v2009.01.01 - Convex Optimization

More documents

Recommendations

Info

440 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX c i = = = [ ] −1 0 1 2 N−2 (N −2)! 2 det T 1 −D i (−1) N 2 N−2 (N −2)! 2 detD i (1072) ( 1 T D −1 i 1 ) (1073) (−1) N 2 N−2 (N −2)! 2 1T cof(D i ) T 1 (1074) where D i is the i th principal N −1×N −1 submatrix 5.57 of D ∈ EDM N , and cof(D i ) is the N −1×N −1 matrix of cofactors [287,4] corresponding to D i . The number of principal 3 × 3 submatrices in D is, of course, equal to the number of triangular facets in the tetrahedron; four (N!/(3!(N −3)!)) when N = 4. 5.14.3.1.1 Exercise. Sufficiency conditions for an EDM of four points. Triangle inequality (property 4) and area inequality (1070) are conditions necessary for D to be an EDM. Prove their sufficiency in conjunction with the remaining three Euclidean metric properties. 5.14.3.2 N = 5 Moving to the next level, we might encounter a Euclidean body called polychoron: a bounded polyhedron in four dimensions. 5.58 The polychoron has five (N!/(4!(N −4)!)) facets, each of them a general tetrahedron whose volume-squared c i is calculated using the same formula; (1072) where D is the EDM corresponding to the polychoron, and D i is the EDM corresponding to the i th facet (the principal 4 × 4 submatrix of D ∈ EDM N corresponding to the i th tetrahedron). The analogue to triangle & distance is now polychoron & facet-volume. We could then write another generalized “triangle” inequality like (1070) but in terms of facet volume; [331,IV] √ ci ≤ √ c j + √ c k + √ c l + √ c m , i≠j ≠k≠l≠m∈{1... 5} (1075) 5.57 Every principal submatrix of an EDM remains an EDM. [202,4.1] 5.58 The simplest polychoron is called a pentatope [326]; a regular simplex hence convex. (A pentahedron is a three-dimensional body having five vertices.)
5.14. FIFTH PROPERTY OF EUCLIDEAN METRIC 441 . Figure 113: Length of one-dimensional face a equals height h=a=1 of this nonsimplicial pyramid in R 3 with square base inscribed in a circle of radius R centered at the origin. [326, Pyramid] 5.14.3.2.1 Exercise. Sufficiency for an EDM of five points. For N = 5, triangle (distance) inequality (5.2), area inequality (1070), and volume inequality (1075) are conditions necessary for D to be an EDM. Prove their sufficiency. 5.14.3.3 Volume of simplices There is no known formula for the volume of a bounded general convex polyhedron expressed either by halfspace or vertex-description. [342,2.1] [243, p.173] [200] [147] [148] Volume is a concept germane to R 3 ; in higher dimensions it is called content. Applying the EDM assertion (5.9.1.0.4) and a result from [53, p.407], a general nonempty simplex (2.12.3) in R N−1 corresponding to an EDM D ∈ S N h has content √ c = content(S) √ det(−V T N DV N) (1076) where the content-squared of the unit simplex S ⊂ R N−1 is proportional to its Cayley-Menger determinant; content(S) 2 = [ ] −1 0 1 2 N−1 (N −1)! 2 det T 1 −D([0 e 1 e 2 · · · e N−1 ]) where e i ∈ R N−1 and the EDM operator used is D(X) (798). (1077)
Page 1 and 2:
DATTORRO CONVEX OPTIMIZATION & EUCL
Page 3 and 4:
Convex Optimization & Euclidean Dis
Page 5 and 6:
for Jennie Columba ♦ Antonio ♦
Page 7 and 8:
Prelude The constant demands of my
Page 9 and 10:
Page 11 and 12:
CONVEX OPTIMIZATION & EUCLIDEAN DIS
Page 13 and 14:
List of Figures 1 Overview 19 1 Ori
Page 15 and 16:
LIST OF FIGURES 15 3 Geometry of co
Page 17 and 18:
LIST OF FIGURES 17 126 Decomposing
Page 19 and 20:
Chapter 1 Overview Convex Optimizat
Page 21 and 22:
ˇx 4 ˇx 3 ˇx 2 Figure 2: Applica
Page 23 and 24:
23 Figure 4: This coarsely discreti
Page 25 and 26:
ases (biorthogonal expansion). We e
Page 27 and 28:
27 Figure 7: These bees construct a
Page 29 and 30:
that establish its membership to th
Page 31 and 32:
31 appendices Provided so as to be
Page 33 and 34:
Chapter 2 Convex geometry Convexity
Page 35 and 36:
2.1. CONVEX SET 35 2.1.2 linear ind
Page 37 and 38:
2.1. CONVEX SET 37 2.1.6 empty set
Page 39 and 40:
2.1. CONVEX SET 39 2.1.7.1 Line int
Page 41 and 42:
2.1. CONVEX SET 41 (a) R 2 (b) R 3
Page 43 and 44:
2.1. CONVEX SET 43 This theorem in
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:
Page 55 and 56:
2.3. HULLS 55 Figure 16: Convex hul
Page 57 and 58:
2.3. HULLS 57 The affine hull of tw
Page 59 and 60:
2.3. HULLS 59 2.3.2 Convex hull The
Page 61 and 62:
2.3. HULLS 61 In case k = N , the F
Page 63 and 64:
2.3. HULLS 63 2.3.2.0.3 Exercise. C
Page 65 and 66:
2.3. HULLS 65 Figure 20: A simplici
Page 67 and 68:
2.4. HALFSPACE, HYPERPLANE 67 H + a
Page 69 and 70:
2.4. HALFSPACE, HYPERPLANE 69 1 1
Page 71 and 72:
2.4. HALFSPACE, HYPERPLANE 71 Recal
Page 73 and 74:
2.4. HALFSPACE, HYPERPLANE 73 C H
Page 75 and 76:
2.4. HALFSPACE, HYPERPLANE 75 2.4.2
Page 77 and 78:
2.4. HALFSPACE, HYPERPLANE 77 (conf
Page 79 and 80:
2.5. SUBSPACE REPRESENTATIONS 79 Ra
Page 81 and 82:
2.5. SUBSPACE REPRESENTATIONS 81 If
Page 83 and 84:
2.6. EXTREME, EXPOSED 83 2.6 Extrem
Page 85 and 86:
2.6. EXTREME, EXPOSED 85 A B C D Fi
Page 87 and 88:
2.7. CONES 87 2.6.1.3.1 Definition.
Page 89 and 90:
2.7. CONES 89 0 Figure 30: Boundary
Page 91 and 92:
2.7. CONES 91 2.7.2 Convex cone We
Page 93 and 94:
2.7. CONES 93 Then a pointed closed
Page 95 and 96:
2.7. CONES 95 A pointed closed conv
Page 97 and 98:
2.8. CONE BOUNDARY 97 So the ray th
Page 99 and 100:
2.8. CONE BOUNDARY 99 2.8.1.1 extre
Page 101 and 102:
2.8. CONE BOUNDARY 101 2.8.2 Expose
Page 103 and 104:
2.8. CONE BOUNDARY 103 From Theorem
Page 105 and 106:
2.9. POSITIVE SEMIDEFINITE (PSD) CO
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:
Page 123 and 124:
Page 125 and 126:
Page 127 and 128:
2.10. CONIC INDEPENDENCE (C.I.) 127
Page 129 and 130:
Page 131 and 132:
Page 133 and 134:
2.12. CONVEX POLYHEDRA 133 It follo
Page 135 and 136:
2.12. CONVEX POLYHEDRA 135 Coeffici
Page 137 and 138:
2.12. CONVEX POLYHEDRA 137 2.12.3 U
Page 139 and 140:
2.12. CONVEX POLYHEDRA 139
Page 141 and 142:
2.13. DUAL CONE & GENERALIZED INEQU
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:
Page 185 and 186:
Page 187 and 188:
Page 189 and 190:
Page 191 and 192:
Page 193 and 194:
Page 195 and 196:
Chapter 3 Geometry of convex functi
Page 197 and 198:
3.1. CONVEX FUNCTION 197 f 1 (x) f
Page 199 and 200:
3.1. CONVEX FUNCTION 199 3.1.3 norm
Page 201 and 202:
3.1. CONVEX FUNCTION 201 A B 1 Figu
Page 203 and 204:
3.1. CONVEX FUNCTION 203 k/m 1 0.9
Page 205 and 206:
k∑ i=1 3.1. CONVEX FUNCTION 205 S
Page 207 and 208:
3.1. CONVEX FUNCTION 207 rather x >
Page 209 and 210:
3.1. CONVEX FUNCTION 209 rather ] x
Page 211 and 212:
3.1. CONVEX FUNCTION 211 3.1.6.0.2
Page 213 and 214:
3.1. CONVEX FUNCTION 213 q(x) f(x)
Page 215 and 216:
3.1. CONVEX FUNCTION 215 3.1.7.0.2
Page 217 and 218:
3.1. CONVEX FUNCTION 217 We learned
Page 219 and 220:
3.1. CONVEX FUNCTION 219 Since opti
Page 221 and 222:
3.1. CONVEX FUNCTION 221 2 1.5 1 0.
Page 223 and 224:
3.1. CONVEX FUNCTION 223 Setting th
Page 225 and 226:
3.1. CONVEX FUNCTION 225 Similarly,
Page 227 and 228:
3.1. CONVEX FUNCTION 227 For vector
Page 229 and 230:
3.1. CONVEX FUNCTION 229 This means
Page 231 and 232:
3.1. CONVEX FUNCTION 231 f(Y ) −
Page 233 and 234:
3.2. MATRIX-VALUED CONVEX FUNCTION
Page 235 and 236:
Page 237 and 238:
Page 239 and 240:
3.3. QUASICONVEX 239 exponential al
Page 241 and 242:
3.3. QUASICONVEX 241 Unlike convex
Page 243 and 244:
3.4. SALIENT PROPERTIES 243 6. (af
Page 245 and 246:
Chapter 4 Semidefinite programming
Page 247 and 248:
4.1. CONIC PROBLEM 247 where K is a
Page 249 and 250:
4.1. CONIC PROBLEM 249 4.1.1.2 Redu
Page 251 and 252:
4.1. CONIC PROBLEM 251 In any SDP f
Page 253 and 254:
4.1. CONIC PROBLEM 253 Proposition
Page 255 and 256:
4.2. FRAMEWORK 255 sets are closed
Page 257 and 258:
4.2. FRAMEWORK 257 4.2.1.1.3 Exampl
Page 259 and 260:
4.2. FRAMEWORK 259 4.2.2 Duals The
Page 261 and 262:
4.2. FRAMEWORK 261 When equality is
Page 263 and 264:
4.2. FRAMEWORK 263 The pseudoinvers
Page 265 and 266:
4.2. FRAMEWORK 265 For the data giv
Page 267 and 268:
4.2. FRAMEWORK 267 minimizes an aff
Page 269 and 270:
4.3. RANK REDUCTION 269 whose rank
Page 271 and 272:
4.3. RANK REDUCTION 271 and where m
Page 273 and 274:
4.3. RANK REDUCTION 273 4.3.3 Optim
Page 275 and 276:
4.3. RANK REDUCTION 275 Initialize:
Page 277 and 278:
4.4. RANK-CONSTRAINED SEMIDEFINITE
Page 279 and 280:
Page 281 and 282:
Page 283 and 284:
Page 285 and 286:
Page 287 and 288:
Page 289 and 290:
Page 291 and 292:
Page 293 and 294:
Page 295 and 296:
4.5. CONSTRAINING CARDINALITY 295 m
Page 297 and 298:
4.5. CONSTRAINING CARDINALITY 297 m
Page 299 and 300:
4.5. CONSTRAINING CARDINALITY 299 a
Page 301 and 302:
4.5. CONSTRAINING CARDINALITY 301 f
Page 303 and 304:
4.5. CONSTRAINING CARDINALITY 303 n
Page 305 and 306:
4.5. CONSTRAINING CARDINALITY 305 W
Page 307 and 308:
4.5. CONSTRAINING CARDINALITY 307 t
Page 309 and 310:
4.6. CARDINALITY AND RANK CONSTRAIN
Page 311 and 312:
Page 313 and 314:
Page 315 and 316:
Page 317 and 318:
Page 319 and 320:
Page 321 and 322:
Page 323 and 324:
Page 325 and 326:
Page 327 and 328:
Page 329 and 330:
Page 331 and 332:
Page 333 and 334:
Page 335 and 336:
Page 337 and 338:
Page 339 and 340:
Page 341 and 342:
4.7. CONVEX ITERATION RANK-1 341 fi
Page 343 and 344:
4.7. CONVEX ITERATION RANK-1 343 Gi
Page 345 and 346:
Chapter 5 Euclidean Distance Matrix
Page 347 and 348:
5.2. FIRST METRIC PROPERTIES 347 co
Page 349 and 350:
5.3. ∃ FIFTH EUCLIDEAN METRIC PRO
Page 351 and 352:
5.3. ∃ FIFTH EUCLIDEAN METRIC PRO
Page 353 and 354:
5.4. EDM DEFINITION 353 The collect
Page 355 and 356:
5.4. EDM DEFINITION 355 5.4.2 Gram-
Page 357 and 358:
5.4. EDM DEFINITION 357 D ∈ EDM N
Page 359 and 360:
5.4. EDM DEFINITION 359 5.4.2.2.1 E
Page 361 and 362:
5.4. EDM DEFINITION 361 ten affine
Page 363 and 364:
5.4. EDM DEFINITION 363 spheres: Th
Page 365 and 366:
5.4. EDM DEFINITION 365 By eliminat
Page 367 and 368:
5.4. EDM DEFINITION 367 where Φ ij
Page 369 and 370:
5.4. EDM DEFINITION 369 5.4.2.2.6 D
Page 371 and 372:
5.4. EDM DEFINITION 371 10 5 ˇx 4
Page 373 and 374:
5.4. EDM DEFINITION 373 corrected b
Page 375 and 376:
5.4. EDM DEFINITION 375 by translat
Page 377 and 378:
5.4. EDM DEFINITION 377 Crippen & H
Page 379 and 380:
5.4. EDM DEFINITION 379 where ([√
Page 381 and 382:
5.4. EDM DEFINITION 381 because (A.
Page 383 and 384:
5.5. INVARIANCE 383 5.5.1.0.1 Examp
Page 385 and 386:
5.5. INVARIANCE 385 x 2 x 2 x 3 x 1
Page 387 and 388:
5.6. INJECTIVITY OF D & UNIQUE RECO
Page 389 and 390: 5.6. INJECTIVITY OF D & UNIQUE RECO
Page 391 and 392: 5.6. INJECTIVITY OF D & UNIQUE RECO
Page 393 and 394: 5.7. EMBEDDING IN AFFINE HULL 393 5
Page 395 and 396: 5.7. EMBEDDING IN AFFINE HULL 395 F
Page 397 and 398: 5.7. EMBEDDING IN AFFINE HULL 397 5
Page 399 and 400: 5.8. EUCLIDEAN METRIC VERSUS MATRIX
Page 407 and 408: 5.9. BRIDGE: CONVEX POLYHEDRA TO ED
Page 413 and 414: 5.10. EDM-ENTRY COMPOSITION 413 of
Page 415 and 416: 5.10. EDM-ENTRY COMPOSITION 415 The
Page 417 and 418: 5.11. EDM INDEFINITENESS 417 5.11.1
Page 419 and 420: 5.11. EDM INDEFINITENESS 419 we hav
Page 421 and 422: 5.11. EDM INDEFINITENESS 421 So bec
Page 423 and 424: 5.11. EDM INDEFINITENESS 423 where
Page 425 and 426: 5.12. LIST RECONSTRUCTION 425 where
Page 427 and 428: 5.12. LIST RECONSTRUCTION 427 (a) (
Page 429 and 430: 5.13. RECONSTRUCTION EXAMPLES 429 D
Page 431 and 432: 5.13. RECONSTRUCTION EXAMPLES 431 T
Page 433 and 434: 5.13. RECONSTRUCTION EXAMPLES 433 w
Page 435 and 436: 5.14. FIFTH PROPERTY OF EUCLIDEAN M
Page 439: 5.14. FIFTH PROPERTY OF EUCLIDEAN M
Page 445 and 446: Chapter 6 Cone of distance matrices
Page 447 and 448: 6.1. DEFINING EDM CONE 447 6.1 Defi
Page 449 and 450: 6.2. POLYHEDRAL BOUNDS 449 This con
Page 451 and 452: 6.3. √ EDM CONE IS NOT CONVEX 451
Page 453 and 454: 6.4. A GEOMETRY OF COMPLETION 453 (
Page 455 and 456: 6.4. A GEOMETRY OF COMPLETION 455 (
Page 457 and 458: 6.4. A GEOMETRY OF COMPLETION 457 F
Page 459 and 460: 6.5. EDM DEFINITION IN 11 T 459 by
Page 461 and 462: 6.5. EDM DEFINITION IN 11 T 461 6.5
Page 463 and 464: 6.5. EDM DEFINITION IN 11 T 463 1 0
Page 465 and 466: 6.5. EDM DEFINITION IN 11 T 465 6.5
Page 467 and 468: 6.6. CORRESPONDENCE TO PSD CONE S N
Page 473 and 474: 6.7. VECTORIZATION & PROJECTION INT
Page 475 and 476: 6.7. VECTORIZATION & PROJECTION INT
Page 477 and 478: 6.8. DUAL EDM CONE 477 When the Fin
Page 479 and 480: 6.8. DUAL EDM CONE 479 Proof. First
Page 481 and 482: 6.8. DUAL EDM CONE 481 EDM 2 = S 2
Page 483 and 484: 6.8. DUAL EDM CONE 483 whose veraci
Page 485 and 486: 6.8. DUAL EDM CONE 485 6.8.1.3.1 Ex
Page 487 and 488: 6.8. DUAL EDM CONE 487 has dual aff
Page 489 and 490: 6.8. DUAL EDM CONE 489 6.8.1.7 Scho
Page 491 and 492:
6.9. THEOREM OF THE ALTERNATIVE 491
Page 493 and 494:
6.10. POSTSCRIPT 493 When D is an E
Page 495 and 496:
Chapter 7 Proximity problems In sum
Page 497 and 498:
497 project on the subspace, then p
Page 499 and 500:
499 H S N h 0 EDM N K = S N h ∩ R
Page 501 and 502:
501 7.0.3 Problem approach Problems
Page 503 and 504:
7.1. FIRST PREVALENT PROBLEM: 503 f
Page 505 and 506:
7.1. FIRST PREVALENT PROBLEM: 505 7
Page 507 and 508:
7.1. FIRST PREVALENT PROBLEM: 507 d
Page 509 and 510:
7.1. FIRST PREVALENT PROBLEM: 509 7
Page 511 and 512:
7.1. FIRST PREVALENT PROBLEM: 511 w
Page 513 and 514:
7.1. FIRST PREVALENT PROBLEM: 513 T
Page 515 and 516:
7.2. SECOND PREVALENT PROBLEM: 515
Page 517 and 518:
Page 519 and 520:
Page 521 and 522:
Page 523 and 524:
Page 525 and 526:
7.3. THIRD PREVALENT PROBLEM: 525 g
Page 527 and 528:
7.3. THIRD PREVALENT PROBLEM: 527 w
Page 529 and 530:
7.3. THIRD PREVALENT PROBLEM: 529 7
Page 531 and 532:
7.3. THIRD PREVALENT PROBLEM: 531 7
Page 533 and 534:
7.3. THIRD PREVALENT PROBLEM: 533 O
Page 535 and 536:
7.4. CONCLUSION 535 The rank constr
Page 537 and 538:
Appendix A Linear algebra A.1 Main-
Page 539 and 540:
A.1. MAIN-DIAGONAL δ OPERATOR, λ
Page 541 and 542:
A.1. MAIN-DIAGONAL δ OPERATOR, λ
Page 543 and 544:
A.2. SEMIDEFINITENESS: DOMAIN OF TE
Page 545 and 546:
A.3. PROPER STATEMENTS 545 (AB) T
Page 547 and 548:
A.3. PROPER STATEMENTS 547 A.3.1 Se
Page 549 and 550:
A.3. PROPER STATEMENTS 549 For A di
Page 551 and 552:
A.3. PROPER STATEMENTS 551 Diagonal
Page 553 and 554:
A.3. PROPER STATEMENTS 553 For A,B
Page 555 and 556:
A.3. PROPER STATEMENTS 555 A.3.1.0.
Page 557 and 558:
A.4. SCHUR COMPLEMENT 557 A.4 Schur
Page 559 and 560:
A.4. SCHUR COMPLEMENT 559 A.4.0.0.2
Page 561 and 562:
A.5. EIGEN DECOMPOSITION 561 When B
Page 563 and 564:
A.5. EIGEN DECOMPOSITION 563 dim N(
Page 565 and 566:
A.6. SINGULAR VALUE DECOMPOSITION,
Page 567 and 568:
A.6. SINGULAR VALUE DECOMPOSITION,
Page 569 and 570:
A.7. ZEROS 569 Given symmetric matr
Page 571 and 572:
A.7. ZEROS 571 (TRANSPOSE.) Likewis
Page 573 and 574:
A.7. ZEROS 573 For X,A∈ S M + [31
Page 575 and 576:
A.7. ZEROS 575 A.7.5.0.1 Propositio
Page 577 and 578:
Appendix B Simple matrices Mathemat
Page 579 and 580:
B.1. RANK-ONE MATRIX (DYAD) 579 R(v
Page 581 and 582:
B.1. RANK-ONE MATRIX (DYAD) 581 ran
Page 583 and 584:
B.2. DOUBLET 583 R([u v ]) R(Π)= R
Page 585 and 586:
B.3. ELEMENTARY MATRIX 585 If λ
Page 587 and 588:
B.4. AUXILIARY V -MATRICES 587 the
Page 589 and 590:
B.4. AUXILIARY V -MATRICES 589 18.
Page 591 and 592:
B.5. ORTHOGONAL MATRIX 591 B.5 Orth
Page 593 and 594:
B.5. ORTHOGONAL MATRIX 593 Figure 1
Page 595 and 596:
Appendix C Some analytical optimal
Page 597 and 598:
C.2. TRACE, SINGULAR AND EIGEN VALU
Page 599 and 600:
Page 601 and 602:
Page 603 and 604:
C.3. ORTHOGONAL PROCRUSTES PROBLEM
Page 605 and 606:
C.4. TWO-SIDED ORTHOGONAL PROCRUSTE
Page 607 and 608:
C.4. TWO-SIDED ORTHOGONAL PROCRUSTE
Page 609 and 610:
Appendix D Matrix calculus From too
Page 611 and 612:
D.1. DIRECTIONAL DERIVATIVE, TAYLOR
Page 613 and 614:
Page 615 and 616:
Page 617 and 618:
Page 619 and 620:
Page 621 and 622:
Page 623 and 624:
Page 625 and 626:
Page 627 and 628:
Page 629 and 630:
Page 631 and 632:
D.2. TABLES OF GRADIENTS AND DERIVA
Page 633 and 634:
Page 635 and 636:
Page 637 and 638:
Page 639 and 640:
Appendix E Projection For any A∈
Page 641 and 642:
641 U T = U † for orthonormal (in
Page 643 and 644:
E.1. IDEMPOTENT MATRICES 643 where
Page 645 and 646:
E.1. IDEMPOTENT MATRICES 645 order,
Page 647 and 648:
E.1. IDEMPOTENT MATRICES 647 When t
Page 649 and 650:
E.3. SYMMETRIC IDEMPOTENT MATRICES
Page 651 and 652:
Page 653 and 654:
Page 655 and 656:
E.5. PROJECTION EXAMPLES 655 E.4.0.
Page 657 and 658:
E.5. PROJECTION EXAMPLES 657 a ∗
Page 659 and 660:
E.5. PROJECTION EXAMPLES 659 E.5.0.
Page 661 and 662:
E.6. VECTORIZATION INTERPRETATION,
Page 663 and 664:
Page 665 and 666:
Page 667 and 668:
Page 669 and 670:
E.7. ON VECTORIZED MATRICES OF HIGH
Page 671 and 672:
E.7. ON VECTORIZED MATRICES OF HIGH
Page 673 and 674:
E.8. RANGE/ROWSPACE INTERPRETATION
Page 675 and 676:
E.9. PROJECTION ON CONVEX SET 675 A
Page 677 and 678:
E.9. PROJECTION ON CONVEX SET 677 W
Page 679 and 680:
E.9. PROJECTION ON CONVEX SET 679 P
Page 681 and 682:
E.9. PROJECTION ON CONVEX SET 681 E
Page 683 and 684:
E.9. PROJECTION ON CONVEX SET 683 T
Page 685 and 686:
E.9. PROJECTION ON CONVEX SET 685
Page 687 and 688:
E.10. ALTERNATING PROJECTION 687 E.
Page 689 and 690:
E.10. ALTERNATING PROJECTION 689 b
Page 691 and 692:
E.10. ALTERNATING PROJECTION 691 a
Page 693 and 694:
E.10. ALTERNATING PROJECTION 693 (a
Page 695 and 696:
E.10. ALTERNATING PROJECTION 695 wh
Page 697 and 698:
Page 699 and 700:
E.10. ALTERNATING PROJECTION 699 10
Page 701 and 702:
Page 703 and 704:
E.10. ALTERNATING PROJECTION 703 E
Page 705 and 706:
Appendix F Notation and a few defin
Page 707 and 708:
707 a.i. c.i. l.i. w.r.t affinely i
Page 709 and 710:
709 is or ← → t → 0 + as in
Page 711 and 712:
711 ∑ π(γ) Ξ Π ∏ ψ(Z) D D
Page 713 and 714:
713 R m×n Euclidean vector space o
Page 715 and 716:
715 H − H + ∂H ∂H ∂H −
Page 717 and 718:
717 O O sort-index matrix order of
Page 719 and 720:
(x,y) angle between vectors x and y
Page 721 and 722:
Bibliography [1] Suliman Al-Homidan
Page 723 and 724:
BIBLIOGRAPHY 723 [24] Alexander I.
Page 725 and 726:
BIBLIOGRAPHY 725 [52] Stephen Boyd,
Page 727 and 728:
BIBLIOGRAPHY 727 [78] Frank Critchl
Page 729 and 730:
BIBLIOGRAPHY 729 [105] Richard L. D
Page 731 and 732:
BIBLIOGRAPHY 731 [132] Michel X. Go
Page 733 and 734:
BIBLIOGRAPHY 733 [162] T. Herrmann,
Page 735 and 736:
BIBLIOGRAPHY 735 [191] Mark Kahrs a
Page 737 and 738:
BIBLIOGRAPHY 737 [220] K. V. Mardia
Page 739 and 740:
BIBLIOGRAPHY 739 [250] Pythagoras P
Page 741 and 742:
BIBLIOGRAPHY 741 [277] Anthony Man-
Page 743 and 744:
BIBLIOGRAPHY 743 [306] Michael W. T
Page 745 and 746:
[333] Margaret H. Wright. The inter
Page 747 and 748:
Index 0-norm, 203, 261, 294, 296, 2
Page 749 and 750:
INDEX 749 product, 43, 92, 147, 254
Page 751 and 752:
INDEX 751 coordinates, 140, 170, 17
Page 753 and 754:
INDEX 753 affine dimension, 485 fea
Page 755 and 756:
INDEX 755 affine, 209 nonlinear, 19
Page 757 and 758:
INDEX 757 of point, 37 ray, 90 rela
Page 759 and 760:
INDEX 759 normal, 47, 548, 563 norm
Page 761 and 762:
INDEX 761 strictly, 515, 520 functi
Page 763 and 764:
INDEX 763 vector, 45, 241, 248, 325
Page 765 and 766:
INDEX 765 convex envelope, see conv
Page 767 and 768:
INDEX 767 cone, 418, 420, 507 dual,
Page 769 and 770:
INDEX 769 trilateration, 21, 42, 36
Page 772:
show all

v2009.01.01 - Convex Optimization

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

Delete template?

Save as template?