v2010.10.26 - Convex Optimization
v2010.10.26 - Convex Optimization v2010.10.26 - Convex Optimization
544 CHAPTER 6. CONE OF DISTANCE MATRICESWhen N = 1, the EDM cone and its dual in ambient S h each comprisethe origin in isomorphic R 0 ; thus, selfdual in this dimension. (confer (104))When N = 2, the EDM cone is the nonnegative real line in isomorphic R .(Figure 142) EDM 2∗ in S 2 h is identical, thus selfdual in this dimension.This [ result]is in agreement [ with ](1294), verified directly: for all κ∈ R ,11z = κ and δ(zz−1T ) = κ 2 ⇒ d ∗ 12 ≥ 0.1The first case adverse to selfdualness N = 3 may be deduced fromFigure 138; the EDM cone is a circular cone in isomorphic R 3 correspondingto no rotation of Lorentz cone (178) (the selfdual circular cone). Figure 151illustrates the EDM cone and its dual in ambient S 3 h ; no longer selfdual.6.9 Theorem of the alternativeIn2.13.2.1.1 we showed how alternative systems of generalized inequalitycan be derived from closed convex cones and their duals. This section is,therefore, a fitting postscript to the discussion of the dual EDM cone.6.9.0.0.1 Theorem. EDM alternative. [163,1]Given D ∈ S N hD ∈ EDM Nor in the alternative{1 T z = 1∃ z such thatDz = 0(1297)In words, either N(D) intersects hyperplane {z | 1 T z =1} or D is an EDM;the alternatives are incompatible.⋄When D is an EDM [259,2]N(D) ⊂ N(1 T ) = {z | 1 T z = 0} (1298)Because [163,2] (E.0.1)thenDD † 1 = 11 T D † D = 1 T (1299)R(1) ⊂ R(D) (1300)
6.10. POSTSCRIPT 5456.10 PostscriptWe provided an equality (1268) relating the convex cone of Euclidean distancematrices to the convex cone of positive semidefinite matrices. Projection ona positive semidefinite cone, constrained by an upper bound on rank, is easyand well known; [129] simply, a matter of truncating a list of eigenvalues.Projection on a positive semidefinite cone with such a rank constraint is, infact, a convex optimization problem. (7.1.4)In the past, it was difficult to project on the EDM cone under aconstraint on rank or affine dimension. A surrogate method was to invokethe Schoenberg criterion (910) and then project on a positive semidefinitecone under a rank constraint bounding affine dimension from above. But asolution acquired that way is necessarily suboptimal.In7.3.3 we present a method for projecting directly on the EDM coneunder a constraint on rank or affine dimension.
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- 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
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6.10. POSTSCRIPT 5456.10 PostscriptWe provided an equality (1268) relating the convex cone of Euclidean distancematrices to the convex cone of positive semidefinite matrices. Projection ona positive semidefinite cone, constrained by an upper bound on rank, is easyand well known; [129] simply, a matter of truncating a list of eigenvalues.Projection on a positive semidefinite cone with such a rank constraint is, infact, a convex optimization problem. (7.1.4)In the past, it was difficult to project on the EDM cone under aconstraint on rank or affine dimension. A surrogate method was to invokethe Schoenberg criterion (910) and then project on a positive semidefinitecone under a rank constraint bounding affine dimension from above. But asolution acquired that way is necessarily suboptimal.In7.3.3 we present a method for projecting directly on the EDM coneunder a constraint on rank or affine dimension.
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