418:
5.11. EDM INDEFINITENESS 417 5.11.1
Page 419 and 420:
5.11. EDM INDEFINITENESS 419 we hav
Page 421 and 422:
5.11. EDM INDEFINITENESS 421 So bec
Page 423 and 424:
5.11. EDM INDEFINITENESS 423 where
Page 425 and 426:
5.12. LIST RECONSTRUCTION 425 where
Page 427 and 428:
5.12. LIST RECONSTRUCTION 427 (a) (
Page 429 and 430:
5.13. RECONSTRUCTION EXAMPLES 429 D
Page 431 and 432:
5.13. RECONSTRUCTION EXAMPLES 431 T
Page 433 and 434:
5.13. RECONSTRUCTION EXAMPLES 433 w
Page 435 and 436:
5.14. FIFTH PROPERTY OF EUCLIDEAN M
Page 437 and 438:
Page 439 and 440:
Page 441 and 442:
Page 443 and 444:
Page 445 and 446:
Chapter 6 Cone of distance matrices
Page 447 and 448:
6.1. DEFINING EDM CONE 447 6.1 Defi
Page 449 and 450:
6.2. POLYHEDRAL BOUNDS 449 This con
Page 451 and 452:
6.3. √ EDM CONE IS NOT CONVEX 451
Page 453 and 454:
6.4. A GEOMETRY OF COMPLETION 453 (
Page 455 and 456:
6.4. A GEOMETRY OF COMPLETION 455 (
Page 457 and 458:
6.4. A GEOMETRY OF COMPLETION 457 F
Page 459 and 460:
6.5. EDM DEFINITION IN 11 T 459 by
Page 461 and 462:
6.5. EDM DEFINITION IN 11 T 461 6.5
Page 463 and 464:
6.5. EDM DEFINITION IN 11 T 463 1 0
Page 465 and 466:
6.5. EDM DEFINITION IN 11 T 465 6.5
Page 467 and 468:
6.6. CORRESPONDENCE TO PSD CONE S N
Page 469 and 470:
Page 471 and 472:
Page 473 and 474:
6.7. VECTORIZATION & PROJECTION INT
Page 475 and 476:
6.7. VECTORIZATION & PROJECTION INT
Page 477 and 478:
6.8. DUAL EDM CONE 477 When the Fin
Page 479 and 480:
6.8. DUAL EDM CONE 479 Proof. First
Page 481 and 482:
6.8. DUAL EDM CONE 481 EDM 2 = S 2
Page 483 and 484:
6.8. DUAL EDM CONE 483 whose veraci
Page 485 and 486:
6.8. DUAL EDM CONE 485 6.8.1.3.1 Ex
Page 487 and 488:
6.8. DUAL EDM CONE 487 has dual aff
Page 489 and 490:
6.8. DUAL EDM CONE 489 6.8.1.7 Scho
Page 491 and 492:
6.9. THEOREM OF THE ALTERNATIVE 491
Page 493 and 494:
6.10. POSTSCRIPT 493 When D is an E
Page 495 and 496:
Chapter 7 Proximity problems In sum
Page 497 and 498:
497 project on the subspace, then p
Page 499 and 500:
499 H S N h 0 EDM N K = S N h ∩ R
Page 501 and 502:
501 7.0.3 Problem approach Problems
Page 503 and 504:
7.1. FIRST PREVALENT PROBLEM: 503 f
Page 505 and 506:
7.1. FIRST PREVALENT PROBLEM: 505 7
Page 507 and 508:
7.1. FIRST PREVALENT PROBLEM: 507 d
Page 509 and 510:
7.1. FIRST PREVALENT PROBLEM: 509 7
Page 511 and 512:
7.1. FIRST PREVALENT PROBLEM: 511 w
Page 513 and 514:
7.1. FIRST PREVALENT PROBLEM: 513 T
Page 515 and 516:
7.2. SECOND PREVALENT PROBLEM: 515
Page 517 and 518:
Page 519 and 520:
Page 521 and 522:
Page 523 and 524:
Page 525 and 526:
7.3. THIRD PREVALENT PROBLEM: 525 g
Page 527 and 528:
7.3. THIRD PREVALENT PROBLEM: 527 w
Page 529 and 530:
7.3. THIRD PREVALENT PROBLEM: 529 7
Page 531 and 532:
7.3. THIRD PREVALENT PROBLEM: 531 7
Page 533 and 534:
7.3. THIRD PREVALENT PROBLEM: 533 O
Page 535 and 536:
7.4. CONCLUSION 535 The rank constr
Page 537 and 538:
Appendix A Linear algebra A.1 Main-
Page 539 and 540:
A.1. MAIN-DIAGONAL δ OPERATOR, λ
Page 541 and 542:
A.1. MAIN-DIAGONAL δ OPERATOR, λ
Page 543 and 544:
A.2. SEMIDEFINITENESS: DOMAIN OF TE
Page 545 and 546:
A.3. PROPER STATEMENTS 545 (AB) T
Page 547 and 548:
A.3. PROPER STATEMENTS 547 A.3.1 Se
Page 549 and 550:
A.3. PROPER STATEMENTS 549 For A di
Page 551 and 552:
A.3. PROPER STATEMENTS 551 Diagonal
Page 553 and 554:
A.3. PROPER STATEMENTS 553 For A,B
Page 555 and 556:
A.3. PROPER STATEMENTS 555 A.3.1.0.
Page 557 and 558:
A.4. SCHUR COMPLEMENT 557 A.4 Schur
Page 559 and 560:
A.4. SCHUR COMPLEMENT 559 A.4.0.0.2
Page 561 and 562:
A.5. EIGEN DECOMPOSITION 561 When B
Page 563 and 564:
A.5. EIGEN DECOMPOSITION 563 dim N(
Page 565 and 566:
A.6. SINGULAR VALUE DECOMPOSITION,
Page 567 and 568:
A.6. SINGULAR VALUE DECOMPOSITION,
Page 569 and 570:
A.7. ZEROS 569 Given symmetric matr
Page 571 and 572:
A.7. ZEROS 571 (TRANSPOSE.) Likewis
Page 573 and 574:
A.7. ZEROS 573 For X,A∈ S M + [31
Page 575 and 576:
A.7. ZEROS 575 A.7.5.0.1 Propositio
Page 577 and 578:
Appendix B Simple matrices Mathemat
Page 579 and 580:
B.1. RANK-ONE MATRIX (DYAD) 579 R(v
Page 581 and 582:
B.1. RANK-ONE MATRIX (DYAD) 581 ran
Page 583 and 584:
B.2. DOUBLET 583 R([u v ]) R(Π)= R
Page 585 and 586:
B.3. ELEMENTARY MATRIX 585 If λ
Page 587 and 588:
B.4. AUXILIARY V -MATRICES 587 the
Page 589 and 590:
B.4. AUXILIARY V -MATRICES 589 18.
Page 591 and 592:
B.5. ORTHOGONAL MATRIX 591 B.5 Orth
Page 593 and 594:
B.5. ORTHOGONAL MATRIX 593 Figure 1
Page 595 and 596:
Appendix C Some analytical optimal
Page 597 and 598:
C.2. TRACE, SINGULAR AND EIGEN VALU
Page 599 and 600:
Page 601 and 602:
Page 603 and 604:
C.3. ORTHOGONAL PROCRUSTES PROBLEM
Page 605 and 606:
C.4. TWO-SIDED ORTHOGONAL PROCRUSTE
Page 607 and 608:
C.4. TWO-SIDED ORTHOGONAL PROCRUSTE
Page 609 and 610:
Appendix D Matrix calculus From too
Page 611 and 612:
D.1. DIRECTIONAL DERIVATIVE, TAYLOR
Page 613 and 614:
Page 615 and 616:
Page 617 and 618:
Page 619 and 620:
Page 621 and 622:
Page 623 and 624:
Page 625 and 626:
Page 627 and 628:
Page 629 and 630:
Page 631 and 632:
D.2. TABLES OF GRADIENTS AND DERIVA
Page 633 and 634:
Page 635 and 636:
Page 637 and 638:
Page 639 and 640:
Appendix E Projection For any A∈
Page 641 and 642:
641 U T = U † for orthonormal (in
Page 643 and 644:
E.1. IDEMPOTENT MATRICES 643 where
Page 645 and 646:
E.1. IDEMPOTENT MATRICES 645 order,
Page 647 and 648:
E.1. IDEMPOTENT MATRICES 647 When t
Page 649 and 650:
E.3. SYMMETRIC IDEMPOTENT MATRICES
Page 651 and 652:
Page 653 and 654:
Page 655 and 656:
E.5. PROJECTION EXAMPLES 655 E.4.0.
Page 657 and 658:
E.5. PROJECTION EXAMPLES 657 a ∗
Page 659 and 660:
E.5. PROJECTION EXAMPLES 659 E.5.0.
Page 661 and 662:
E.6. VECTORIZATION INTERPRETATION,
Page 663 and 664:
Page 665 and 666:
Page 667 and 668:
Page 669 and 670:
E.7. ON VECTORIZED MATRICES OF HIGH
Page 671 and 672:
E.7. ON VECTORIZED MATRICES OF HIGH
Page 673 and 674:
E.8. RANGE/ROWSPACE INTERPRETATION
Page 675 and 676:
E.9. PROJECTION ON CONVEX SET 675 A
Page 677 and 678:
E.9. PROJECTION ON CONVEX SET 677 W
Page 679 and 680:
E.9. PROJECTION ON CONVEX SET 679 P
Page 681 and 682:
E.9. PROJECTION ON CONVEX SET 681 E
Page 683 and 684:
E.9. PROJECTION ON CONVEX SET 683 T
Page 685 and 686:
E.9. PROJECTION ON CONVEX SET 685
Page 687 and 688:
E.10. ALTERNATING PROJECTION 687 E.
Page 689 and 690:
E.10. ALTERNATING PROJECTION 689 b
Page 691 and 692:
E.10. ALTERNATING PROJECTION 691 a
Page 693 and 694:
E.10. ALTERNATING PROJECTION 693 (a
Page 695 and 696:
E.10. ALTERNATING PROJECTION 695 wh
Page 697 and 698:
Page 699 and 700:
E.10. ALTERNATING PROJECTION 699 10
Page 701 and 702:
Page 703 and 704:
E.10. ALTERNATING PROJECTION 703 E
Page 705 and 706:
Appendix F Notation and a few defin
Page 707 and 708:
707 a.i. c.i. l.i. w.r.t affinely i
Page 709 and 710:
709 is or ← → t → 0 + as in
Page 711 and 712:
711 ∑ π(γ) Ξ Π ∏ ψ(Z) D D
Page 713 and 714:
713 R m×n Euclidean vector space o
Page 715 and 716:
715 H − H + ∂H ∂H ∂H −
Page 717 and 718:
717 O O sort-index matrix order of
Page 719 and 720:
(x,y) angle between vectors x and y
Page 721 and 722:
Bibliography [1] Suliman Al-Homidan
Page 723 and 724:
BIBLIOGRAPHY 723 [24] Alexander I.
Page 725 and 726:
BIBLIOGRAPHY 725 [52] Stephen Boyd,
Page 727 and 728:
BIBLIOGRAPHY 727 [78] Frank Critchl
Page 729 and 730:
BIBLIOGRAPHY 729 [105] Richard L. D
Page 731 and 732:
BIBLIOGRAPHY 731 [132] Michel X. Go
Page 733 and 734:
BIBLIOGRAPHY 733 [162] T. Herrmann,
Page 735 and 736:
BIBLIOGRAPHY 735 [191] Mark Kahrs a
Page 737 and 738:
BIBLIOGRAPHY 737 [220] K. V. Mardia
Page 739 and 740:
BIBLIOGRAPHY 739 [250] Pythagoras P
Page 741 and 742:
BIBLIOGRAPHY 741 [277] Anthony Man-
Page 743 and 744:
BIBLIOGRAPHY 743 [306] Michael W. T
Page 745 and 746:
[333] Margaret H. Wright. The inter
Page 747 and 748:
Index 0-norm, 203, 261, 294, 296, 2
Page 749 and 750:
INDEX 749 product, 43, 92, 147, 254
Page 751 and 752:
INDEX 751 coordinates, 140, 170, 17
Page 753 and 754:
INDEX 753 affine dimension, 485 fea
Page 755 and 756:
INDEX 755 affine, 209 nonlinear, 19
Page 757 and 758:
INDEX 757 of point, 37 ray, 90 rela
Page 759 and 760:
INDEX 759 normal, 47, 548, 563 norm
Page 761 and 762:
INDEX 761 strictly, 515, 520 functi
Page 763 and 764:
INDEX 763 vector, 45, 241, 248, 325
Page 765 and 766:
INDEX 765 convex envelope, see conv
Page 767 and 768:
INDEX 767 cone, 418, 420, 507 dual,
Page 769 and 770:
INDEX 769 trilateration, 21, 42, 36
Page 772:
show all

v2009.01.01 - Convex Optimization

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

Delete template?

Save as template?