v2010.10.26 - Convex Optimization
v2010.10.26 - Convex Optimization v2010.10.26 - Convex Optimization
514 CHAPTER 6. CONE OF DISTANCE MATRICES6.5.0.0.2 Example. ⎡ Extreme ⎤ rays versus rays on the boundary.0 1 4The EDM D = ⎣ 1 0 1 ⎦ is an extreme direction of EDM 3 where[ ]4 1 01u = in (1215). Because −VN 2TDV N has eigenvalues {0, 5} , the raywhose direction is D also lies on[ the relative ] boundary of EDM 3 .0 1In exception, EDM D = κ , for any particular κ > 0, is an1 0extreme direction of EDM 2 but −VN TDV N has only one eigenvalue: {κ}.Because EDM 2 is a ray whose relative boundary (2.6.1.4.1) is the origin,this conventional boundary does not include D which belongs to the relativeinterior in this dimension. (2.7.0.0.1)6.5.1 Gram-form correspondence to S N−1+With respect to D(G)=δ(G)1 T + 1δ(G) T − 2G (903) the linear Gram-formEDM operator, results in5.6.1 provide [2,2.6]EDM N = D ( V(EDM N ) ) ≡ D ( )V N S N−1+ VNT(1220)V N S N−1+ VN T ≡ V ( D ( ))V N S N−1+ VN T = V(EDM N ) −V EDM N V 1 = 2 SN c ∩ S N +(1221)a one-to-one correspondence between EDM N and S N−1+ .6.5.2 EDM cone by elliptopeHaving defined the elliptope parametrized by scalar t>0then following Alfakih [9] we haveE N t = S N + ∩ {Φ∈ S N | δ(Φ)=t1} (1097)EDM N = cone{11 T − E N 1 } = {t(11 T − E N 1 ) | t ≥ 0} (1222)Identification E N = E N 1 equates the standard elliptope (5.9.1.0.1,Figure 130) to our parametrized elliptope.
6.5. CORRESPONDENCE TO PSD CONE S N−1+ 515dvec rel ∂ EDM 3dvec(11 T − E 3 )EDM N = cone{11 T − E N } = {t(11 T − E N ) | t ≥ 0} (1222)Figure 141: Three views of translated negated elliptope 11 T − E 3 1(confer Figure 130) shrouded by truncated EDM cone. Fractal on EDMcone relative boundary is numerical artifact belonging to intersection withelliptope relative boundary. The fractal is trying to convey existence of aneighborhood about the origin where the translated elliptope boundary andEDM cone boundary intersect.
- 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
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- Page 497 and 498: Chapter 6Cone of distance matricesF
- 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.
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- Page 517 and 518: 6.6. VECTORIZATION & PROJECTION INT
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- 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
- Page 547 and 548: Chapter 7Proximity problemsIn the
- Page 549 and 550: 549project on the subspace, then pr
- Page 551 and 552: 551HS N h0EDM NK = S N h ∩ R N×N
- Page 553 and 554: 5537.0.3 Problem approachProblems t
- Page 555 and 556: 7.1. FIRST PREVALENT PROBLEM: 555fi
- Page 557 and 558: 7.1. FIRST PREVALENT PROBLEM: 5577.
- Page 559 and 560: 7.1. FIRST PREVALENT PROBLEM: 559di
- Page 561 and 562: 7.1. FIRST PREVALENT PROBLEM: 5617.
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6.5. CORRESPONDENCE TO PSD CONE S N−1+ 515dvec rel ∂ EDM 3dvec(11 T − E 3 )EDM N = cone{11 T − E N } = {t(11 T − E N ) | t ≥ 0} (1222)Figure 141: Three views of translated negated elliptope 11 T − E 3 1(confer Figure 130) shrouded by truncated EDM cone. Fractal on EDMcone relative boundary is numerical artifact belonging to intersection withelliptope relative boundary. The fractal is trying to convey existence of aneighborhood about the origin where the translated elliptope boundary andEDM cone boundary intersect.
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