v2007.11.26 - Convex Optimization

More documents

Recommendations

Info

714 BIBLIOGRAPHY[85] Carl Eckart and Gale Young. The approximation of one matrix byanother of lower rank. Psychometrika, 1(3):211–218, September 1936.www.convexoptimization.com/TOOLS/eckart&young.1936.pdf[86] Alan Edelman, Tomás A. Arias, and Steven T. Smith. The geometryof algorithms with orthogonality constraints. SIAM Journal on MatrixAnalysis and Applications, 20(2):303–353, 1998.[87] John J. Edgell. Graphics calculator applications on 4-D constructs,1996.http://archives.math.utk.edu/ICTCM/EP-9/C47/pdf/paper.pdf[88] Ivar Ekeland and Roger Témam. Convex Analysis and VariationalProblems. SIAM, 1999.[89] Julius Farkas. Theorie der einfachen Ungleichungen. Journal für diereine und angewandte Mathematik, 124:1–27, 1902.dz-srv1.sub.uni-goettingen.de/sub/digbib/loader?ht=VIEW&did=D261364[90] Maryam Fazel. Matrix Rank Minimization with Applications. PhDthesis, Stanford University, Department of Electrical Engineering,March 2002.http://www.cds.caltech.edu/∼maryam/thesis-final.pdf[91] Maryam Fazel, Haitham Hindi, and Stephen P. Boyd. A rankminimization heuristic with application to minimum order systemapproximation. In Proceedings of the American Control Conference,volume 6, pages 4734–4739. American Automatic Control Council(AACC), June 2001.http://www.cds.caltech.edu/∼maryam/nucnorm.html[92] Maryam Fazel, Haitham Hindi, and Stephen P. Boyd. Log-detheuristic for matrix rank minimization with applications to Hankel andEuclidean distance matrices. In Proceedings of the American ControlConference. American Automatic Control Council (AACC), June 2003.http://www.cds.caltech.edu/∼maryam/acc03 final.pdf[93] Maryam Fazel, Haitham Hindi, and Stephen P. Boyd. Rankminimization and applications in system theory. In Proceedings of theAmerican Control Conference. American Automatic Control Council
BIBLIOGRAPHY 715(AACC), June 2004.http://www.cds.caltech.edu/∼maryam/acc04-tutorial.pdf[94] J. T. Feddema, R. H. Byrne, J. J. Harrington, D. M. Kilman,C. L. Lewis, R. D. Robinett, B. P. Van Leeuwen, and J. G. Young.Advanced mobile networking, sensing, and controls. Sandia ReportSAND2005-1661, Sandia National Laboratories, Albuquerque,New Mexico, March 2005.www.prod.sandia.gov/cgi-bin/techlib/access-control.pl/2005/051661.pdf[95] César Fernández, Thomas Szyperski, Thierry Bruyère, Paul Ramage,Egon Mösinger, and Kurt Wüthrich. NMR solution structure ofthe pathogenesis-related protein P14a. Journal of Molecular Biology,266:576–593, 1997.[96] Richard Phillips Feynman, Robert B. Leighton, and Matthew L. Sands.The Feynman Lectures on Physics: Commemorative Issue, volume I.Addison-Wesley, 1989.[97] Paul Finsler. Über das Vorkommen definiter und semidefiniter Formenin Scharen quadratischer Formen. Commentarii Mathematici Helvetici,9:188–192, 1937.[98] Anders Forsgren, Philip E. Gill, and Margaret H. Wright. Interiormethods for nonlinear optimization. SIAM Review, 44(4):525–597,2002.[99] Norbert Gaffke and Rudolf Mathar. A cyclic projection algorithm viaduality. Metrika, 36:29–54, 1989.[100] Jérôme Galtier. Semi-definite programming as a simple extension tolinear programming: convex optimization with queueing, equity andother telecom functionals. In 3ème Rencontres Francophones sur lesAspects Algorithmiques des Télécommunications (AlgoTel). INRIA,France, 2001.www-sop.inria.fr/mascotte/Jerome.Galtier/misc/Galtier01b.pdf[101] Laurent El Ghaoui. EE 227A: Convex Optimization and Applications,Lecture 11 − October 3. University of California, Berkeley, Fall 2006.Scribe: Nikhil Shetty.http://www.convexoptimization.com/TOOLS/Ghaoui.pdf
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 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 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 47 and 48:
Page 49 and 50:
Page 51 and 52:
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: BIBLIOGRAPHY 713[76] Frank R. Deuts
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

Create successful ePaper yourself

Delete template?

Save as template?