v2007.11.26 - Convex Optimization

More documents

Recommendations

Info

542 APPENDIX B. SIMPLE MATRICESDue to symmetry of H , the matrix 2-norm (the spectral norm) is equal to thelargest eigenvalue-magnitude. A Householder matrix is thus characterized,H T = H , H −1 = H T , ‖H‖ 2 = 1, H 0 (1454)For example, the permutation matrix⎡ ⎤1 0 0Ξ = ⎣ 0 0 1 ⎦ (1455)0 1 0is a Householder matrix having u=[ 0 1 −1 ] T / √ 2 . Not all permutationmatrices are Householder matrices, although all permutation matrices areorthogonal matrices [251,3.4] because they are made by permuting rowsand columns of the identity matrix. Neither are all symmetric permutationmatrices Householder matrices; e.g., Ξ = ⎢⎣Householder matrix.B.4 Auxiliary V -matricesB.4.1Auxiliary projector matrix V⎡0 0 0 10 0 1 00 1 0 01 0 0 0⎤⎥⎦(1533) is not aIt is convenient to define a matrix V that arises naturally as a consequence oftranslating the geometric center α c (5.5.1.0.1) of some list X to the origin.In place of X − α c 1 T we may write XV as in (815) whereV ∆ = I − 1 N 11T ∈ S N (757)is an elementary matrix called the geometric centering matrix.Any elementary matrix in R N×N has N −1 eigenvalues equal to 1. For theparticular elementary matrix V , the N th eigenvalue equals 0. The numberof 0 eigenvalues must equal dim N(V ) = 1, by the 0 eigenvalues theorem(A.7.3.0.1), because V =V T is diagonalizable. BecauseV 1 = 0 (1456)
B.4. AUXILIARY V -MATRICES 543the nullspace N(V )= R(1) is spanned by the eigenvector 1. The remainingeigenvectors span R(V ) ≡ 1 ⊥ = N(1 T ) that has dimension N −1.BecauseV 2 = V (1457)and V T = V , elementary matrix V is also a projection matrix (E.3)projecting orthogonally on its range N(1 T ) which is a hyperplane containingthe origin in R N V = I − 1(1 T 1) −1 1 T (1458)The {0, 1} eigenvalues also indicate diagonalizable V is a projectionmatrix. [303,4.1, thm.4.1] Symmetry of V denotes orthogonal projection;from (1709),V T = V , V † = V , ‖V ‖ 2 = 1, V ≽ 0 (1459)Matrix V is also circulant [118].B.4.1.0.1 Example. Relationship of auxiliary to Householder matrix.Let H ∈ S N be a Householder matrix (1452) defined by⎡ ⎤u = ⎢1.⎥⎣ 11 + √ ⎦ ∈ RN (1460)NThen we have [106,2]Let D ∈ S N h and define[ I 0V = H0 T 0[−HDH = ∆ A b−b T c]H (1461)](1462)where b is a vector. Then because H is nonsingular (A.3.1.0.5) [133,3][ ] A 0−V DV = −H0 T H ≽ 0 ⇔ −A ≽ 0 (1463)0and affine dimension is r = rankA when D is a Euclidean distance matrix.
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 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: B.3. ELEMENTARY MATRIX 541If λ ≠
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 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 587 and 588: D.2. TABLES OF GRADIENTS AND DERIVA
Page 593 and 594:
D.2. TABLES OF GRADIENTS AND DERIVA
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

Create successful ePaper yourself

Delete template?

Save as template?