v2010.10.26 - Convex Optimization
v2010.10.26 - Convex Optimization v2010.10.26 - Convex Optimization
496 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX
Chapter 6Cone of distance matricesFor N > 3, the cone of EDMs is no longer a circular cone andthe geometry becomes complicated...−Hayden, Wells, Liu, & Tarazaga (1991) [185,3]In the subspace of symmetric matrices S N , we know that 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 or negative semidefinite EDM. (1099) [235]EDM N ∩ S N + = 0 (1175)Even so, the two convex cones can be related. In6.8.1 we prove the equalityEDM N = S N h ∩ ( )S N⊥c − S N + (1268)2001 Jon Dattorro. co&edg version 2010.10.26. All rights reserved.citation: Dattorro, Convex Optimization & Euclidean Distance Geometry,Mεβoo Publishing USA, 2005, v2010.10.26.497
- Page 445 and 446: 5.7. EMBEDDING IN AFFINE HULL 4455.
- Page 447 and 448: 5.7. EMBEDDING IN AFFINE HULL 447Fo
- Page 449 and 450: 5.7. EMBEDDING IN AFFINE HULL 4495.
- Page 451 and 452: 5.8. EUCLIDEAN METRIC VERSUS MATRIX
- Page 453 and 454: 5.8. EUCLIDEAN METRIC VERSUS MATRIX
- Page 455 and 456: 5.8. EUCLIDEAN METRIC VERSUS MATRIX
- Page 457 and 458: 5.8. EUCLIDEAN METRIC VERSUS MATRIX
- Page 459 and 460: 5.9. BRIDGE: CONVEX POLYHEDRA TO ED
- Page 461 and 462: 5.9. BRIDGE: CONVEX POLYHEDRA TO ED
- Page 463 and 464: 5.9. BRIDGE: CONVEX POLYHEDRA TO ED
- Page 465 and 466: 5.10. EDM-ENTRY COMPOSITION 465of a
- Page 467 and 468: 5.10. EDM-ENTRY COMPOSITION 467Then
- Page 469 and 470: 5.11. EDM INDEFINITENESS 4695.11.1
- Page 471 and 472: 5.11. EDM INDEFINITENESS 471we have
- Page 473 and 474: 5.11. EDM INDEFINITENESS 473So beca
- Page 475 and 476: 5.11. EDM INDEFINITENESS 475holds o
- Page 477 and 478: 5.12. LIST RECONSTRUCTION 477where
- Page 479 and 480: 5.12. LIST RECONSTRUCTION 479(a)(c)
- Page 481 and 482: 5.13. RECONSTRUCTION EXAMPLES 481Wi
- Page 483 and 484: 5.13. RECONSTRUCTION EXAMPLES 483d
- Page 485 and 486: 5.13. RECONSTRUCTION EXAMPLES 485Th
- Page 487 and 488: 5.14. FIFTH PROPERTY OF EUCLIDEAN M
- Page 489 and 490: 5.14. FIFTH PROPERTY OF EUCLIDEAN M
- Page 491 and 492: 5.14. FIFTH PROPERTY OF EUCLIDEAN M
- Page 493 and 494: 5.14. FIFTH PROPERTY OF EUCLIDEAN M
- Page 495: 5.14. FIFTH PROPERTY OF EUCLIDEAN M
- Page 499 and 500: 6.1. DEFINING EDM CONE 4996.1 Defin
- Page 501 and 502: 6.2. POLYHEDRAL BOUNDS 501This cone
- Page 503 and 504: 6.4. EDM DEFINITION IN 11 T 503That
- Page 505 and 506: 6.4. EDM DEFINITION IN 11 T 505N(1
- Page 507 and 508: 6.4. EDM DEFINITION IN 11 T 507Then
- Page 509 and 510: 6.4. EDM DEFINITION IN 11 T 5096.4.
- Page 511 and 512: 6.5. CORRESPONDENCE TO PSD CONE S N
- Page 513 and 514: 6.5. CORRESPONDENCE TO PSD CONE S N
- Page 515 and 516: 6.5. CORRESPONDENCE TO PSD CONE S N
- Page 517 and 518: 6.6. VECTORIZATION & PROJECTION INT
- Page 519 and 520: 6.6. VECTORIZATION & PROJECTION INT
- Page 521 and 522: 6.6. VECTORIZATION & PROJECTION INT
- Page 523 and 524: 6.7. A GEOMETRY OF COMPLETION 523(b
- Page 525 and 526: 6.7. A GEOMETRY OF COMPLETION 525[3
- Page 527 and 528: 6.7. A GEOMETRY OF COMPLETION 527wh
- Page 529 and 530: 6.8. DUAL EDM CONE 529to the geomet
- Page 531 and 532: 6.8. DUAL EDM CONE 531Proof. First,
- Page 533 and 534: 6.8. DUAL EDM CONE 533EDM 2 = S 2 h
- Page 535 and 536: 6.8. DUAL EDM CONE 535therefore the
- Page 537 and 538: 6.8. DUAL EDM CONE 537Elegance of t
- Page 539 and 540: 6.8. DUAL EDM CONE 5396.8.1.5 Affin
- Page 541 and 542: 6.8. DUAL EDM CONE 5416.8.1.7 Schoe
- Page 543 and 544: 6.8. DUAL EDM CONE 5430dvec rel ∂
- Page 545 and 546: 6.10. POSTSCRIPT 5456.10 Postscript
Chapter 6Cone of distance matricesFor N > 3, the cone of EDMs is no longer a circular cone andthe geometry becomes complicated...−Hayden, Wells, Liu, & Tarazaga (1991) [185,3]In the subspace of symmetric matrices S N , we know that 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 or negative semidefinite EDM. (1099) [235]EDM N ∩ S N + = 0 (1175)Even so, the two convex cones can be related. In6.8.1 we prove the equalityEDM N = S N h ∩ ( )S N⊥c − S N + (1268)2001 Jon Dattorro. co&edg version 2010.10.26. All rights reserved.citation: Dattorro, <strong>Convex</strong> <strong>Optimization</strong> & Euclidean Distance Geometry,Mεβoo Publishing USA, 2005, <strong>v2010.10.26</strong>.497
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