v2007.11.26 - Convex Optimization

More documents

Recommendations

Info

20 CHAPTER 1. OVERVIEWFigure 1: Orion nebula. (astrophotography by Massimo Robberto)Euclidean distance geometry is, fundamentally, a determination of pointconformation (configuration, relative position or location) by inference frominterpoint distance information. By inference we mean: e.g., given onlydistance information, determine whether there corresponds a realizableconformation of points; a list of points in some dimension that attains thegiven interpoint distances. Each point may represent simply location or,abstractly, any entity expressible as a vector in finite-dimensional Euclideanspace; e.g., distance geometry of music [72].It is a common misconception to presume that some desired pointconformation cannot be recovered in the absence of complete interpointdistance information. We might, for example, want to realize a constellationgiven only interstellar distance (or, equivalently, distance from Earth andrelative angular measurement; the Earth as vertex to two stars). At first itmay seem O(N 2 ) data is required, yet there are many circumstances wherethis can be reduced to O(N).If we agree that a set of points can have a shape (three points can forma triangle and its interior, for example, four points a tetrahedron), then
ˇx 4ˇx 3ˇx 2Figure 2: Application of trilateration (5.4.2.2.4) is localization (determiningposition) of a radio signal source in 2 dimensions; more commonly known byradio engineers as the process “triangulation”. In this scenario, anchorsˇx 2 , ˇx 3 , ˇx 4 are illustrated as fixed antennae. [154] The radio signal source(a sensor • x 1 ) anywhere in affine hull of three antenna bases can beuniquely localized by measuring distance to each (dashed white arrowed linesegments). Ambiguity of lone distance measurement to sensor is representedby circle about each antenna. So & Ye proved trilateration is expressible asa semidefinite program; hence, a convex optimization problem. [241]we can ascribe shape of a set of points to their convex hull. It should beapparent: from distance, these shapes can be determined only to within arigid transformation (rotation, reflection, translation).Absolute position information is generally lost, given only distanceinformation, but we can determine the smallest possible dimension in whichan unknown list of points can exist; that attribute is their affine dimension(a triangle in any ambient space has affine dimension 2, for example). Incircumstances where fixed points of reference are also provided, it becomespossible to determine absolute position or location; e.g., Figure 2.21
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 Epigraph . . .
Page 17 and 18: LIST OF FIGURES 17120 Four fundamen
Page 19: Chapter 1OverviewConvex Optimizatio
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 11: 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 53 and 54: 2.3. HULLS 53Figure 14: 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 73nor
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 26: 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 95 and 96:
2.8. CONE BOUNDARY 95For a proper c
Page 97 and 98:
2.9. POSITIVE SEMIDEFINITE (PSD) CO
Page 99 and 100:
Page 101 and 102:
Page 103 and 104:
Page 105 and 106:
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:
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 189n∑π(x) i
Page 191 and 192:
3.1. CONVEX FUNCTION 191This means
Page 193 and 194:
3.1. CONVEX FUNCTION 1933.1.5.3 pos
Page 195 and 196:
3.1. CONVEX FUNCTION 1953.1.6.0.1 E
Page 197 and 198:
3.1. CONVEX FUNCTION 1973.1.7 epigr
Page 199 and 200:
Page 201 and 202:
3.1. CONVEX FUNCTION 201Although th
Page 203 and 204:
3.1. CONVEX FUNCTION 203minimize tX
Page 205 and 206:
3.1. CONVEX FUNCTION 2053.1.8 gradi
Page 207 and 208:
Page 209 and 210:
3.1. CONVEX FUNCTION 209f(Y )[ ∇f
Page 211 and 212:
3.1. CONVEX FUNCTION 211g is convex
Page 213 and 214:
3.1. CONVEX FUNCTION 213αβα ≥
Page 215 and 216:
3.1. CONVEX FUNCTION 2153.1.11 seco
Page 217 and 218:
3.2. MATRIX-VALUED CONVEX FUNCTION
Page 219 and 220:
3.2. MATRIX-VALUED CONVEX FUNCTION
Page 221 and 222:
3.3. QUASICONVEX 221matrices in the
Page 223 and 224:
3.3. QUASICONVEX 223on R M + . Alth
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 2414.2.3.1.1 Example
Page 243 and 244:
4.2. FRAMEWORK 243where δ is the m
Page 245 and 246:
4.2. FRAMEWORK 2454.2.3.1.2 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:
4.5. CONSTRAINING CARDINALITY 2734.
Page 275 and 276:
4.5. CONSTRAINING CARDINALITY 275Th
Page 277 and 278:
4.5. CONSTRAINING CARDINALITY 277wh
Page 279 and 280:
4.5. CONSTRAINING CARDINALITY 279f
Page 281 and 282:
4.5. CONSTRAINING CARDINALITY 281al
Page 283 and 284:
4.6. CARDINALITY AND RANK CONSTRAIN
Page 285 and 286:
Page 287 and 288:
Page 289 and 290:
Page 291 and 292:
Page 293 and 294:
Page 295 and 296:
Page 297 and 298:
Page 299 and 300:
Page 301 and 302:
4.7. CONVEX ITERATION RANK-1 301whi
Page 303 and 304:
4.7. CONVEX ITERATION RANK-1 303the
Page 305 and 306:
Chapter 5Euclidean Distance MatrixT
Page 307 and 308:
5.2. FIRST METRIC PROPERTIES 307cor
Page 309 and 310:
5.3. ∃ FIFTH EUCLIDEAN METRIC PRO
Page 311 and 312:
5.3. ∃ FIFTH EUCLIDEAN METRIC PRO
Page 313 and 314:
5.4. EDM DEFINITION 313The collecti
Page 315 and 316:
5.4. EDM DEFINITION 3155.4.2 Gram-f
Page 317 and 318:
5.4. EDM DEFINITION 317D ∈ EDM N
Page 319 and 320:
5.4. EDM DEFINITION 3195.4.2.2.1 Ex
Page 321 and 322:
5.4. EDM DEFINITION 321ten affine e
Page 323 and 324:
5.4. EDM DEFINITION 323spheres:Then
Page 325 and 326:
5.4. EDM DEFINITION 325By eliminati
Page 327 and 328:
5.4. EDM DEFINITION 327whereΦ ij =
Page 329 and 330:
5.4. EDM DEFINITION 3295.4.2.2.5 De
Page 331 and 332:
5.4. EDM DEFINITION 331105ˇx 4ˇx
Page 333 and 334:
5.4. EDM DEFINITION 333corrected by
Page 335 and 336:
5.4. EDM DEFINITION 335by translate
Page 337 and 338:
5.4. EDM DEFINITION 337Crippen & Ha
Page 339 and 340:
5.4. EDM DEFINITION 339where ([√t
Page 341 and 342:
5.4. EDM DEFINITION 341because (A.3
Page 343 and 344:
5.5. INVARIANCE 3435.5.1.0.1 Exampl
Page 345 and 346:
5.6. INJECTIVITY OF D & UNIQUE RECO
Page 347 and 348:
Page 349 and 350:
Page 351 and 352:
5.7. EMBEDDING IN AFFINE HULL 3515.
Page 353 and 354:
5.7. EMBEDDING IN AFFINE HULL 353Fo
Page 355 and 356:
5.7. EMBEDDING IN AFFINE HULL 3555.
Page 357 and 358:
5.8. EUCLIDEAN METRIC VERSUS MATRIX
Page 359 and 360:
Page 361 and 362:
Page 363 and 364:
Page 365 and 366:
5.9. BRIDGE: CONVEX POLYHEDRA TO ED
Page 367 and 368:
Page 369 and 370:
Page 371 and 372:
Page 373 and 374:
5.10. EDM-ENTRY COMPOSITION 373(ii)
Page 375 and 376:
5.11. EDM INDEFINITENESS 3755.11.1
Page 377 and 378:
5.11. EDM INDEFINITENESS 377(confer
Page 379 and 380:
5.11. EDM INDEFINITENESS 379we have
Page 381 and 382:
5.11. EDM INDEFINITENESS 381For pre
Page 383 and 384:
5.12. LIST RECONSTRUCTION 383where
Page 385 and 386:
5.12. LIST RECONSTRUCTION 385(a)(c)
Page 387 and 388:
5.13. RECONSTRUCTION EXAMPLES 387D
Page 389 and 390:
5.13. RECONSTRUCTION EXAMPLES 389Th
Page 391 and 392:
5.13. RECONSTRUCTION EXAMPLES 391wh
Page 393 and 394:
5.14. FIFTH PROPERTY OF EUCLIDEAN M
Page 395 and 396:
Page 397 and 398:
Page 399 and 400:
Page 401 and 402:
Page 403 and 404:
Chapter 6Cone of distance matricesF
Page 405 and 406:
6.1. DEFINING EDM CONE 4056.1 Defin
Page 407 and 408:
6.2. POLYHEDRAL BOUNDS 407This cone
Page 409 and 410:
6.3.√EDM CONE IS NOT CONVEX 409N
Page 411 and 412:
6.4. A GEOMETRY OF COMPLETION 4116.
Page 413 and 414:
6.4. A GEOMETRY OF COMPLETION 413(a
Page 415 and 416:
6.4. A GEOMETRY OF COMPLETION 415Fi
Page 417 and 418:
6.5. EDM DEFINITION IN 11 T 417and
Page 419 and 420:
6.5. EDM DEFINITION IN 11 T 419then
Page 421 and 422:
6.5. EDM DEFINITION IN 11 T 4216.5.
Page 423 and 424:
6.5. EDM DEFINITION IN 11 T 423D =
Page 425 and 426:
6.6. CORRESPONDENCE TO PSD CONE S N
Page 427 and 428:
Page 429 and 430:
Page 431 and 432:
6.7. VECTORIZATION & PROJECTION INT
Page 433 and 434:
6.7. VECTORIZATION & PROJECTION INT
Page 435 and 436:
6.8. DUAL EDM CONE 435When the Fins
Page 437 and 438:
6.8. DUAL EDM CONE 437Proof. First,
Page 439 and 440:
6.8. DUAL EDM CONE 439EDM 2 = S 2 h
Page 441 and 442:
6.8. DUAL EDM CONE 441whose veracit
Page 443 and 444:
6.8. DUAL EDM CONE 4436.8.1.3.1 Exe
Page 445 and 446:
6.8. DUAL EDM CONE 445has dual affi
Page 447 and 448:
6.8. DUAL EDM CONE 4476.8.1.7 Schoe
Page 449 and 450:
6.9. THEOREM OF THE ALTERNATIVE 449
Page 451 and 452:
6.10. POSTSCRIPT 451When D is an ED
Page 453 and 454:
Chapter 7Proximity problemsIn summa
Page 455 and 456:
In contrast, order of projection on
Page 457 and 458:
457HS N h0EDM NK = S N h ∩ R N×N
Page 459 and 460:
4597.0.3 Problem approachProblems t
Page 461 and 462:
7.1. FIRST PREVALENT PROBLEM: 461fi
Page 463 and 464:
7.1. FIRST PREVALENT PROBLEM: 4637.
Page 465 and 466:
7.1. FIRST PREVALENT PROBLEM: 465di
Page 467 and 468:
7.1. FIRST PREVALENT PROBLEM: 4677.
Page 469 and 470:
7.1. FIRST PREVALENT PROBLEM: 469wh
Page 471 and 472:
7.1. FIRST PREVALENT PROBLEM: 471Th
Page 473 and 474:
7.2. SECOND PREVALENT PROBLEM: 473O
Page 475 and 476:
7.2. SECOND PREVALENT PROBLEM: 475S
Page 477 and 478:
7.2. SECOND PREVALENT PROBLEM: 477r
Page 479 and 480:
7.2. SECOND PREVALENT PROBLEM: 479c
Page 481 and 482:
7.2. SECOND PREVALENT PROBLEM: 4817
Page 483 and 484:
7.3. THIRD PREVALENT PROBLEM: 483fo
Page 485 and 486:
7.3. THIRD PREVALENT PROBLEM: 485a
Page 487 and 488:
7.3. THIRD PREVALENT PROBLEM: 4877.
Page 489 and 490:
7.3. THIRD PREVALENT PROBLEM: 4897.
Page 491 and 492:
7.3. THIRD PREVALENT PROBLEM: 491Ou
Page 493 and 494:
7.4. CONCLUSION 493The rank constra
Page 495 and 496:
Appendix ALinear algebraA.1 Main-di
Page 497 and 498:
A.1. MAIN-DIAGONAL δ OPERATOR, λ
Page 499 and 500:
A.2. SEMIDEFINITENESS: DOMAIN OF TE
Page 501 and 502:
A.2. SEMIDEFINITENESS: DOMAIN OF TE
Page 503 and 504:
A.3. PROPER STATEMENTS 503A.3.0.0.1
Page 505 and 506:
A.3. PROPER STATEMENTS 505By simila
Page 507 and 508:
A.3. PROPER STATEMENTS 507Because R
Page 509 and 510:
For A,B ∈ R n×n x T Ax ≥ x T B
Page 511 and 512:
A.3. PROPER STATEMENTS 511A.3.1.0.2
Page 513 and 514:
A.3. PROPER STATEMENTS 513We can de
Page 515 and 516:
A.4. SCHUR COMPLEMENT 515Origin of
Page 517 and 518:
A.4. SCHUR COMPLEMENT 517Schur-form
Page 519 and 520:
A.5. EIGEN DECOMPOSITION 519A.5.0.1
Page 521 and 522:
A.6. SINGULAR VALUE DECOMPOSITION,
Page 523 and 524:
Page 525 and 526:
Page 527 and 528:
A.7. ZEROS 527For diagonalizable ma
Page 529 and 530:
A.7. ZEROS 529A.7.4For X,A∈ S M +
Page 531 and 532:
A.7. ZEROS 531A.7.5.0.1 Proposition
Page 533 and 534:
Appendix BSimple matricesMathematic
Page 535 and 536:
B.1. RANK-ONE MATRIX (DYAD) 535R(v)
Page 537 and 538:
B.1. RANK-ONE MATRIX (DYAD) 537rang
Page 539 and 540:
B.2. DOUBLET 539R([u v ])R(Π)= R([
Page 541 and 542:
B.3. ELEMENTARY MATRIX 541If λ ≠
Page 543 and 544:
B.4. AUXILIARY V -MATRICES 543the n
Page 545 and 546:
B.4. AUXILIARY V -MATRICES 54518. V
Page 547 and 548:
B.5. ORTHOGONAL MATRIX 547B.5 Ortho
Page 549 and 550:
B.5. ORTHOGONAL MATRIX 549B.5.3.0.1
Page 551 and 552:
Appendix CSome analytical optimal r
Page 553 and 554:
C.2. DIAGONAL, TRACE, SINGULAR AND
Page 555 and 556:
Page 557 and 558:
Page 559 and 560:
C.3. ORTHOGONAL PROCRUSTES PROBLEM
Page 561 and 562:
C.4. TWO-SIDED ORTHOGONAL PROCRUSTE
Page 563 and 564:
C.4. TWO-SIDED ORTHOGONAL PROCRUSTE
Page 565 and 566:
Appendix DMatrix calculusFrom too m
Page 567 and 568:
D.1. DIRECTIONAL DERIVATIVE, TAYLOR
Page 569 and 570:
Page 571 and 572:
Page 573 and 574:
Page 575 and 576:
Page 577 and 578:
Page 579 and 580:
Page 581 and 582:
Page 583 and 584:
Page 585 and 586:
Page 587 and 588:
D.2. TABLES OF GRADIENTS AND DERIVA
Page 589 and 590:
Page 591 and 592:
Page 593 and 594:
Page 595 and 596:
Appendix EProjectionFor any A∈ R
Page 597 and 598:
597Equivalent to the corresponding
Page 599 and 600:
E.1. IDEMPOTENT MATRICES 599where R
Page 601 and 602:
E.1. IDEMPOTENT MATRICES 601TxT ⊥
Page 603 and 604:
E.2. I − P , PROJECTION ON ALGEBR
Page 605 and 606:
E.3. SYMMETRIC IDEMPOTENT MATRICES
Page 607 and 608:
Page 609 and 610:
Page 611 and 612:
E.5. PROJECTION EXAMPLES 611E.5.0.0
Page 613 and 614:
E.5. PROJECTION EXAMPLES 613of rela
Page 615 and 616:
E.5. PROJECTION EXAMPLES 615E.5.0.0
Page 617 and 618:
E.6. VECTORIZATION INTERPRETATION,
Page 619 and 620:
Page 621 and 622:
Page 623 and 624:
Page 625 and 626:
E.7. ON VECTORIZED MATRICES OF HIGH
Page 627 and 628:
E.7. ON VECTORIZED MATRICES OF HIGH
Page 629 and 630:
E.9. PROJECTION ON CONVEX SET 629E.
Page 631 and 632:
E.9. PROJECTION ON CONVEX SET 631
Page 633 and 634:
E.9. PROJECTION ON CONVEX SET 633Pr
Page 635 and 636:
Page 637 and 638:
E.9. PROJECTION ON CONVEX SET 637wh
Page 639 and 640:
E.9. PROJECTION ON CONVEX SET 639Un
Page 641 and 642:
Page 643 and 644:
E.10. ALTERNATING PROJECTION 643bC
Page 645 and 646:
E.10. ALTERNATING PROJECTION 6450
Page 647 and 648:
E.10. ALTERNATING PROJECTION 647E.1
Page 649 and 650:
E.10. ALTERNATING PROJECTION 649y 2
Page 651 and 652:
E.10. ALTERNATING PROJECTION 651By
Page 653 and 654:
E.10. ALTERNATING PROJECTION 653Bar
Page 655 and 656:
E.10. ALTERNATING PROJECTION 655bH
Page 657 and 658:
E.10. ALTERNATING PROJECTION 657K
Page 659 and 660:
E.10. ALTERNATING PROJECTION 659Whe
Page 661 and 662:
Appendix FMatlab programsMade by Th
Page 663 and 664:
F.1. ISEDM() 663% is nonnegativeif
Page 665 and 666:
F.1. ISEDM() 665F.1.1Subroutines fo
Page 667 and 668:
F.2. CONIC INDEPENDENCE, CONICI() 6
Page 669 and 670:
F.2. CONIC INDEPENDENCE, CONICI() 6
Page 671 and 672:
F.3. MAP OF THE USA 671% plot origi
Page 673 and 674:
F.3. MAP OF THE USA 673statelat = d
Page 675 and 676:
F.4. RANK REDUCTION SUBROUTINE, RRF
Page 677 and 678:
F.4. RANK REDUCTION SUBROUTINE, RRF
Page 679 and 680:
F.5. STURM’S PROCEDURE 679F.5 Stu
Page 681 and 682:
F.6. CONVEX ITERATION DEMONSTRATION
Page 683 and 684:
F.6. CONVEX ITERATION DEMONSTRATION
Page 685 and 686:
F.7. FAST MAX CUT 685endoldtrace =
Page 687 and 688:
F.8. SIGNAL DROPOUT PROBLEM 687whil
Page 689 and 690:
Appendix GNotation and a few defini
Page 691 and 692:
691l.i.w.r.tlinearly independentwit
Page 693 and 694:
693t → 0 +t goes to 0 from above;
Page 695 and 696:
695ψ(Z)DDD T (X)D(X) TD −1 (X)D(
Page 697 and 698:
697R n −or R n×n−S nS n⊥S n
Page 699 and 700:
699∂H −∂H +da supporting hype
Page 701 and 702:
701maximizexargsup Xarg supf(x)subj
Page 703 and 704:
703⊁≥not positive definitescala
Page 705 and 706:
Bibliography[1] Suliman Al-Homidan
Page 707 and 708:
BIBLIOGRAPHY 707[16] Keith Ball. An
Page 709 and 710:
BIBLIOGRAPHY 709[38] A. W. Bojanczy
Page 711 and 712:
BIBLIOGRAPHY 711[57] Steven Chu. Au
Page 713 and 714:
BIBLIOGRAPHY 713[76] Frank R. Deuts
Page 715 and 716:
BIBLIOGRAPHY 715(AACC), June 2004.h
Page 717 and 718:
BIBLIOGRAPHY 717[114] John Clifford
Page 719 and 720:
BIBLIOGRAPHY 719[135] Uwe Helmke an
Page 721 and 722:
BIBLIOGRAPHY 721[157] Joakim Jaldé
Page 723 and 724:
BIBLIOGRAPHY 723[178] Jung Rye Lee.
Page 725 and 726:
BIBLIOGRAPHY 725[200] T. S. Motzkin
Page 727 and 728:
BIBLIOGRAPHY 727[220] Teemu Pennane
Page 729 and 730:
BIBLIOGRAPHY 729[240] Joshua A. Sin
Page 731 and 732:
BIBLIOGRAPHY 731[264] Warren S. Tor
Page 733 and 734:
BIBLIOGRAPHY 733[285] Bernard Widro
Page 735 and 736:
Index0-norm, 241, 273, 275, 2941-no
Page 737 and 738:
INDEX 737clipping, 190, 454, 638, 6
Page 739 and 740:
INDEX 739hull, 53, 56, 306of outer
Page 741 and 742:
INDEX 741unique, 519elbow, 457, 458
Page 743 and 744:
INDEX 743polarity, 62handoff, 335ov
Page 745 and 746:
INDEX 745manifold, 23, 24, 105, 410
Page 747 and 748:
INDEX 747objectivebilinear, 256, 27
Page 749 and 750:
INDEX 749Hadamard, 46, 472, 571, 69
Page 751 and 752:
INDEX 751trace, 477, 478heuristic,
Page 753 and 754:
INDEX 753symmetric hollow, 51tangen
Page 755 and 756:
INDEX 755matrix, 65slack, 227vec, 4
Page 757 and 758:
757
Page 760:
Convex Optimization & Euclidean Dis
show all

v2007.11.26 - Convex Optimization

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

Delete template?

Save as template?