v2010.10.26 - Convex Optimization
v2010.10.26 - Convex Optimization v2010.10.26 - Convex Optimization
500 CHAPTER 6. CONE OF DISTANCE MATRICESdvec rel∂EDM 3d 0 13 0.20.2d 120.40.40.60.60.80.8111d 230.80.60.4d 23(a)0.20(d)d 12d 130.80.8√d23 0.60.6√ √ 0 0 d13 0.20.2 d120.20.20.40.4 0.40.40.60.6 0.60.60.80.8 0.80.811 11110.40.40.20.2(b)00(c)Figure 138: Relative boundary (tiled) of EDM cone EDM 3 drawn truncatedin isometrically isomorphic subspace R 3 . (a) EDM cone drawn in usualdistance-square coordinates d ij . View is from interior toward origin. Unlikepositive semidefinite cone, EDM cone is not selfdual; neither is it properin ambient symmetric subspace (dual EDM cone for this example belongsto isomorphic R 6 ). (b) Drawn in its natural coordinates √ d ij (absolutedistance), cone remains convex (confer5.10); intersection of three halfspaces(1062) whose partial boundaries each contain origin. Cone geometry becomesnonconvex (nonpolyhedral) in higher dimension. (6.3) (c) Two coordinatesystems √ artificially superimposed. Coordinate transformation from d ij todij appears a topological contraction. (d) Sitting on its vertex 0, pointedEDM 3 is a circular cone having axis of revolution dvec(−E)= dvec(11 T − I)(1095) (73). (Rounded vertex is plot artifact.)
6.2. POLYHEDRAL BOUNDS 501This cone is more easily visualized in the isomorphic vector subspaceR N(N−1)/2 corresponding to S N h :In the case N = 1 point, the EDM cone is the origin in R 0 .In the case N = 2, the EDM cone is the nonnegative real line in R ; ahalfline in a subspace of the realization in Figure 142.The EDM cone in the case N = 3 is a circular cone in R 3 illustrated inFigure 138(a)(d); rather, the set of all matrices⎡D = ⎣0 d 12 d 13d 12 0 d 23d 13 d 23 0⎤⎦ ∈ EDM 3 (1180)makes a circular cone in this dimension. In this case, the first four Euclideanmetric properties are necessary and sufficient tests to certify realizabilityof triangles; (1156). Thus triangle inequality property 4 describes threehalfspaces (1062) whose intersection makes a polyhedral cone in R 3 ofrealizable √ d ij (absolute distance); an isomorphic subspace representationof the set of all EDMs D in the natural coordinates⎡ √ √ ⎤0 d12 d13◦√ √d12 √ D ⎣ 0 d23 ⎦√d13 √(1181)d23 0illustrated in Figure 138b.6.2 Polyhedral boundsThe convex cone of EDMs is nonpolyhedral in d ij for N > 2 ; e.g.,Figure 138a. Still we found necessary and sufficient bounding polyhedralrelations consistent with EDM cones for cardinality N = 1, 2, 3, 4:N = 3. Transforming distance-square coordinates d ij by taking their positivesquare root provides the polyhedral cone in Figure 138b; polyhedral√because an intersection of three halfspaces in natural coordinatesdij is provided by triangle inequalities (1062). This polyhedralcone implicitly encompasses necessary and sufficient metric properties:nonnegativity, self-distance, symmetry, and triangle inequality.
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- 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
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- Page 507 and 508: 6.4. EDM DEFINITION IN 11 T 507Then
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- 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
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- Page 543 and 544: 6.8. DUAL EDM CONE 5430dvec rel ∂
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500 CHAPTER 6. CONE OF DISTANCE MATRICESdvec rel∂EDM 3d 0 13 0.20.2d 120.40.40.60.60.80.8111d 230.80.60.4d 23(a)0.20(d)d 12d 130.80.8√d23 0.60.6√ √ 0 0 d13 0.20.2 d120.20.20.40.4 0.40.40.60.6 0.60.60.80.8 0.80.811 11110.40.40.20.2(b)00(c)Figure 138: Relative boundary (tiled) of EDM cone EDM 3 drawn truncatedin isometrically isomorphic subspace R 3 . (a) EDM cone drawn in usualdistance-square coordinates d ij . View is from interior toward origin. Unlikepositive semidefinite cone, EDM cone is not selfdual; neither is it properin ambient symmetric subspace (dual EDM cone for this example belongsto isomorphic R 6 ). (b) Drawn in its natural coordinates √ d ij (absolutedistance), cone remains convex (confer5.10); intersection of three halfspaces(1062) whose partial boundaries each contain origin. Cone geometry becomesnonconvex (nonpolyhedral) in higher dimension. (6.3) (c) Two coordinatesystems √ artificially superimposed. Coordinate transformation from d ij todij appears a topological contraction. (d) Sitting on its vertex 0, pointedEDM 3 is a circular cone having axis of revolution dvec(−E)= dvec(11 T − I)(1095) (73). (Rounded vertex is plot artifact.)
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