v2007.09.13 - Convex Optimization

More documents

Recommendations

Info

552 APPENDIX D. MATRIX CALCULUSGradient of vector-valued function g(X) : R K×L →R N on matrix domainis a cubix[∇X(:,1) g 1 (X) ∇ X(:,1) g 2 (X) · · · ∇ X(:,1) g N (X)∇g(X) ∆ =∇ X(:,2) g 1 (X) ∇ X(:,2) g 2 (X) · · · ∇ X(:,2) g N (X).........∇ X(:,L) g 1 (X) ∇ X(:,L) g 2 (X) · · · ∇ X(:,L) g N (X) ]= [ ∇g 1 (X) ∇g 2 (X) · · · ∇g N (X) ] ∈ R K×N×L (1539)while the second-order gradient has a five-dimensional representation;[∇∇X(:,1) g 1 (X) ∇∇ X(:,1) g 2 (X) · · · ∇∇ X(:,1) g N (X)∇ 2 g(X) ∆ =∇∇ X(:,2) g 1 (X) ∇∇ X(:,2) g 2 (X) · · · ∇∇ X(:,2) g N (X).........∇∇ X(:,L) g 1 (X) ∇∇ X(:,L) g 2 (X) · · · ∇∇ X(:,L) g N (X) ]= [ ∇ 2 g 1 (X) ∇ 2 g 2 (X) · · · ∇ 2 g N (X) ] ∈ R K×N×L×K×L (1540)The gradient of matrix-valued function g(X) : R K×L →R M×N on matrixdomain has a four-dimensional representation called quartix⎡∇g(X) =∆ ⎢⎣∇g 11 (X) ∇g 12 (X) · · · ∇g 1N (X)∇g 21 (X) ∇g 22 (X) · · · ∇g 2N (X). .∇g M1 (X) ∇g M2 (X) · · ·.∇g MN (X)⎤⎥⎦ ∈ RM×N×K×L (1541)while the second-order gradient has six-dimensional representation⎡∇ 2 g(X) =∆ ⎢⎣and so on.∇ 2 g 11 (X) ∇ 2 g 12 (X) · · · ∇ 2 g 1N (X)∇ 2 g 21 (X) ∇ 2 g 22 (X) · · · ∇ 2 g 2N (X). .∇ 2 g M1 (X) ∇ 2 g M2 (X) · · ·.∇ 2 g MN (X)⎤⎥⎦ ∈ RM×N×K×L×K×L(1542)
D.1. DIRECTIONAL DERIVATIVE, TAYLOR SERIES 553D.1.2Product rules for matrix-functionsGiven dimensionally compatible matrix-valued functions of matrix variablef(X) and g(X)while [39,8.3] [235]∇ X(f(X) T g(X) ) = ∇ X (f)g + ∇ X (g)f (1543)∇ X tr ( f(X) T g(X) ) = ∇ X(tr ( f(X) T g(Z) ) + tr ( g(X)f(Z) T))∣ ∣∣Z←X (1544)These expressions implicitly apply as well to scalar-, vector-, or matrix-valuedfunctions of scalar, vector, or matrix arguments.D.1.2.0.1 Example. Cubix.Suppose f(X) : R 2×2 →R 2 = X T a and g(X) : R 2×2 →R 2 = Xb . We wishto find∇ X(f(X) T g(X) ) = ∇ X a T X 2 b (1545)using the product rule. Formula (1543) calls for∇ X a T X 2 b = ∇ X (X T a)Xb + ∇ X (Xb)X T a (1546)Consider the first of the two terms:∇ X (f)g = ∇ X (X T a)Xb= [ ∇(X T a) 1 ∇(X T a) 2]Xb(1547)The gradient of X T a forms a cubix in R 2×2×2 .⎡∇ X (X T a)Xb =⎢⎣∂(X T a) 1∂X 11 ∂(X T a) 1∂X 21 ∂(X T a) 2∂X 11 ∂(X T a) 1∂(X T a) 2∂X 12 ∂X 12∂(X T a) 2∂X 21 ∂(X T a) 1∂(X T a) 2∂X 22 ∂X 22⎤⎡⎢⎣⎥⎦(1548)⎤(Xb) 1⎥⎦ ∈ R 2×1×2(Xb) 2
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 Quadratic func
Page 17 and 18:
LIST OF FIGURES 17E Projection 5791
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 9: 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 12: 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 732.5
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 24: 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 96 and 97:
96 CHAPTER 2. CONVEX GEOMETRYBCADFi
Page 98 and 99:
98 CHAPTER 2. CONVEX GEOMETRYThe po
Page 100 and 101:
100 CHAPTER 2. CONVEX GEOMETRY2.9.0
Page 102 and 103:
102 CHAPTER 2. CONVEX GEOMETRYwhere
Page 104 and 105:
104 CHAPTER 2. CONVEX GEOMETRY√2
Page 106 and 107:
106 CHAPTER 2. CONVEX GEOMETRYwhich
Page 108 and 109:
108 CHAPTER 2. CONVEX GEOMETRY2.9.2
Page 110:
110 CHAPTER 2. CONVEX GEOMETRYA con
Page 115 and 116:
2.9. POSITIVE SEMIDEFINITE (PSD) CO
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 189where the n
Page 191 and 192:
3.1. CONVEX FUNCTION 191where k ∈
Page 193 and 194:
3.1. CONVEX FUNCTION 193f(z)Az 2z 1
Page 195 and 196:
3.1. CONVEX FUNCTION 195{a T z 1 +
Page 197 and 198:
3.1. CONVEX FUNCTION 197When an epi
Page 199 and 200:
3.1. CONVEX FUNCTION 199orthonormal
Page 201 and 202:
3.1. CONVEX FUNCTION 201[30,1.1] Ex
Page 203 and 204:
3.1. CONVEX FUNCTION 20321.510.5Y 2
Page 205 and 206:
3.1. CONVEX FUNCTION 2053.1.8.0.1 E
Page 207 and 208:
3.1. CONVEX FUNCTION 207This equiva
Page 209 and 210:
3.1. CONVEX FUNCTION 2093.1.8.1 mon
Page 211 and 212:
3.1. CONVEX FUNCTION 211[ Yt]∈ ep
Page 213 and 214:
3.1. CONVEX FUNCTION 213→Y −Xwh
Page 215 and 216:
3.2. MATRIX-VALUED CONVEX FUNCTION
Page 217 and 218:
Page 219 and 220:
Page 221 and 222:
3.3. QUASICONVEX 221A quasiconcave
Page 223 and 224:
3.4. SALIENT PROPERTIES 2236.A nonn
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 2414.2.3.0.2 Example
Page 243 and 244:
4.2. FRAMEWORK 243where δ is the m
Page 245 and 246:
4.2. FRAMEWORK 2454.2.3.0.3 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:
Page 275 and 276:
Page 277 and 278:
Page 279 and 280:
Page 281 and 282:
Page 283 and 284:
Page 285 and 286:
Page 287 and 288:
Page 289 and 290:
Chapter 5Euclidean Distance MatrixT
Page 291 and 292:
5.2. FIRST METRIC PROPERTIES 291cor
Page 293 and 294:
5.3. ∃ FIFTH EUCLIDEAN METRIC PRO
Page 295 and 296:
5.3. ∃ FIFTH EUCLIDEAN METRIC PRO
Page 297 and 298:
5.4. EDM DEFINITION 297The collecti
Page 299 and 300:
5.4. EDM DEFINITION 2995.4.2 Gram-f
Page 301 and 302:
5.4. EDM DEFINITION 301D ∈ EDM N
Page 303 and 304:
5.4. EDM DEFINITION 3035.4.2.2.1 Ex
Page 305 and 306:
5.4. EDM DEFINITION 305ten affine e
Page 307 and 308:
5.4. EDM DEFINITION 307spheres:Then
Page 309 and 310:
5.4. EDM DEFINITION 309By eliminati
Page 311 and 312:
5.4. EDM DEFINITION 311whereΦ ij =
Page 313 and 314:
5.4. EDM DEFINITION 3135.4.2.2.5 De
Page 315 and 316:
5.4. EDM DEFINITION 315105ˇx 4ˇx
Page 317 and 318:
5.4. EDM DEFINITION 317corrected by
Page 319 and 320:
5.4. EDM DEFINITION 319aptly be app
Page 321 and 322:
5.4. EDM DEFINITION 321As before, a
Page 323 and 324:
5.4. EDM DEFINITION 323where ([√t
Page 325 and 326:
5.4. EDM DEFINITION 325because (A.3
Page 327 and 328:
5.5. INVARIANCE 3275.5.1.0.1 Exampl
Page 329 and 330:
5.6. INJECTIVITY OF D & UNIQUE RECO
Page 331 and 332:
Page 333 and 334:
Page 335 and 336:
5.7. EMBEDDING IN AFFINE HULL 3355.
Page 337 and 338:
5.7. EMBEDDING IN AFFINE HULL 337Fo
Page 339 and 340:
5.7. EMBEDDING IN AFFINE HULL 3395.
Page 341 and 342:
5.8. EUCLIDEAN METRIC VERSUS MATRIX
Page 343 and 344:
Page 345 and 346:
Page 347 and 348:
Page 349 and 350:
5.9. BRIDGE: CONVEX POLYHEDRA TO ED
Page 351 and 352:
Page 353 and 354:
Page 355 and 356:
Page 357 and 358:
5.10. EDM-ENTRY COMPOSITION 357(ii)
Page 359 and 360:
5.11. EDM INDEFINITENESS 3595.11.1
Page 361 and 362:
5.11. EDM INDEFINITENESS 361(confer
Page 363 and 364:
5.11. EDM INDEFINITENESS 363we have
Page 365 and 366:
5.11. EDM INDEFINITENESS 365For pre
Page 367 and 368:
5.12. LIST RECONSTRUCTION 367where
Page 369 and 370:
5.12. LIST RECONSTRUCTION 369(a)(c)
Page 371 and 372:
5.13. RECONSTRUCTION EXAMPLES 371D
Page 373 and 374:
5.13. RECONSTRUCTION EXAMPLES 373Th
Page 375 and 376:
5.13. RECONSTRUCTION EXAMPLES 375wh
Page 377 and 378:
5.14. FIFTH PROPERTY OF EUCLIDEAN M
Page 379 and 380:
Page 381 and 382:
Page 383 and 384:
Page 385 and 386:
Page 387 and 388:
Chapter 6EDM coneFor N > 3, the con
Page 389 and 390:
6.1. DEFINING EDM CONE 3896.1 Defin
Page 391 and 392:
6.2. POLYHEDRAL BOUNDS 391This cone
Page 393 and 394:
6.3.√EDM CONE IS NOT CONVEX 393N
Page 395 and 396:
6.4. A GEOMETRY OF COMPLETION 3956.
Page 397 and 398:
6.4. A GEOMETRY OF COMPLETION 397(a
Page 399 and 400:
6.4. A GEOMETRY OF COMPLETION 399Fi
Page 401 and 402:
6.5. EDM DEFINITION IN 11 T 401and
Page 403 and 404:
6.5. EDM DEFINITION IN 11 T 403then
Page 405 and 406:
6.5. EDM DEFINITION IN 11 T 4056.5.
Page 407 and 408:
6.5. EDM DEFINITION IN 11 T 407D =
Page 409 and 410:
6.6. CORRESPONDENCE TO PSD CONE S N
Page 411 and 412:
Page 413 and 414:
Page 415 and 416:
6.7. VECTORIZATION & PROJECTION INT
Page 417 and 418:
6.7. VECTORIZATION & PROJECTION INT
Page 419 and 420:
6.8. DUAL EDM CONE 419When the Fins
Page 421 and 422:
6.8. DUAL EDM CONE 421Proof. First,
Page 423 and 424:
6.8. DUAL EDM CONE 423EDM 2 = S 2 h
Page 425 and 426:
6.8. DUAL EDM CONE 425whose veracit
Page 427 and 428:
6.8. DUAL EDM CONE 4276.8.1.3.1 Exe
Page 429 and 430:
6.8. DUAL EDM CONE 429has dual affi
Page 431 and 432:
6.8. DUAL EDM CONE 4316.8.1.7 Schoe
Page 433 and 434:
6.9. THEOREM OF THE ALTERNATIVE 433
Page 435 and 436:
6.10. POSTSCRIPT 435When D is an ED
Page 437 and 438:
Chapter 7Proximity problemsIn summa
Page 439 and 440:
In contrast, order of projection on
Page 441 and 442:
441HS N h0EDM NK = S N h ∩ R N×N
Page 443 and 444:
4437.0.3 Problem approachProblems t
Page 445 and 446:
7.1. FIRST PREVALENT PROBLEM: 445fi
Page 447 and 448:
7.1. FIRST PREVALENT PROBLEM: 4477.
Page 449 and 450:
7.1. FIRST PREVALENT PROBLEM: 449di
Page 451 and 452:
7.1. FIRST PREVALENT PROBLEM: 4517.
Page 453 and 454:
7.1. FIRST PREVALENT PROBLEM: 453wh
Page 455 and 456:
7.1. FIRST PREVALENT PROBLEM: 455Th
Page 457 and 458:
7.2. SECOND PREVALENT PROBLEM: 457O
Page 459 and 460:
7.2. SECOND PREVALENT PROBLEM: 459S
Page 461 and 462:
7.2. SECOND PREVALENT PROBLEM: 461r
Page 463 and 464:
7.2. SECOND PREVALENT PROBLEM: 463c
Page 465 and 466:
7.2. SECOND PREVALENT PROBLEM: 4657
Page 467 and 468:
7.3. THIRD PREVALENT PROBLEM: 467fo
Page 469 and 470:
7.3. THIRD PREVALENT PROBLEM: 469a
Page 471 and 472:
7.3. THIRD PREVALENT PROBLEM: 4717.
Page 473 and 474:
7.3. THIRD PREVALENT PROBLEM: 4737.
Page 475 and 476:
7.3. THIRD PREVALENT PROBLEM: 475Ou
Page 478 and 479:
478 CHAPTER 7. PROXIMITY PROBLEMSth
Page 480 and 481:
480 APPENDIX A. LINEAR ALGEBRAA.1.1
Page 482 and 483:
Page 484 and 485:
484 APPENDIX A. LINEAR ALGEBRAonly
Page 486 and 487:
486 APPENDIX A. LINEAR ALGEBRA(AB)
Page 488 and 489:
Page 490 and 491:
490 APPENDIX A. LINEAR ALGEBRAFor A
Page 492 and 493:
492 APPENDIX A. LINEAR ALGEBRADiago
Page 494 and 495:
494 APPENDIX A. LINEAR ALGEBRAFor A
Page 496 and 497:
Page 498 and 499:
498 APPENDIX A. LINEAR ALGEBRAA.4 S
Page 500 and 501:
Page 502 and 503: 502 APPENDIX A. LINEAR ALGEBRAA.5 e
Page 504 and 505: 504 APPENDIX A. LINEAR ALGEBRAs i w
Page 506 and 507: 506 APPENDIX A. LINEAR ALGEBRAA.6.2
Page 508 and 509: 508 APPENDIX A. LINEAR ALGEBRAΣq 2
Page 510 and 511: 510 APPENDIX A. LINEAR ALGEBRAA.7 Z
Page 512 and 513: 512 APPENDIX A. LINEAR ALGEBRAThere
Page 514 and 515: 514 APPENDIX A. LINEAR ALGEBRAA.7.5
Page 516 and 517: 516 APPENDIX A. LINEAR ALGEBRA
Page 518 and 519: 518 APPENDIX B. SIMPLE MATRICESB.1
Page 520 and 521: 520 APPENDIX B. SIMPLE MATRICESProo
Page 522 and 523: 522 APPENDIX B. SIMPLE MATRICESB.1.
Page 524 and 525: 524 APPENDIX B. SIMPLE MATRICESN(u
Page 526 and 527: 526 APPENDIX B. SIMPLE MATRICESDue
Page 530 and 531: 530 APPENDIX B. SIMPLE MATRICEShas
Page 532 and 533: 532 APPENDIX B. SIMPLE MATRICESFigu
Page 536 and 537: 536 APPENDIX C. SOME ANALYTICAL OPT
Page 550 and 551: 550 APPENDIX D. MATRIX CALCULUSThe
Page 554 and 555: 554 APPENDIX D. MATRIX CALCULUSBeca
Page 556 and 557: 556 APPENDIX D. MATRIX CALCULUSwhic
Page 558 and 559: 558 APPENDIX D. MATRIX CALCULUS⎡
Page 560 and 561: 560 APPENDIX D. MATRIX CALCULUS→Y
Page 562 and 563: 562 APPENDIX D. MATRIX CALCULUSD.1.
Page 564 and 565: 564 APPENDIX D. MATRIX CALCULUSwhic
Page 566 and 567: 566 APPENDIX D. MATRIX CALCULUSIn t
Page 570 and 571: 570 APPENDIX D. MATRIX CALCULUSD.2
Page 572 and 573: 572 APPENDIX D. MATRIX CALCULUSalge
Page 574 and 575: 574 APPENDIX D. MATRIX CALCULUStrac
Page 578 and 579: 578 APPENDIX D. MATRIX CALCULUS
Page 580 and 581: 580 APPENDIX E. PROJECTIONThe follo
Page 582 and 583: 582 APPENDIX E. PROJECTIONFor matri
Page 584 and 585: 584 APPENDIX E. PROJECTION(⇐) To
Page 586 and 587: 586 APPENDIX E. PROJECTIONNonorthog
Page 588 and 589: 588 APPENDIX E. PROJECTIONE.2.0.0.1
Page 590 and 591: 590 APPENDIX E. PROJECTIONE.3.2Orth
Page 592 and 593: 592 APPENDIX E. PROJECTIONE.3.5Unif
Page 594 and 595: 594 APPENDIX E. PROJECTIONE.4 Algeb
Page 596 and 597: 596 APPENDIX E. PROJECTIONa ∗ 2K
Page 598 and 599: 598 APPENDIX E. PROJECTIONwhere Y =
Page 600 and 601: 600 APPENDIX E. PROJECTION(B.4.2).
Page 602 and 603:
602 APPENDIX E. PROJECTIONis a nono
Page 604 and 605:
604 APPENDIX E. PROJECTIONE.6.4.1Or
Page 606 and 607:
606 APPENDIX E. PROJECTIONq i q T i
Page 608 and 609:
608 APPENDIX E. PROJECTIONThe test
Page 610 and 611:
610 APPENDIX E. PROJECTIONPerpendic
Page 612 and 613:
612 APPENDIX E. PROJECTIONE.8 Range
Page 614 and 615:
614 APPENDIX E. PROJECTIONAs for su
Page 616 and 617:
616 APPENDIX E. PROJECTIONWith refe
Page 618 and 619:
618 APPENDIX E. PROJECTIONProjectio
Page 620 and 621:
620 APPENDIX E. PROJECTIONE.9.2.2.2
Page 622 and 623:
622 APPENDIX E. PROJECTIONThe foreg
Page 624 and 625:
624 APPENDIX E. PROJECTION❇❇❇
Page 626 and 627:
626 APPENDIX E. PROJECTIONE.10 Alte
Page 628 and 629:
628 APPENDIX E. PROJECTIONbH 1H 2P
Page 630 and 631:
630 APPENDIX E. PROJECTIONa(a){y |
Page 632 and 633:
632 APPENDIX E. PROJECTION(a feasib
Page 634 and 635:
634 APPENDIX E. PROJECTIONwhile, th
Page 636 and 637:
636 APPENDIX E. PROJECTIONE.10.2.1.
Page 638 and 639:
638 APPENDIX E. PROJECTION10 0dist(
Page 640 and 641:
640 APPENDIX E. PROJECTIONE.10.3.1D
Page 642 and 643:
642 APPENDIX E. PROJECTIONE 3K ⊥
Page 644 and 645:
644 APPENDIX E. PROJECTION
Page 646 and 647:
646 APPENDIX F. MATLAB PROGRAMSif n
Page 648 and 649:
648 APPENDIX F. MATLAB PROGRAMSend%
Page 650 and 651:
650 APPENDIX F. MATLAB PROGRAMSF.1.
Page 652 and 653:
652 APPENDIX F. MATLAB PROGRAMScoun
Page 654 and 655:
654 APPENDIX F. MATLAB PROGRAMSF.3
Page 656 and 657:
Page 658 and 659:
658 APPENDIX F. MATLAB PROGRAMS% so
Page 660 and 661:
660 APPENDIX F. MATLAB PROGRAMS% tr
Page 662 and 663:
Page 664 and 665:
664 APPENDIX F. MATLAB PROGRAMSbrea
Page 666 and 667:
666 APPENDIX F. MATLAB PROGRAMSwhil
Page 668 and 669:
668 APPENDIX F. MATLAB PROGRAMSF.7
Page 670 and 671:
670 APPENDIX F. MATLAB PROGRAMS
Page 672 and 673:
672 APPENDIX G. NOTATION AND A FEW
Page 674 and 675:
Page 676 and 677:
Page 678 and 679:
Page 680 and 681:
Page 682 and 683:
Page 684 and 685:
Page 686 and 687:
Page 688 and 689:
688 BIBLIOGRAPHY[7] Abdo Y. Alfakih
Page 690 and 691:
690 BIBLIOGRAPHY[27] Aharon Ben-Tal
Page 692 and 693:
692 BIBLIOGRAPHY[48] Lev M. Brègma
Page 694 and 695:
694 BIBLIOGRAPHY[67] Joel Dawson, S
Page 696 and 697:
696 BIBLIOGRAPHY[85] Alan Edelman,
Page 698 and 699:
698 BIBLIOGRAPHY[102] Philip E. Gil
Page 700 and 701:
700 BIBLIOGRAPHYWeiss, editors, Pol
Page 702 and 703:
702 BIBLIOGRAPHY[146] Jean-Baptiste
Page 704 and 705:
704 BIBLIOGRAPHY[168] Jean B. Lasse
Page 706 and 707:
706 BIBLIOGRAPHY[189] Rudolf Mathar
Page 708 and 709:
708 BIBLIOGRAPHY[211] M. L. Overton
Page 710 and 711:
710 BIBLIOGRAPHY[229] C. K. Rushfor
Page 712 and 713:
712 BIBLIOGRAPHY[252] Jos F. Sturm
Page 714 and 715:
714 BIBLIOGRAPHY[274] È. B. Vinber
Page 716 and 717:
[294] Yinyu Ye. Semidefinite progra
Page 718 and 719:
718 INDEXobtuse, 62positive semidef
Page 720 and 721:
720 INDEXelliptope, 642orthant, 177
Page 722 and 723:
722 INDEXdistancegeometry, 20, 317m
Page 724 and 725:
724 INDEX-dimensional, 37, 89rank,
Page 726 and 727:
726 INDEXGram form, 331is, 674isedm
Page 728 and 729:
728 INDEXdiscretized, 152, 431in su
Page 730 and 731:
730 INDEXboundary, 115dimension, 10
Page 732 and 733:
732 INDEXlinear operator, 587, 591,
Page 734 and 735:
734 INDEXlargest entries, 188monoto
Page 736:
736 INDEXsimilarity, 606translation
show all

v2007.09.13 - Convex Optimization

Create successful ePaper yourself

Delete template?

Save as template?