v2010.10.26 - Convex Optimization
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v2010.10.26 - Convex Optimization

v2010.10.26 - Convex Optimization v2010.10.26 - Convex Optimization

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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)

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.<strong>Convex</strong> 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 .

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