v2007.09.13 - Convex Optimization
v2007.09.13 - Convex Optimization v2007.09.13 - Convex Optimization
386 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX
Chapter 6EDM coneFor N > 3, the cone of EDMs is no longer a circular cone andthe geometry becomes complicated...−Hayden, Wells, Liu, & Tarazaga (1991) [133,3]In the subspace of symmetric matrices S N , we know the convex cone ofEuclidean distance matrices EDM N (the EDM cone) does not intersect thepositive semidefinite cone S N + (PSD cone) except at the origin, their onlyvertex; there can be no positive nor negative semidefinite EDM. (902) [170]EDM N ∩ S N + = 0 (978)Even so, the two convex cones can be related. In6.8.1 we prove the newequalityEDM N = S N h ∩ ( )S N⊥c − S N + (1070)2001 Jon Dattorro. CO&EDG version 2007.09.13. All rights reserved.Citation: Jon Dattorro, Convex Optimization & Euclidean Distance Geometry,Meboo Publishing USA, 2005.387
- 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
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- Page 349 and 350: 5.9. BRIDGE: CONVEX POLYHEDRA TO ED
- Page 351 and 352: 5.9. BRIDGE: CONVEX POLYHEDRA TO ED
- Page 353 and 354: 5.9. BRIDGE: CONVEX POLYHEDRA TO ED
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- 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
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- 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
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- 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
386 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX
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