v2010.10.26 - Convex Optimization
v2010.10.26 - Convex Optimization v2010.10.26 - Convex Optimization
516 CHAPTER 6. CONE OF DISTANCE MATRICES6.5.2.0.1 Expository. Normal cone, tangent cone, elliptope.Define T E (11 T ) to be the tangent cone to the elliptope E at point 11 T ;id est,T E (11 T ) {t(E − 11 T ) | t≥0} (1223)The normal cone K ⊥ E (11T ) to the elliptope at 11 T is a closed convex conedefined (E.10.3.2.1, Figure 175)K ⊥ E (11 T ) {B | 〈B , Φ − 11 T 〉 ≤ 0, Φ∈ E } (1224)The polar cone of any set K is the closed convex cone (confer (297))K ◦ {B | 〈B , A〉≤0, for all A∈ K} (1225)The normal cone is well known to be the polar of the tangent cone,and vice versa; [199,A.5.2.4]K ⊥ E (11 T ) = T E (11 T ) ◦ (1226)K ⊥ E (11 T ) ◦ = T E (11 T ) (1227)From Deza & Laurent [113, p.535] we have the EDM coneEDM = −T E (11 T ) (1228)The polar EDM cone is also expressible in terms of the elliptope. From (1226)we haveEDM ◦ = −K ⊥ E (11 T ) (1229)⋆In5.10.1 we proposed the expression for EDM DD = t11 T − E ∈ EDM N (1098)where t∈ R + and E belongs to the parametrized elliptope E N t . We furtherpropose, for any particular t>0EDM N = cone{t11 T − E N t } (1230)Proof. Pending.6.5.2.0.2 Exercise. EDM cone from elliptope.Relationship of the translated negated elliptope with the EDM cone isillustrated in Figure 141. Prove whether it holds thatEDM N = limt→∞t11 T − E N t (1231)
6.6. VECTORIZATION & PROJECTION INTERPRETATION 5176.6 Vectorization & projection interpretationInE.7.2.0.2 we learn: −V DV can be interpreted as orthogonal projection[6,2] of vectorized −D ∈ S N h on the subspace of geometrically centeredsymmetric matricesS N c = {G∈ S N | G1 = 0} (1998)= {G∈ S N | N(G) ⊇ 1} = {G∈ S N | R(G) ⊆ N(1 T )}= {V Y V | Y ∈ S N } ⊂ S N (1999)≡ {V N AVN T | A ∈ SN−1 }(990)because elementary auxiliary matrix V is an orthogonal projector (B.4.1).Yet there is another useful projection interpretation:Revising the fundamental matrix criterion for membership to the EDMcone (886), 6.9〈zz T , −D〉 ≥ 0 ∀zz T | 11 T zz T = 0D ∈ S N h}⇔ D ∈ EDM N (1232)this is equivalent, of course, to the Schoenberg criterion−VN TDV }N ≽ 0⇔ D ∈ EDM N (910)D ∈ S N hbecause N(11 T )= R(V N ). When D ∈ EDM N , correspondence (1232)means −z T Dz is proportional to a nonnegative coefficient of orthogonalprojection (E.6.4.2, Figure 143) of −D in isometrically isomorphicR N(N+1)/2 on the range of each and every vectorized (2.2.2.1) symmetricdyad (B.1) in the nullspace of 11 T ; id est, on each and every member ofT { svec(zz T ) | z ∈ N(11 T )= R(V N ) } ⊂ svec ∂ S N += { svec(V N υυ T V T N ) | υ ∈ RN−1} (1233)whose dimension isdim T = N(N − 1)/2 (1234)6.9 N(11 T )= N(1 T ) and R(zz T )= R(z)
- 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 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
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- 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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6.6. VECTORIZATION & PROJECTION INTERPRETATION 5176.6 Vectorization & projection interpretationInE.7.2.0.2 we learn: −V DV can be interpreted as orthogonal projection[6,2] of vectorized −D ∈ S N h on the subspace of geometrically centeredsymmetric matricesS N c = {G∈ S N | G1 = 0} (1998)= {G∈ S N | N(G) ⊇ 1} = {G∈ S N | R(G) ⊆ N(1 T )}= {V Y V | Y ∈ S N } ⊂ S N (1999)≡ {V N AVN T | A ∈ SN−1 }(990)because elementary auxiliary matrix V is an orthogonal projector (B.4.1).Yet there is another useful projection interpretation:Revising the fundamental matrix criterion for membership to the EDMcone (886), 6.9〈zz T , −D〉 ≥ 0 ∀zz T | 11 T zz T = 0D ∈ S N h}⇔ D ∈ EDM N (1232)this is equivalent, of course, to the Schoenberg criterion−VN TDV }N ≽ 0⇔ D ∈ EDM N (910)D ∈ S N hbecause N(11 T )= R(V N ). When D ∈ EDM N , correspondence (1232)means −z T Dz is proportional to a nonnegative coefficient of orthogonalprojection (E.6.4.2, Figure 143) of −D in isometrically isomorphicR N(N+1)/2 on the range of each and every vectorized (2.2.2.1) symmetricdyad (B.1) in the nullspace of 11 T ; id est, on each and every member ofT { svec(zz T ) | z ∈ N(11 T )= R(V N ) } ⊂ svec ∂ S N += { svec(V N υυ T V T N ) | υ ∈ RN−1} (1233)whose dimension isdim T = N(N − 1)/2 (1234)6.9 N(11 T )= N(1 T ) and R(zz T )= R(z)