v2006.03.09 - Convex Optimization

More documents

Recommendations

Info

24 CONVEX OPTIMIZATION & EUCLIDEAN DISTANCE GEOMETRYE.3.4.4 nonorthogonal projector, biorthogonal . . . . 536E.4 Algebra of projection on affine subsets . . . . . . . . . . . . . 536E.5 Projection examples . . . . . . . . . . . . . . . . . . . . . . . 537E.6 Vectorization interpretation, . . . . . . . . . . . . . . . . . . . 543E.6.1 Nonorthogonal projection on a vector . . . . . . . . . . 543E.6.2 Nonorthogonal projection on vectorized matrix . . . . . 544E.6.2.1 Nonorthogonal projection on dyad . . . . . . 544E.6.3 Orthogonal projection on a vector . . . . . . . . . . . . 546E.6.4 Orthogonal projection on a vectorized matrix . . . . . 546E.6.4.1 Orthogonal projection on dyad . . . . . . . . 546E.6.4.2 Positive semidefiniteness test as projection . . 549E.6.4.3 PXP ≽ 0 . . . . . . . . . . . . . . . . . . . . 551E.7 on vectorized matrices of higher rank . . . . . . . . . . . . . . 551E.7.1 PXP misinterpretation for higher-rank P . . . . . . . 551E.7.2 Orthogonal projection on matrix subspaces . . . . . . . 551E.8 Range/Rowspace interpretation . . . . . . . . . . . . . . . . . 555E.9 Projection on convex set . . . . . . . . . . . . . . . . . . . . . 555E.9.1 Dual interpretation of projection on convex set . . . . . 556E.9.1.1 Dual interpretation as optimization . . . . . . 557E.9.2 Projection on cone . . . . . . . . . . . . . . . . . . . . 559E.9.2.1 Relation to subspace projection . . . . . . . . 560E.9.2.2 Salient properties: Projection on cone . . . . 561E.9.3 nonexpansivity . . . . . . . . . . . . . . . . . . . . . . 563E.9.4 Easy projections . . . . . . . . . . . . . . . . . . . . . 564E.9.5 Projection on convex set in subspace . . . . . . . . . . 567E.10 Alternating projection . . . . . . . . . . . . . . . . . . . . . . 568E.10.0.1 commutative projectors . . . . . . . . . . . . 568E.10.0.2 noncommutative projectors . . . . . . . . . . 569E.10.0.3 the bullets . . . . . . . . . . . . . . . . . . . . 571E.10.1 Distance and existence . . . . . . . . . . . . . . . . . . 573E.10.2 Feasibility and convergence . . . . . . . . . . . . . . . 573E.10.2.1 Relative measure of convergence . . . . . . . 576E.10.3 <strong>Optimization</strong> and projection . . . . . . . . . . . . . . . 580E.10.3.1 Dykstra’s algorithm . . . . . . . . . . . . . . 580E.10.3.2 Normal cone . . . . . . . . . . . . . . . . . . 582F Proof of EDM composition 585F.1 EDM-entry exponential . . . . . . . . . . . . . . . . . . . . . . 585
CONVEX OPTIMIZATION & EUCLIDEAN DISTANCE GEOMETRY 25G Matlab programs 589G.1 isedm() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589G.1.1 Subroutines for isedm() . . . . . . . . . . . . . . . . . 593G.1.1.1 chop() . . . . . . . . . . . . . . . . . . . . . 593G.1.1.2 Vn() . . . . . . . . . . . . . . . . . . . . . . . 593G.1.1.3 Vm() . . . . . . . . . . . . . . . . . . . . . . . 593G.1.1.4 signeig() . . . . . . . . . . . . . . . . . . . 594G.1.1.5 Dx() . . . . . . . . . . . . . . . . . . . . . . . 594G.2 conic independence, conici() . . . . . . . . . . . . . . . . . . 595G.2.1 lp() . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597G.3 Map of the USA . . . . . . . . . . . . . . . . . . . . . . . . . . 598G.3.1 EDM, mapusa() . . . . . . . . . . . . . . . . . . . . . . 598G.3.1.1 USA map input-data decimation, decimate() 600G.3.2 EDM using ordinal data, omapusa() . . . . . . . . . . 600G.4 Rank reduction subroutine, RRf() . . . . . . . . . . . . . . . . 603G.4.1 svec() . . . . . . . . . . . . . . . . . . . . . . . . . . . 605G.4.2 svecinv() . . . . . . . . . . . . . . . . . . . . . . . . . 606H Notation and a few definitions 607Bibliography 621
Page 1 and 2: DATTORROCONVEXOPTIMIZATION&EUCLIDEA
Page 3 and 4: Mεβoo Publishing
Page 5 and 6: Convex Optimization&Euclidean Dista
Page 9 and 10: for Jennie Columba♦Antonio♦♦&
Page 11 and 12: PreludeThe constant demands of my d
Page 15 and 16: CONVEX OPTIMIZATION & EUCLIDEAN DIS
Page 23: CONVEX OPTIMIZATION & EUCLIDEAN DIS
Page 27 and 28: List of Figures1 Overview 331 Cocoo
Page 29 and 30: LIST OF FIGURES 294 Euclidean Dista
Page 31: List of Tables2 Convex GeometryTabl
Page 34 and 35: 34 CHAPTER 1. OVERVIEWFigure 1: Coc
Page 36 and 37: 36 CHAPTER 1. OVERVIEWAbsolute posi
Page 38 and 39: 38 CHAPTER 1. OVERVIEWFigure 4: Thi
Page 40 and 41: 40 CHAPTER 1. OVERVIEWoriginalrecon
Page 42 and 43: 42 CHAPTER 1. OVERVIEWWe characteri
Page 44 and 45: 44 CHAPTER 1. OVERVIEWEight appendi
Page 46 and 47: 46 CHAPTER 2. CONVEX GEOMETRY2.1 Co
Page 48 and 49: 48 CHAPTER 2. CONVEX GEOMETRY2.1.3
Page 50 and 51: 50 CHAPTER 2. CONVEX GEOMETRY(a)R(b
Page 52 and 53: 52 CHAPTER 2. CONVEX GEOMETRYwhere
Page 54 and 55: 54 CHAPTER 2. CONVEX GEOMETRYNow le
Page 58 and 59: 58 CHAPTER 2. CONVEX GEOMETRYand wh
Page 60 and 61: 60 CHAPTER 2. CONVEX GEOMETRY2.2.1.
Page 62 and 63: 62 CHAPTER 2. CONVEX GEOMETRYwhere
Page 64 and 65: 64 CHAPTER 2. CONVEX GEOMETRYAny ma
Page 66 and 67: 66 CHAPTER 2. CONVEX GEOMETRYIn par
Page 70 and 71: 70 CHAPTER 2. CONVEX GEOMETRY2.3.2.
Page 72 and 73: 72 CHAPTER 2. CONVEX GEOMETRYBy con
Page 74 and 75:
74 CHAPTER 2. CONVEX GEOMETRYH −
Page 76 and 77:
76 CHAPTER 2. CONVEX GEOMETRY2.4.2
Page 78 and 79:
78 CHAPTER 2. CONVEX GEOMETRYA 1A 2
Page 80 and 81:
80 CHAPTER 2. CONVEX GEOMETRYCH −
Page 82 and 83:
82 CHAPTER 2. CONVEX GEOMETRY2.4.2.
Page 84 and 85:
84 CHAPTER 2. CONVEX GEOMETRYMany o
Page 86 and 87:
Page 88 and 89:
Page 90 and 91:
90 CHAPTER 2. CONVEX GEOMETRYABCDFi
Page 92 and 93:
92 CHAPTER 2. CONVEX GEOMETRY2.6.1.
Page 94 and 95:
94 CHAPTER 2. CONVEX GEOMETRYX(a)00
Page 96 and 97:
96 CHAPTER 2. CONVEX GEOMETRYXXFigu
Page 98 and 99:
98 CHAPTER 2. CONVEX GEOMETRYFamili
Page 100 and 101:
100 CHAPTER 2. CONVEX GEOMETRYThe v
Page 102 and 103:
102 CHAPTER 2. CONVEX GEOMETRYPrope
Page 104 and 105:
104 CHAPTER 2. CONVEX GEOMETRYthe s
Page 106 and 107:
Page 108 and 109:
108 CHAPTER 2. CONVEX GEOMETRYFrom
Page 110 and 111:
110 CHAPTER 2. CONVEX GEOMETRYThe o
Page 112 and 113:
112 CHAPTER 2. CONVEX GEOMETRYCXFig
Page 114 and 115:
114 CHAPTER 2. CONVEX GEOMETRYis a
Page 116 and 117:
Page 118 and 119:
118 CHAPTER 2. CONVEX GEOMETRYmatri
Page 120 and 121:
Page 122 and 123:
122 CHAPTER 2. CONVEX GEOMETRY0-1-0
Page 124 and 125:
124 CHAPTER 2. CONVEX GEOMETRYposit
Page 126 and 127:
Page 128 and 129:
Page 130 and 131:
130 CHAPTER 2. CONVEX GEOMETRY000(a
Page 132 and 133:
132 CHAPTER 2. CONVEX GEOMETRYK(a)K
Page 134 and 135:
134 CHAPTER 2. CONVEX GEOMETRYHx 2x
Page 136 and 137:
136 CHAPTER 2. CONVEX GEOMETRY2.12
Page 138 and 139:
138 CHAPTER 2. CONVEX GEOMETRYsubsp
Page 140 and 141:
140 CHAPTER 2. CONVEX GEOMETRY2.12.
Page 142 and 143:
142 CHAPTER 2. CONVEX GEOMETRYS= co
Page 144 and 145:
144 CHAPTER 2. CONVEX GEOMETRY
Page 146 and 147:
146 CHAPTER 2. CONVEX GEOMETRYis a
Page 148 and 149:
148 CHAPTER 2. CONVEX GEOMETRYR 2 R
Page 150 and 151:
150 CHAPTER 2. CONVEX GEOMETRYwhich
Page 152 and 153:
152 CHAPTER 2. CONVEX GEOMETRYK is
Page 154 and 155:
154 CHAPTER 2. CONVEX GEOMETRYKK
Page 156 and 157:
156 CHAPTER 2. CONVEX GEOMETRYWhen
Page 158 and 159:
158 CHAPTER 2. CONVEX GEOMETRYAy
Page 160 and 161:
160 CHAPTER 2. CONVEX GEOMETRYwhere
Page 162 and 163:
Page 164 and 165:
Page 166 and 167:
Page 168 and 169:
168 CHAPTER 2. CONVEX GEOMETRYΓ 4
Page 170 and 171:
170 CHAPTER 2. CONVEX GEOMETRYCoord
Page 172 and 173:
172 CHAPTER 2. CONVEX GEOMETRYunder
Page 174 and 175:
174 CHAPTER 2. CONVEX GEOMETRYWhen
Page 176:
176 CHAPTER 2. CONVEX GEOMETRYFor e
Page 184 and 185:
Page 186 and 187:
Page 188 and 189:
188 CHAPTER 2. CONVEX GEOMETRYwhere
Page 190 and 191:
190 CHAPTER 2. CONVEX GEOMETRY
Page 192 and 193:
192 CHAPTER 3. GEOMETRY OF CONVEX F
Page 194 and 195:
Page 196 and 197:
Page 198 and 199:
Page 200 and 201:
Page 202 and 203:
Page 204 and 205:
Page 206 and 207:
Page 208 and 209:
Page 210 and 211:
Page 212 and 213:
Page 214 and 215:
Page 216 and 217:
216 CHAPTER 4. EUCLIDEAN DISTANCE M
Page 218 and 219:
Page 220 and 221:
Page 222 and 223:
Page 224 and 225:
Page 226 and 227:
Page 228 and 229:
Page 230 and 231:
Page 232 and 233:
Page 234 and 235:
Page 236 and 237:
Page 238 and 239:
Page 240 and 241:
Page 242 and 243:
Page 244 and 245:
Page 246 and 247:
Page 248 and 249:
Page 250 and 251:
Page 252 and 253:
Page 254 and 255:
Page 256 and 257:
Page 258 and 259:
Page 260 and 261:
Page 262 and 263:
Page 264 and 265:
Page 266 and 267:
Page 268 and 269:
Page 270 and 271:
Page 272 and 273:
Page 274 and 275:
Page 276 and 277:
Page 278 and 279:
Page 280 and 281:
Page 282 and 283:
Page 284 and 285:
Page 286 and 287:
Page 288 and 289:
Page 290 and 291:
Page 292 and 293:
Page 294 and 295:
Page 296 and 297:
Page 298 and 299:
Page 300 and 301:
Page 302 and 303:
Page 304 and 305:
Page 306 and 307:
306 CHAPTER 5. EDM CONE5.1 Defining
Page 308 and 309:
308 CHAPTER 5. EDM CONEThis cone is
Page 310 and 311:
310 CHAPTER 5. EDM CONEN = 4. Relat
Page 312 and 313:
312 CHAPTER 5. EDM CONE5.4 a geomet
Page 314 and 315:
314 CHAPTER 5. EDM CONE(a)(b)Figure
Page 316 and 317:
316 CHAPTER 5. EDM CONEFigure 78: T
Page 318 and 319:
318 CHAPTER 5. EDM CONEProof. Next
Page 320 and 321:
320 CHAPTER 5. EDM CONETo prove tha
Page 322 and 323:
322 CHAPTER 5. EDM CONE{D ∈ EDM N
Page 324 and 325:
324 CHAPTER 5. EDM CONED = δ ( −
Page 326 and 327:
326 CHAPTER 5. EDM CONE5.6 Correspo
Page 328 and 329:
328 CHAPTER 5. EDM CONE5.6.0.0.2 Ex
Page 330 and 331:
330 CHAPTER 5. EDM CONE5.6.2.0.1 Ex
Page 332 and 333:
332 CHAPTER 5. EDM CONEThe set of a
Page 334 and 335:
334 CHAPTER 5. EDM CONEsvec ∂ S 2
Page 336 and 337:
336 CHAPTER 5. EDM CONEWhen the Fin
Page 338 and 339:
338 CHAPTER 5. EDM CONEProof. First
Page 340 and 341:
340 CHAPTER 5. EDM CONEEDM 2 = S 2
Page 342 and 343:
342 CHAPTER 5. EDM CONEwhose veraci
Page 344 and 345:
344 CHAPTER 5. EDM CONED ◦ = δ(D
Page 346 and 347:
346 CHAPTER 5. EDM CONEcone; id est
Page 348 and 349:
348 CHAPTER 5. EDM CONE5.8.1.7 Scho
Page 350 and 351:
350 CHAPTER 5. EDM CONETo find the
Page 352 and 353:
352 CHAPTER 5. EDM CONEWhen D is an
Page 354 and 355:
354 CHAPTER 6. SEMIDEFINITE PROGRAM
Page 356 and 357:
Page 358 and 359:
Page 360 and 361:
Page 362 and 363:
Page 364 and 365:
Page 366 and 367:
Page 368 and 369:
Page 370 and 371:
Page 372 and 373:
Page 374 and 375:
Page 376 and 377:
Page 378 and 379:
Page 380 and 381:
Page 382 and 383:
Page 384 and 385:
Page 386 and 387:
Page 388 and 389:
388 CHAPTER 7. EDM PROXIMITY7.0.1 M
Page 390 and 391:
390 CHAPTER 7. EDM PROXIMITY......
Page 392 and 393:
392 CHAPTER 7. EDM PROXIMITYAny pro
Page 394 and 395:
394 CHAPTER 7. EDM PROXIMITYSubstit
Page 396 and 397:
396 CHAPTER 7. EDM PROXIMITY7.1.1 C
Page 398 and 399:
398 CHAPTER 7. EDM PROXIMITYdimensi
Page 400 and 401:
400 CHAPTER 7. EDM PROXIMITYGiven
Page 402 and 403:
402 CHAPTER 7. EDM PROXIMITYProblem
Page 404 and 405:
404 CHAPTER 7. EDM PROXIMITYBecause
Page 406 and 407:
406 CHAPTER 7. EDM PROXIMITY7.2.1 C
Page 408 and 409:
408 CHAPTER 7. EDM PROXIMITY7.2.1.2
Page 410 and 411:
Page 412 and 413:
412 CHAPTER 7. EDM PROXIMITYTo see
Page 414 and 415:
414 CHAPTER 7. EDM PROXIMITY∑mini
Page 416 and 417:
416 CHAPTER 7. EDM PROXIMITYcharact
Page 418 and 419:
Page 420 and 421:
420 CHAPTER 7. EDM PROXIMITYwhere
Page 422 and 423:
422 CHAPTER 7. EDM PROXIMITY
Page 424 and 425:
424 APPENDIX A. LINEAR ALGEBRAA.1.1
Page 426 and 427:
426 APPENDIX A. LINEAR ALGEBRA27. T
Page 428 and 429:
428 APPENDIX A. LINEAR ALGEBRAonly
Page 430 and 431:
430 APPENDIX A. LINEAR ALGEBRA(AB)
Page 432 and 433:
Page 434 and 435:
434 APPENDIX A. LINEAR ALGEBRA(Fan)
Page 436 and 437:
436 APPENDIX A. LINEAR ALGEBRAFor A
Page 438 and 439:
438 APPENDIX A. LINEAR ALGEBRAFor A
Page 440 and 441:
Page 442 and 443:
442 APPENDIX A. LINEAR ALGEBRAA.4 S
Page 444 and 445:
Page 446 and 447:
Page 448 and 449:
448 APPENDIX A. LINEAR ALGEBRAA.5 e
Page 450 and 451:
450 APPENDIX A. LINEAR ALGEBRAs i w
Page 452 and 453:
Page 454 and 455:
454 APPENDIX A. LINEAR ALGEBRAΣq 2
Page 456 and 457:
456 APPENDIX A. LINEAR ALGEBRAA.7 Z
Page 458 and 459:
458 APPENDIX A. LINEAR ALGEBRAThe r
Page 460 and 461:
460 APPENDIX A. LINEAR ALGEBRAThe p
Page 462 and 463:
462 APPENDIX B. SIMPLE MATRICESB.1
Page 464 and 465:
464 APPENDIX B. SIMPLE MATRICESB.1.
Page 466 and 467:
Page 468 and 469:
468 APPENDIX B. SIMPLE MATRICESN(u
Page 470 and 471:
470 APPENDIX B. SIMPLE MATRICESDue
Page 472 and 473:
Page 474 and 475:
474 APPENDIX B. SIMPLE MATRICEShas
Page 476 and 477:
476 APPENDIX B. SIMPLE MATRICESFigu
Page 478 and 479:
Page 480 and 481:
480 APPENDIX C. SOME ANALYTICAL OPT
Page 482 and 483:
Page 484 and 485:
Page 486 and 487:
Page 488 and 489:
Page 490 and 491:
Page 493:
C.5. TWO-SIDED ORTHOGONAL PROCRUSTE
Page 496 and 497:
496 APPENDIX D. MATRIX CALCULUSThe
Page 498 and 499:
498 APPENDIX D. MATRIX CALCULUSGrad
Page 500 and 501:
500 APPENDIX D. MATRIX CALCULUSBeca
Page 502 and 503:
502 APPENDIX D. MATRIX CALCULUSfrom
Page 504 and 505:
504 APPENDIX D. MATRIX CALCULUSin m
Page 506 and 507:
506 APPENDIX D. MATRIX CALCULUSFigu
Page 508 and 509:
508 APPENDIX D. MATRIX CALCULUS⎡
Page 510 and 511:
510 APPENDIX D. MATRIX CALCULUSD.1.
Page 512 and 513:
Page 514 and 515:
514 APPENDIX D. MATRIX CALCULUSD.2
Page 516 and 517:
516 APPENDIX D. MATRIX CALCULUSAlge
Page 518 and 519:
518 APPENDIX D. MATRIX CALCULUSTrac
Page 520 and 521:
Page 522 and 523:
522 APPENDIX D. MATRIX CALCULUS
Page 524 and 525:
524 APPENDIX E. PROJECTIONThe follo
Page 526 and 527:
526 APPENDIX E. PROJECTIONE.1.1Bior
Page 528 and 529:
528 APPENDIX E. PROJECTIONTxT ⊥ R
Page 530 and 531:
530 APPENDIX E. PROJECTIONE.2 I−P
Page 532 and 533:
532 APPENDIX E. PROJECTIONIn any ca
Page 534 and 535:
534 APPENDIX E. PROJECTIONProof. We
Page 536 and 537:
536 APPENDIX E. PROJECTIONthe direc
Page 538 and 539:
538 APPENDIX E. PROJECTIONand defin
Page 540 and 541:
540 APPENDIX E. PROJECTIONnonorthog
Page 542 and 543:
542 APPENDIX E. PROJECTIONwhere y p
Page 544 and 545:
544 APPENDIX E. PROJECTIONE.6.2Nono
Page 546 and 547:
546 APPENDIX E. PROJECTIONE.6.3Orth
Page 548 and 549:
548 APPENDIX E. PROJECTIONE.6.4.1.1
Page 550 and 551:
550 APPENDIX E. PROJECTIONBy (1537)
Page 552 and 553:
552 APPENDIX E. PROJECTIONAs for al
Page 554 and 555:
554 APPENDIX E. PROJECTIONcharacter
Page 556 and 557:
Page 558 and 559:
558 APPENDIX E. PROJECTION∂H −C
Page 560 and 561:
Page 562 and 563:
562 APPENDIX E. PROJECTIONNow assum
Page 564 and 565:
564 APPENDIX E. PROJECTIONProof. [3
Page 566 and 567:
566 APPENDIX E. PROJECTION❇❇❇
Page 568 and 569:
568 APPENDIX E. PROJECTIONE.10 Alte
Page 570 and 571:
570 APPENDIX E. PROJECTIONbH 1H 2P
Page 572 and 573:
572 APPENDIX E. PROJECTIONa(a)C 1C
Page 574 and 575:
574 APPENDIX E. PROJECTION(a feasib
Page 576 and 577:
576 APPENDIX E. PROJECTIONwhile, th
Page 578 and 579:
578 APPENDIX E. PROJECTIONThe dista
Page 580 and 581:
580 APPENDIX E. PROJECTION⎡⎢⎣
Page 582 and 583:
582 APPENDIX E. PROJECTIONK ⊥ H 1
Page 584 and 585:
584 APPENDIX E. PROJECTION
Page 586 and 587:
586 APPENDIX F. PROOF OF EDM COMPOS
Page 588 and 589:
588 APPENDIX F. PROOF OF EDM COMPOS
Page 590 and 591:
590 APPENDIX G. MATLAB PROGRAMSif n
Page 592 and 593:
592 APPENDIX G. MATLAB PROGRAMSend%
Page 594 and 595:
594 APPENDIX G. MATLAB PROGRAMSG.1.
Page 596 and 597:
596 APPENDIX G. MATLAB PROGRAMScoun
Page 598 and 599:
598 APPENDIX G. MATLAB PROGRAMSG.3
Page 600 and 601:
Page 602 and 603:
602 APPENDIX G. MATLAB PROGRAMSf(id
Page 604 and 605:
604 APPENDIX G. MATLAB PROGRAMS%try
Page 606 and 607:
Page 608 and 609:
608 APPENDIX H. NOTATION AND A FEW
Page 610 and 611:
Page 612 and 613:
Page 614 and 615:
Page 616 and 617:
Page 618 and 619:
Page 620 and 621:
Page 622 and 623:
622 BIBLIOGRAPHY[7] Abdo Y. Alfakih
Page 624 and 625:
624 BIBLIOGRAPHY[29] Pratik Biswas,
Page 626 and 627:
626 BIBLIOGRAPHY[51] Frank Critchle
Page 628 and 629:
628 BIBLIOGRAPHY[70] Ivar Ekeland a
Page 630 and 631:
630 BIBLIOGRAPHY[91] John Clifford
Page 632 and 633:
632 BIBLIOGRAPHYnew software CANDID
Page 634 and 635:
634 BIBLIOGRAPHY[133] Florian Jarre
Page 636 and 637:
636 BIBLIOGRAPHY[155] David G. Luen
Page 638 and 639:
638 BIBLIOGRAPHY[178] Brad Osgood.
Page 640 and 641:
640 BIBLIOGRAPHY[198] Uwe Schäfer.
Page 642 and 643:
642 BIBLIOGRAPHY[222] Akimichi Take
Page 644 and 645:
644 BIBLIOGRAPHY[244] Eric W. Weiss
Page 646 and 647:
646 BIBLIOGRAPHY
Page 648 and 649:
648
Page 650 and 651:
650
Page 652 and 653:
652
Page 654:
654
show all

v2006.03.09 - Convex Optimization

Create successful ePaper yourself

Delete template?

Save as template?