v2007.09.13 - Convex Optimization

More documents

Recommendations

Info

130 CHAPTER 2. CONVEX GEOMETRYS = {s | s ≽ 0, 1 T s ≤ 1}1Figure 40: Unit simplex S in R 3 is a unique solid tetrahedron, but is notregular.will contain at least one line and there can be no vertex; id est, the conecannot otherwise be pointed.For any pointed polyhedral cone, there is a one-to-one correspondence ofone-dimensional faces with extreme directions.Examples of pointed closed convex cones K are not limited to polyhedralcones: the origin, any 0-based ray in a subspace, any two-dimensionalV-shaped cone in a subspace, the Lorentz (ice-cream) cone and its polyhedralflavors, the cone of Euclidean distance matrices EDM N in S N h , the propercones: S M + in ambient S M , any orthant in R n or R m×n ; e.g., the nonnegativereal line R + in vector space R .
2.12. CONVEX POLYHEDRA 1312.12.3 Unit simplexA peculiar convex subset of the nonnegative orthant havinghalfspace-descriptionS ∆ = {s | s ≽ 0, 1 T s ≤ 1} ⊆ R n + (253)is a unique bounded convex polyhedron called unit simplex (Figure 40)having nonempty interior, n + 1 vertices, and dimension [46,2.2.4]dim S = n (254)The origin supplies one vertex while heads of the standard basis [149][247] {e i , i=1... n} in R n constitute those remaining; 2.42 thus itsvertex-description:S2.12.3.1 Simplex= conv {0, {e i , i=1... n}}= { [0 e 1 e 2 · · · e n ]a | a T 1 = 1, a ≽ 0 } (255)The unit simplex comes from a class of general polyhedra called simplex,having vertex-description: [64] [228] [277] [77]conv{x l ∈ R n } | l = 1... k+1, dim aff{x l } = k , n ≥ k (256)So defined, a simplex is a closed bounded convex set having possibly emptyinterior. Examples of simplices, by increasing affine dimension, are: a point,any line segment, any triangle and its relative interior, a general tetrahedron,polychoron, and so on.2.12.3.1.1 Definition. Simplicial cone.A polyhedral proper (2.7.2.2.1) cone K in R n is called simplicial iff Khas exactly n extreme directions; [17,II.A] equivalently, iff proper K hasexactly n linearly independent generators contained in any given set ofgenerators.△There are an infinite variety of simplicial cones in R n ; e.g., Figure 15,Figure 41, Figure 50. Any orthant is simplicial, as is any rotation thereof.2.42 In R 0 the unit simplex is the point at the origin, in R the unit simplex is the linesegment [0,1], in R 2 it is a triangle and its relative interior, in R 3 it is the convex hull ofa tetrahedron (Figure 40), in R 4 it is the convex hull of a pentatope [279], and so on.
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 191 Orion
Page 15 and 16:
LIST OF FIGURES 1559 Quadratic func
Page 17 and 18:
LIST OF FIGURES 17E Projection 5791
Page 19 and 20:
Chapter 1OverviewConvex Optimizatio
Page 21 and 22:
ˇx 4ˇx 3ˇx 2Figure 2: Applicatio
Page 23 and 24:
23Figure 4: This coarsely discretiz
Page 25 and 26:
ases (biorthogonal expansion). We e
Page 27 and 28:
27Figure 7: These bees construct a
Page 29 and 30:
its membership to the EDM cone. The
Page 31 and 32:
31appendicesProvided so as to be mo
Page 33 and 34:
Chapter 2Convex geometryConvexity h
Page 35 and 36:
2.1. CONVEX SET 35Figure 9: A slab
Page 37 and 38:
2.1. CONVEX SET 372.1.6 empty set v
Page 39 and 40:
2.1. CONVEX SET 392.1.7.1 Line inte
Page 41 and 42:
2.1. CONVEX SET 41(a)R 2(b)R 3(c)(d
Page 43 and 44:
2.1. CONVEX SET 43This theorem in c
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:
2.3. HULLS 53Figure 12: Convex hull
Page 55 and 56:
2.3. HULLS 55Aaffine hull (drawn tr
Page 57 and 58:
2.3. HULLS 57The union of relative
Page 59 and 60:
2.4. HALFSPACE, HYPERPLANE 59of dim
Page 61 and 62:
2.4. HALFSPACE, HYPERPLANE 61H +ay
Page 63 and 64:
2.4. HALFSPACE, HYPERPLANE 63Inters
Page 65 and 66:
2.4. HALFSPACE, HYPERPLANE 65Conver
Page 67 and 68:
2.4. HALFSPACE, HYPERPLANE 67A 1A 2
Page 69 and 70:
2.4. HALFSPACE, HYPERPLANE 69tradit
Page 71 and 72:
2.4. HALFSPACE, HYPERPLANE 71There
Page 73 and 74:
2.5. SUBSPACE REPRESENTATIONS 732.5
Page 75 and 76:
2.5. SUBSPACE REPRESENTATIONS 752.5
Page 77 and 78:
2.6. EXTREME, EXPOSED 77In other wo
Page 79 and 80: 2.6. EXTREME, EXPOSED 792.6.1 Expos
Page 81 and 82: 2.7. CONES 812.6.1.3.1 Definition.
Page 83 and 84: 2.7. CONES 830Figure 24: Boundary o
Page 85 and 86: 2.7. CONES 852.7.2 Convex coneWe ca
Page 87 and 88: 2.7. CONES 87Thus the simplest and
Page 89 and 90: 2.7. CONES 89nomenclature generaliz
Page 91 and 92: 2.8. CONE BOUNDARY 91Proper cone {0
Page 93 and 94: 2.8. CONE BOUNDARY 93the same extre
Page 96 and 97: 96 CHAPTER 2. CONVEX GEOMETRYBCADFi
Page 98 and 99: 98 CHAPTER 2. CONVEX GEOMETRYThe po
Page 100 and 101: 100 CHAPTER 2. CONVEX GEOMETRY2.9.0
Page 102 and 103: 102 CHAPTER 2. CONVEX GEOMETRYwhere
Page 104 and 105: 104 CHAPTER 2. CONVEX GEOMETRY√2
Page 106 and 107: 106 CHAPTER 2. CONVEX GEOMETRYwhich
Page 108 and 109: 108 CHAPTER 2. CONVEX GEOMETRY2.9.2
Page 110: 110 CHAPTER 2. CONVEX GEOMETRYA con
Page 115 and 116: 2.9. POSITIVE SEMIDEFINITE (PSD) CO
Page 121 and 122: 2.10. CONIC INDEPENDENCE (C.I.) 121
Page 127 and 128: 2.12. CONVEX POLYHEDRA 127It follow
Page 129: 2.12. CONVEX POLYHEDRA 129Coefficie
Page 133 and 134: 2.12. CONVEX POLYHEDRA 133
Page 135 and 136: 2.13. DUAL CONE & GENERALIZED INEQU
Page 181 and 182:
2.13. DUAL CONE & GENERALIZED INEQU
Page 183 and 184:
Chapter 3Geometry of convex functio
Page 185 and 186:
3.1. CONVEX FUNCTION 185f 1 (x)f 2
Page 187 and 188:
3.1. CONVEX FUNCTION 1873.1.3 norm
Page 189 and 190:
3.1. CONVEX FUNCTION 189where the n
Page 191 and 192:
3.1. CONVEX FUNCTION 191where k ∈
Page 193 and 194:
3.1. CONVEX FUNCTION 193f(z)Az 2z 1
Page 195 and 196:
3.1. CONVEX FUNCTION 195{a T z 1 +
Page 197 and 198:
3.1. CONVEX FUNCTION 197When an epi
Page 199 and 200:
3.1. CONVEX FUNCTION 199orthonormal
Page 201 and 202:
3.1. CONVEX FUNCTION 201[30,1.1] Ex
Page 203 and 204:
3.1. CONVEX FUNCTION 20321.510.5Y 2
Page 205 and 206:
3.1. CONVEX FUNCTION 2053.1.8.0.1 E
Page 207 and 208:
3.1. CONVEX FUNCTION 207This equiva
Page 209 and 210:
3.1. CONVEX FUNCTION 2093.1.8.1 mon
Page 211 and 212:
3.1. CONVEX FUNCTION 211[ Yt]∈ ep
Page 213 and 214:
3.1. CONVEX FUNCTION 213→Y −Xwh
Page 215 and 216:
3.2. MATRIX-VALUED CONVEX FUNCTION
Page 217 and 218:
Page 219 and 220:
Page 221 and 222:
3.3. QUASICONVEX 221A quasiconcave
Page 223 and 224:
3.4. SALIENT PROPERTIES 2236.A nonn
Page 225 and 226:
Chapter 4Semidefinite programmingPr
Page 227 and 228:
4.1. CONIC PROBLEM 227where K is a
Page 229 and 230:
4.1. CONIC PROBLEM 229C0PΓ 1Γ 2S+
Page 231 and 232:
4.1. CONIC PROBLEM 231faces of S 3
Page 233 and 234:
4.1. CONIC PROBLEM 2334.1.1.3 Previ
Page 235 and 236:
4.2. FRAMEWORK 235Equivalently, pri
Page 237 and 238:
4.2. FRAMEWORK 237is positive semid
Page 239 and 240:
4.2. FRAMEWORK 239Optimal value of
Page 241 and 242:
4.2. FRAMEWORK 2414.2.3.0.2 Example
Page 243 and 244:
4.2. FRAMEWORK 243where δ is the m
Page 245 and 246:
4.2. FRAMEWORK 2454.2.3.0.3 Example
Page 247 and 248:
4.3. RANK REDUCTION 2474.3 Rank red
Page 249 and 250:
4.3. RANK REDUCTION 249A rank-reduc
Page 251 and 252:
4.3. RANK REDUCTION 251(t ⋆ i)
Page 253 and 254:
4.3. RANK REDUCTION 2534.3.3.0.1 Ex
Page 255 and 256:
4.3. RANK REDUCTION 2554.3.3.0.2 Ex
Page 257 and 258:
4.4. RANK-CONSTRAINED SEMIDEFINITE
Page 259 and 260:
Page 261 and 262:
Page 263 and 264:
Page 265 and 266:
Page 267 and 268:
Page 269 and 270:
Page 271 and 272:
Page 273 and 274:
Page 275 and 276:
Page 277 and 278:
Page 279 and 280:
Page 281 and 282:
Page 283 and 284:
Page 285 and 286:
Page 287 and 288:
Page 289 and 290:
Chapter 5Euclidean Distance MatrixT
Page 291 and 292:
5.2. FIRST METRIC PROPERTIES 291cor
Page 293 and 294:
5.3. ∃ FIFTH EUCLIDEAN METRIC PRO
Page 295 and 296:
5.3. ∃ FIFTH EUCLIDEAN METRIC PRO
Page 297 and 298:
5.4. EDM DEFINITION 297The collecti
Page 299 and 300:
5.4. EDM DEFINITION 2995.4.2 Gram-f
Page 301 and 302:
5.4. EDM DEFINITION 301D ∈ EDM N
Page 303 and 304:
5.4. EDM DEFINITION 3035.4.2.2.1 Ex
Page 305 and 306:
5.4. EDM DEFINITION 305ten affine e
Page 307 and 308:
5.4. EDM DEFINITION 307spheres:Then
Page 309 and 310:
5.4. EDM DEFINITION 309By eliminati
Page 311 and 312:
5.4. EDM DEFINITION 311whereΦ ij =
Page 313 and 314:
5.4. EDM DEFINITION 3135.4.2.2.5 De
Page 315 and 316:
5.4. EDM DEFINITION 315105ˇx 4ˇx
Page 317 and 318:
5.4. EDM DEFINITION 317corrected by
Page 319 and 320:
5.4. EDM DEFINITION 319aptly be app
Page 321 and 322:
5.4. EDM DEFINITION 321As before, a
Page 323 and 324:
5.4. EDM DEFINITION 323where ([√t
Page 325 and 326:
5.4. EDM DEFINITION 325because (A.3
Page 327 and 328:
5.5. INVARIANCE 3275.5.1.0.1 Exampl
Page 329 and 330:
5.6. INJECTIVITY OF D & UNIQUE RECO
Page 331 and 332:
Page 333 and 334:
Page 335 and 336:
5.7. EMBEDDING IN AFFINE HULL 3355.
Page 337 and 338:
5.7. EMBEDDING IN AFFINE HULL 337Fo
Page 339 and 340:
5.7. EMBEDDING IN AFFINE HULL 3395.
Page 341 and 342:
5.8. EUCLIDEAN METRIC VERSUS MATRIX
Page 343 and 344:
Page 345 and 346:
Page 347 and 348:
Page 349 and 350:
5.9. BRIDGE: CONVEX POLYHEDRA TO ED
Page 351 and 352:
Page 353 and 354:
Page 355 and 356:
Page 357 and 358:
5.10. EDM-ENTRY COMPOSITION 357(ii)
Page 359 and 360:
5.11. EDM INDEFINITENESS 3595.11.1
Page 361 and 362:
5.11. EDM INDEFINITENESS 361(confer
Page 363 and 364:
5.11. EDM INDEFINITENESS 363we have
Page 365 and 366:
5.11. EDM INDEFINITENESS 365For pre
Page 367 and 368:
5.12. LIST RECONSTRUCTION 367where
Page 369 and 370:
5.12. LIST RECONSTRUCTION 369(a)(c)
Page 371 and 372:
5.13. RECONSTRUCTION EXAMPLES 371D
Page 373 and 374:
5.13. RECONSTRUCTION EXAMPLES 373Th
Page 375 and 376:
5.13. RECONSTRUCTION EXAMPLES 375wh
Page 377 and 378:
5.14. FIFTH PROPERTY OF EUCLIDEAN M
Page 379 and 380:
Page 381 and 382:
Page 383 and 384:
Page 385 and 386:
Page 387 and 388:
Chapter 6EDM coneFor N > 3, the con
Page 389 and 390:
6.1. DEFINING EDM CONE 3896.1 Defin
Page 391 and 392:
6.2. POLYHEDRAL BOUNDS 391This cone
Page 393 and 394:
6.3.√EDM CONE IS NOT CONVEX 393N
Page 395 and 396:
6.4. A GEOMETRY OF COMPLETION 3956.
Page 397 and 398:
6.4. A GEOMETRY OF COMPLETION 397(a
Page 399 and 400:
6.4. A GEOMETRY OF COMPLETION 399Fi
Page 401 and 402:
6.5. EDM DEFINITION IN 11 T 401and
Page 403 and 404:
6.5. EDM DEFINITION IN 11 T 403then
Page 405 and 406:
6.5. EDM DEFINITION IN 11 T 4056.5.
Page 407 and 408:
6.5. EDM DEFINITION IN 11 T 407D =
Page 409 and 410:
6.6. CORRESPONDENCE TO PSD CONE S N
Page 411 and 412:
Page 413 and 414:
Page 415 and 416:
6.7. VECTORIZATION & PROJECTION INT
Page 417 and 418:
6.7. VECTORIZATION & PROJECTION INT
Page 419 and 420:
6.8. DUAL EDM CONE 419When the Fins
Page 421 and 422:
6.8. DUAL EDM CONE 421Proof. First,
Page 423 and 424:
6.8. DUAL EDM CONE 423EDM 2 = S 2 h
Page 425 and 426:
6.8. DUAL EDM CONE 425whose veracit
Page 427 and 428:
6.8. DUAL EDM CONE 4276.8.1.3.1 Exe
Page 429 and 430:
6.8. DUAL EDM CONE 429has dual affi
Page 431 and 432:
6.8. DUAL EDM CONE 4316.8.1.7 Schoe
Page 433 and 434:
6.9. THEOREM OF THE ALTERNATIVE 433
Page 435 and 436:
6.10. POSTSCRIPT 435When D is an ED
Page 437 and 438:
Chapter 7Proximity problemsIn summa
Page 439 and 440:
In contrast, order of projection on
Page 441 and 442:
441HS N h0EDM NK = S N h ∩ R N×N
Page 443 and 444:
4437.0.3 Problem approachProblems t
Page 445 and 446:
7.1. FIRST PREVALENT PROBLEM: 445fi
Page 447 and 448:
7.1. FIRST PREVALENT PROBLEM: 4477.
Page 449 and 450:
7.1. FIRST PREVALENT PROBLEM: 449di
Page 451 and 452:
7.1. FIRST PREVALENT PROBLEM: 4517.
Page 453 and 454:
7.1. FIRST PREVALENT PROBLEM: 453wh
Page 455 and 456:
7.1. FIRST PREVALENT PROBLEM: 455Th
Page 457 and 458:
7.2. SECOND PREVALENT PROBLEM: 457O
Page 459 and 460:
7.2. SECOND PREVALENT PROBLEM: 459S
Page 461 and 462:
7.2. SECOND PREVALENT PROBLEM: 461r
Page 463 and 464:
7.2. SECOND PREVALENT PROBLEM: 463c
Page 465 and 466:
7.2. SECOND PREVALENT PROBLEM: 4657
Page 467 and 468:
7.3. THIRD PREVALENT PROBLEM: 467fo
Page 469 and 470:
7.3. THIRD PREVALENT PROBLEM: 469a
Page 471 and 472:
7.3. THIRD PREVALENT PROBLEM: 4717.
Page 473 and 474:
7.3. THIRD PREVALENT PROBLEM: 4737.
Page 475 and 476:
7.3. THIRD PREVALENT PROBLEM: 475Ou
Page 478 and 479:
478 CHAPTER 7. PROXIMITY PROBLEMSth
Page 480 and 481:
480 APPENDIX A. LINEAR ALGEBRAA.1.1
Page 482 and 483:
Page 484 and 485:
484 APPENDIX A. LINEAR ALGEBRAonly
Page 486 and 487:
486 APPENDIX A. LINEAR ALGEBRA(AB)
Page 488 and 489:
Page 490 and 491:
490 APPENDIX A. LINEAR ALGEBRAFor A
Page 492 and 493:
492 APPENDIX A. LINEAR ALGEBRADiago
Page 494 and 495:
494 APPENDIX A. LINEAR ALGEBRAFor A
Page 496 and 497:
Page 498 and 499:
498 APPENDIX A. LINEAR ALGEBRAA.4 S
Page 500 and 501:
Page 502 and 503:
502 APPENDIX A. LINEAR ALGEBRAA.5 e
Page 504 and 505:
504 APPENDIX A. LINEAR ALGEBRAs i w
Page 506 and 507:
Page 508 and 509:
508 APPENDIX A. LINEAR ALGEBRAΣq 2
Page 510 and 511:
510 APPENDIX A. LINEAR ALGEBRAA.7 Z
Page 512 and 513:
512 APPENDIX A. LINEAR ALGEBRAThere
Page 514 and 515:
Page 516 and 517:
516 APPENDIX A. LINEAR ALGEBRA
Page 518 and 519:
518 APPENDIX B. SIMPLE MATRICESB.1
Page 520 and 521:
520 APPENDIX B. SIMPLE MATRICESProo
Page 522 and 523:
522 APPENDIX B. SIMPLE MATRICESB.1.
Page 524 and 525:
524 APPENDIX B. SIMPLE MATRICESN(u
Page 526 and 527:
526 APPENDIX B. SIMPLE MATRICESDue
Page 528 and 529:
Page 530 and 531:
530 APPENDIX B. SIMPLE MATRICEShas
Page 532 and 533:
532 APPENDIX B. SIMPLE MATRICESFigu
Page 534 and 535:
Page 536 and 537:
536 APPENDIX C. SOME ANALYTICAL OPT
Page 538 and 539:
Page 540 and 541:
Page 542 and 543:
Page 544 and 545:
Page 546 and 547:
Page 548 and 549:
Page 550 and 551:
550 APPENDIX D. MATRIX CALCULUSThe
Page 552 and 553:
552 APPENDIX D. MATRIX CALCULUSGrad
Page 554 and 555:
554 APPENDIX D. MATRIX CALCULUSBeca
Page 556 and 557:
556 APPENDIX D. MATRIX CALCULUSwhic
Page 558 and 559:
558 APPENDIX D. MATRIX CALCULUS⎡
Page 560 and 561:
560 APPENDIX D. MATRIX CALCULUS→Y
Page 562 and 563:
562 APPENDIX D. MATRIX CALCULUSD.1.
Page 564 and 565:
564 APPENDIX D. MATRIX CALCULUSwhic
Page 566 and 567:
566 APPENDIX D. MATRIX CALCULUSIn t
Page 568 and 569:
Page 570 and 571:
570 APPENDIX D. MATRIX CALCULUSD.2
Page 572 and 573:
572 APPENDIX D. MATRIX CALCULUSalge
Page 574 and 575:
574 APPENDIX D. MATRIX CALCULUStrac
Page 576 and 577:
Page 578 and 579:
578 APPENDIX D. MATRIX CALCULUS
Page 580 and 581:
580 APPENDIX E. PROJECTIONThe follo
Page 582 and 583:
582 APPENDIX E. PROJECTIONFor matri
Page 584 and 585:
584 APPENDIX E. PROJECTION(⇐) To
Page 586 and 587:
586 APPENDIX E. PROJECTIONNonorthog
Page 588 and 589:
588 APPENDIX E. PROJECTIONE.2.0.0.1
Page 590 and 591:
590 APPENDIX E. PROJECTIONE.3.2Orth
Page 592 and 593:
592 APPENDIX E. PROJECTIONE.3.5Unif
Page 594 and 595:
594 APPENDIX E. PROJECTIONE.4 Algeb
Page 596 and 597:
596 APPENDIX E. PROJECTIONa ∗ 2K
Page 598 and 599:
598 APPENDIX E. PROJECTIONwhere Y =
Page 600 and 601:
600 APPENDIX E. PROJECTION(B.4.2).
Page 602 and 603:
602 APPENDIX E. PROJECTIONis a nono
Page 604 and 605:
604 APPENDIX E. PROJECTIONE.6.4.1Or
Page 606 and 607:
606 APPENDIX E. PROJECTIONq i q T i
Page 608 and 609:
608 APPENDIX E. PROJECTIONThe test
Page 610 and 611:
610 APPENDIX E. PROJECTIONPerpendic
Page 612 and 613:
612 APPENDIX E. PROJECTIONE.8 Range
Page 614 and 615:
614 APPENDIX E. PROJECTIONAs for su
Page 616 and 617:
616 APPENDIX E. PROJECTIONWith refe
Page 618 and 619:
618 APPENDIX E. PROJECTIONProjectio
Page 620 and 621:
620 APPENDIX E. PROJECTIONE.9.2.2.2
Page 622 and 623:
622 APPENDIX E. PROJECTIONThe foreg
Page 624 and 625:
624 APPENDIX E. PROJECTION❇❇❇
Page 626 and 627:
626 APPENDIX E. PROJECTIONE.10 Alte
Page 628 and 629:
628 APPENDIX E. PROJECTIONbH 1H 2P
Page 630 and 631:
630 APPENDIX E. PROJECTIONa(a){y |
Page 632 and 633:
632 APPENDIX E. PROJECTION(a feasib
Page 634 and 635:
634 APPENDIX E. PROJECTIONwhile, th
Page 636 and 637:
636 APPENDIX E. PROJECTIONE.10.2.1.
Page 638 and 639:
638 APPENDIX E. PROJECTION10 0dist(
Page 640 and 641:
640 APPENDIX E. PROJECTIONE.10.3.1D
Page 642 and 643:
642 APPENDIX E. PROJECTIONE 3K ⊥
Page 644 and 645:
644 APPENDIX E. PROJECTION
Page 646 and 647:
646 APPENDIX F. MATLAB PROGRAMSif n
Page 648 and 649:
648 APPENDIX F. MATLAB PROGRAMSend%
Page 650 and 651:
650 APPENDIX F. MATLAB PROGRAMSF.1.
Page 652 and 653:
652 APPENDIX F. MATLAB PROGRAMScoun
Page 654 and 655:
654 APPENDIX F. MATLAB PROGRAMSF.3
Page 656 and 657:
Page 658 and 659:
658 APPENDIX F. MATLAB PROGRAMS% so
Page 660 and 661:
660 APPENDIX F. MATLAB PROGRAMS% tr
Page 662 and 663:
Page 664 and 665:
664 APPENDIX F. MATLAB PROGRAMSbrea
Page 666 and 667:
666 APPENDIX F. MATLAB PROGRAMSwhil
Page 668 and 669:
668 APPENDIX F. MATLAB PROGRAMSF.7
Page 670 and 671:
670 APPENDIX F. MATLAB PROGRAMS
Page 672 and 673:
672 APPENDIX G. NOTATION AND A FEW
Page 674 and 675:
Page 676 and 677:
Page 678 and 679:
Page 680 and 681:
Page 682 and 683:
Page 684 and 685:
Page 686 and 687:
Page 688 and 689:
688 BIBLIOGRAPHY[7] Abdo Y. Alfakih
Page 690 and 691:
690 BIBLIOGRAPHY[27] Aharon Ben-Tal
Page 692 and 693:
692 BIBLIOGRAPHY[48] Lev M. Brègma
Page 694 and 695:
694 BIBLIOGRAPHY[67] Joel Dawson, S
Page 696 and 697:
696 BIBLIOGRAPHY[85] Alan Edelman,
Page 698 and 699:
698 BIBLIOGRAPHY[102] Philip E. Gil
Page 700 and 701:
700 BIBLIOGRAPHYWeiss, editors, Pol
Page 702 and 703:
702 BIBLIOGRAPHY[146] Jean-Baptiste
Page 704 and 705:
704 BIBLIOGRAPHY[168] Jean B. Lasse
Page 706 and 707:
706 BIBLIOGRAPHY[189] Rudolf Mathar
Page 708 and 709:
708 BIBLIOGRAPHY[211] M. L. Overton
Page 710 and 711:
710 BIBLIOGRAPHY[229] C. K. Rushfor
Page 712 and 713:
712 BIBLIOGRAPHY[252] Jos F. Sturm
Page 714 and 715:
714 BIBLIOGRAPHY[274] È. B. Vinber
Page 716 and 717:
[294] Yinyu Ye. Semidefinite progra
Page 718 and 719:
718 INDEXobtuse, 62positive semidef
Page 720 and 721:
720 INDEXelliptope, 642orthant, 177
Page 722 and 723:
722 INDEXdistancegeometry, 20, 317m
Page 724 and 725:
724 INDEX-dimensional, 37, 89rank,
Page 726 and 727:
726 INDEXGram form, 331is, 674isedm
Page 728 and 729:
728 INDEXdiscretized, 152, 431in su
Page 730 and 731:
730 INDEXboundary, 115dimension, 10
Page 732 and 733:
732 INDEXlinear operator, 587, 591,
Page 734 and 735:
734 INDEXlargest entries, 188monoto
Page 736:
736 INDEXsimilarity, 606translation
show all

v2007.09.13 - Convex Optimization

Create successful ePaper yourself

Delete template?

Save as template?