v2010.10.26 - Convex Optimization

More documents

Recommendations

Info

276 CHAPTER 4. SEMIDEFINITE PROGRAMMING⎡A svec X = ⎣〈A 1 , X〉.〈A m , X〉(651)∑svec −1 (A T y) = m y i A iThe vector inner-product for matrices is defined in the Euclidean/Frobeniussense in the isomorphic vector space R n(n+1)/2 ; id est,i=1〈C , X〉 tr(C T X) = svec(C) T svec X (38)where svec X defined by (56) denotes symmetric vectorization.In a national planning problem of some size, one may easily run into severalhundred variables and perhaps a hundred or more degrees of freedom. ...Itshould always be remembered that any mathematical method and particularlymethods in linear programming must be judged with reference to the type ofcomputing machinery available. Our outlook may perhaps be changed whenwe get used to the super modern, high capacity electronic computor that willbe available here from the middle of next year. −Ragnar Frisch [147]Linear programming, invented by Dantzig in 1947 [94], is now integralto modern technology. The same cannot yet be said of semidefiniteprogramming whose roots trace back to systems of positive semidefinitelinear inequalities studied by Bellman & Fan in 1963. [31] [102] Interior-pointmethods for numerical solution can be traced back to the logarithmic barrierof Frisch in 1954 and Fiacco & McCormick in 1968 [141]. Karmarkar’spolynomial-time interior-point method sparked a log-barrier renaissancein 1984, [275,11] [381] [363] [277, p.3] but numerical performanceof contemporary general-purpose semidefinite program solvers remainslimited: Computational intensity for dense systems varies as O(m 2 n)(m constraints ≪ n variables) based on interior-point methods that producesolutions no more relatively accurate than 1E-8. There are no solverscapable of handling in excess of n=100,000 variables without significant,sometimes crippling, loss of precision or time. 4.3 [35] [276, p.258] [67, p.3]4.3 Heuristics are not ruled out by SIOPT; indeed I would suspect that most successfulmethods have (appropriately described) heuristics under the hood - my codes certainly do....Of course, there are still questions relating to high-accuracy and speed, but for manyapplications a few digits of accuracy suffices and overnight runs for non-real-time deliveryis acceptable.−Nicholas I. M. Gould, Editor-in-Chief, SIOPT⎤⎦
4.1. CONIC PROBLEM 277PCsemidefinite programquadratic programsecond-order cone programlinear programFigure 80: Venn diagram of programming hierarchy. Semidefinite program isa subset of convex program PC. Semidefinite program subsumes other convexprogram classes excepting geometric program. Second-order cone programand quadratic program each subsume linear program. Nonconvex program\PC comprises those for which convex equivalents have not yet been found.
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:
Page 55 and 56:
Page 57 and 58:
Page 59 and 60:
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:
Page 81 and 82:
2.4. HALFSPACE, HYPERPLANE 81tradit
Page 83 and 84:
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:
Page 117 and 118:
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 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 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: 4.1. CONIC PROBLEM 275(confer p.162
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 327 and 328:
4.4. RANK-CONSTRAINED SEMIDEFINITE
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?