v2007.09.13 - Convex Optimization

More documents

Recommendations

Info

384 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX5.14.3.3.1 Example. Pyramid.A formula for volume of a pyramid is known; 5.58 it is 1 the product of its3base area with its height. [163] The pyramid in Figure 93 has volume 1 . 3To find its volume using EDMs, we must first decompose the pyramid intosimplicial parts. Slicing it in half along the plane containing the line segmentscorresponding to radius R and height h we find the vertices of one simplex,⎡X = ⎣1/2 1/2 −1/2 01/2 −1/2 −1/2 00 0 0 1⎤⎦∈ R n×N (976)where N = n + 1 for any nonempty simplex in R n .this simplex is half that of the entire pyramid; id est,evaluating (974).√The volume ofc =1found by6With that, we conclude digression of path.5.14.4 Affine dimension reduction in three dimensions(confer5.8.4) The determinant of any M ×M matrix is equal to the productof its M eigenvalues. [247,5.1] When N = 4 and det(Θ T Θ) is 0, thatmeans one or more eigenvalues of Θ T Θ∈ R 3×3 are 0. The determinant willgo to 0 whenever equality is attained on either side of (699), (965a), or (965b),meaning that a tetrahedron has collapsed to a lower affine dimension; id est,r = rank Θ T Θ = rank Θ is reduced below N −1 exactly by the number of0 eigenvalues (5.7.1.1).In solving completion problems of any size N where one or more entriesof an EDM are unknown, therefore, dimension r of the affine hull requiredto contain the unknown points is potentially reduced by selecting distancesto attain equality in (699) or (965a) or (965b).5.58 Pyramid volume is independent of the paramount vertex position as long as its heightremains constant.
5.14. FIFTH PROPERTY OF EUCLIDEAN METRIC 3855.14.4.1 Exemplum reduxWe now apply the fifth Euclidean metric property to an earlier problem:5.14.4.1.1 Example. Small completion problem, IV. (confer5.9.3.0.1)Returning again to Example 5.3.0.0.2 that pertains to Figure 74 whereN =4, distance-square d 14 is ascertainable from the fifth Euclidean metricproperty. Because all distances in (697) are known except √ d 14 , thencos θ 123 =0 and θ 324 =0 result from identity (766). Applying (699),cos(θ 123 + θ 324 ) ≤ cos θ 124 ≤ cos(θ 123 − θ 324 )0 ≤ cos θ 124 ≤ 0(977)It follows again from (766) that d 14 can only be 2. As explained in thissubsection, affine dimension r cannot exceed N −2 because equality isattained in (977).
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 117 and 118:
Page 119 and 120:
Page 121 and 122:
2.10. CONIC INDEPENDENCE (C.I.) 121
Page 123 and 124:
Page 125 and 126:
Page 127 and 128:
2.12. CONVEX POLYHEDRA 127It follow
Page 129 and 130:
2.12. CONVEX POLYHEDRA 129Coefficie
Page 131 and 132:
2.12. CONVEX POLYHEDRA 1312.12.3 Un
Page 133 and 134:
2.12. CONVEX POLYHEDRA 133
Page 135 and 136:
2.13. DUAL CONE & GENERALIZED INEQU
Page 137 and 138:
Page 139 and 140:
Page 141 and 142:
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:
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:
5.6. INJECTIVITY OF D & UNIQUE RECO
Page 333 and 334: 5.6. INJECTIVITY OF D & UNIQUE RECO
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 349 and 350: 5.9. BRIDGE: CONVEX POLYHEDRA TO ED
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 383: 5.14. FIFTH PROPERTY OF EUCLIDEAN M
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 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?