v2010.10.26 - Convex Optimization
v2010.10.26 - Convex Optimization v2010.10.26 - Convex Optimization
540 CHAPTER 6. CONE OF DISTANCE MATRICESWhen low affine dimension is a desirable result of projection on theEDM cone, projection on the polar EDM cone should be performed instead.Convex polar problem (1283) can be solved for D ◦⋆ by transforming to anequivalent Schur-form semidefinite program (3.5.2). Interior-point methodsfor numerically solving semidefinite programs tend to produce high-ranksolutions. (4.1.2) Then D ⋆ = H − D ◦⋆ ∈ EDM N by Corollary E.9.2.2.1, andD ⋆ will tend to have low affine dimension. This approach breaks whenattempting projection on a cone subset discriminated by affine dimensionor rank, because then we have no complementarity relation like (1285) or(1286) (7.1.4.1).6.8.1.6 EDM cone is not selfdualIn5.6.1.1, via Gram-form EDM operatorD(G) = δ(G)1 T + 1δ(G) T − 2G ∈ EDM N ⇐ G ≽ 0 (903)we established clear connection between the EDM cone and that face (1238)of positive semidefinite cone S N + in the geometric center subspace:whereIn5.6.1 we establishedEDM N = D(S N c ∩ S N +) (1007)V(EDM N ) = S N c ∩ S N + (1008)V(D) = −V DV 1 2(996)S N c ∩ S N + = V N S N−1+ V T N (994)Then from (1274), (1282), and (1248) we can deduceδ(EDM N∗ 1) − EDM N∗ = V N S N−1+ V T N = S N c ∩ S N + (1287)which, by (1007) and (1008), means the EDM cone can be related to the dualEDM cone by an equality:(EDM N = D δ(EDM N∗ 1) − EDM N∗) (1288)V(EDM N ) = δ(EDM N∗ 1) − EDM N∗ (1289)This means projection −V(EDM N ) of the EDM cone on the geometriccenter subspace S N c (E.7.2.0.2) is a linear transformation of the dual EDMcone: EDM N∗ − δ(EDM N∗ 1). Secondarily, it means the EDM cone is notselfdual in S N .
6.8. DUAL EDM CONE 5416.8.1.7 Schoenberg criterion is discretized membership relationWe show the Schoenberg criterion−VN TDV }N ∈ S N−1+D ∈ S N h⇔ D ∈ EDM N (910)to be a discretized membership relation (2.13.4) between a closed convexcone K and its dual K ∗ like〈y , x〉 ≥ 0 for all y ∈ G(K ∗ ) ⇔ x ∈ K (366)where G(K ∗ ) is any set of generators whose conic hull constructs closedconvex dual cone K ∗ :The Schoenberg criterion is the same as}〈zz T , −D〉 ≥ 0 ∀zz T | 11 T zz T = 0⇔ D ∈ EDM N (1232)D ∈ S N hwhich, by (1233), is the same as〈zz T , −D〉 ≥ 0 ∀zz T ∈ { V N υυ T VN T | υ ∈ RN−1}D ∈ S N h}⇔ D ∈ EDM N (1290)where the zz T constitute a set of generators G for the positive semidefinitecone’s smallest face F ( S N + ∋V ) (6.6.1) that contains auxiliary matrix V .From the aggregate in (1248) we get the ordinary membership relation,assuming only D ∈ S N [199, p.58]〈D ∗ , D〉 ≥ 0 ∀D ∗ ∈ EDM N∗ ⇔ D ∈ EDM N(1291)〈D ∗ , D〉 ≥ 0 ∀D ∗ ∈ {δ(u) | u∈ R N } − cone { V N υυ T V T N | υ ∈ RN−1} ⇔ D ∈ EDM NDiscretization (366) yields:〈D ∗ , D〉 ≥ 0 ∀D ∗ ∈ {e i e T i , −e j e T j , −V N υυ T V T N | i, j =1... N , υ ∈ R N−1 } ⇔ D ∈ EDM N(1292)
- 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 and 496: 5.14. FIFTH PROPERTY OF EUCLIDEAN M
- 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.
- 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: 6.8. DUAL EDM CONE 5396.8.1.5 Affin
- 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.
- Page 563 and 564: 7.1. FIRST PREVALENT PROBLEM: 563wh
- Page 565 and 566: 7.1. FIRST PREVALENT PROBLEM: 565Th
- Page 567 and 568: 7.2. SECOND PREVALENT PROBLEM: 567O
- Page 569 and 570: 7.2. SECOND PREVALENT PROBLEM: 569S
- Page 571 and 572: 7.2. SECOND PREVALENT PROBLEM: 571r
- Page 573 and 574: 7.2. SECOND PREVALENT PROBLEM: 573w
- Page 575 and 576: 7.2. SECOND PREVALENT PROBLEM: 5757
- Page 577 and 578: 7.2. SECOND PREVALENT PROBLEM: 577a
- Page 579 and 580: 7.3. THIRD PREVALENT PROBLEM: 579is
- Page 581 and 582: 7.3. THIRD PREVALENT PROBLEM: 581We
- Page 583 and 584: 7.3. THIRD PREVALENT PROBLEM: 583su
- Page 585 and 586: 7.3. THIRD PREVALENT PROBLEM: 585Gi
- Page 587 and 588: 7.3. THIRD PREVALENT PROBLEM: 587Op
- Page 589 and 590: 7.4. CONCLUSION 589filtering, multi
6.8. DUAL EDM CONE 5416.8.1.7 Schoenberg criterion is discretized membership relationWe show the Schoenberg criterion−VN TDV }N ∈ S N−1+D ∈ S N h⇔ D ∈ EDM N (910)to be a discretized membership relation (2.13.4) between a closed convexcone K and its dual K ∗ like〈y , x〉 ≥ 0 for all y ∈ G(K ∗ ) ⇔ x ∈ K (366)where G(K ∗ ) is any set of generators whose conic hull constructs closedconvex dual cone K ∗ :The Schoenberg criterion is the same as}〈zz T , −D〉 ≥ 0 ∀zz T | 11 T zz T = 0⇔ D ∈ EDM N (1232)D ∈ S N hwhich, by (1233), is the same as〈zz T , −D〉 ≥ 0 ∀zz T ∈ { V N υυ T VN T | υ ∈ RN−1}D ∈ S N h}⇔ D ∈ EDM N (1290)where the zz T constitute a set of generators G for the positive semidefinitecone’s smallest face F ( S N + ∋V ) (6.6.1) that contains auxiliary matrix V .From the aggregate in (1248) we get the ordinary membership relation,assuming only D ∈ S N [199, p.58]〈D ∗ , D〉 ≥ 0 ∀D ∗ ∈ EDM N∗ ⇔ D ∈ EDM N(1291)〈D ∗ , D〉 ≥ 0 ∀D ∗ ∈ {δ(u) | u∈ R N } − cone { V N υυ T V T N | υ ∈ RN−1} ⇔ D ∈ EDM NDiscretization (366) yields:〈D ∗ , D〉 ≥ 0 ∀D ∗ ∈ {e i e T i , −e j e T j , −V N υυ T V T N | i, j =1... N , υ ∈ R N−1 } ⇔ D ∈ EDM N(1292)