v2009.01.01 - Convex Optimization
v2009.01.01 - Convex Optimization v2009.01.01 - Convex Optimization
448 CHAPTER 6. CONE OF DISTANCE MATRICES dvec rel∂EDM 3 d 0 13 0.2 0.2 d 12 0.4 0.4 0.6 0.6 0.8 0.8 1 1 1 d 23 0.8 0.6 0.4 d 23 (a) 0.2 0 (d) d 12 d 13 0.8 0.8 √ d23 0.6 0.6 √ √ 0 0 d13 0.2 0.2 d12 0.2 0.2 0.4 0.4 0.4 0.4 0.6 0.6 0.6 0.6 0.8 0.8 0.8 0.8 1 1 1 1 1 1 0.4 0.4 0.2 0.2 (b) 0 0 (c) Figure 114: Relative boundary (tiled) of EDM cone EDM 3 drawn truncated in isometrically isomorphic subspace R 3 . (a) EDM cone drawn in usual distance-square coordinates d ij . View is from interior toward origin. Unlike positive semidefinite cone, EDM cone is not self-dual, neither is it proper in ambient symmetric subspace (dual EDM cone for this example belongs to isomorphic R 6 ). (b) Drawn in its natural coordinates √ d ij (absolute distance), cone remains convex (confer5.10); intersection of three halfspaces (967) whose partial boundaries each contain origin. Cone geometry becomes “complicated” (nonpolyhedral) in higher dimension. [159,3] (c) Two coordinate systems artificially superimposed. Coordinate transformation from d ij to √ d ij appears a topological contraction. (d) Sitting on its vertex 0, pointed EDM 3 is a circular cone having axis of revolution dvec(−E)= dvec(11 T − I) (1000) (66). Rounded vertex is plot artifact.
6.2. POLYHEDRAL BOUNDS 449 This cone is more easily visualized in the isomorphic vector subspace R 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 ; a halfline in a subspace of the realization in Figure 122. The EDM cone in the case N = 3 is a circular cone in R 3 illustrated in Figure 114(a)(d); rather, the set of all matrices ⎡ D = ⎣ 0 d 12 d 13 d 12 0 d 23 d 13 d 23 0 ⎤ ⎦ ∈ EDM 3 (1085) makes a circular cone in this dimension. In this case, the first four Euclidean metric properties are necessary and sufficient tests to certify realizability of triangles; (1061). Thus triangle inequality property 4 describes three halfspaces (967) whose intersection makes a polyhedral cone in R 3 of realizable √ d ij (absolute distance); an isomorphic subspace representation of the set of all EDMs D in the natural coordinates ⎡ √ √ ⎤ 0 d12 d13 ◦√ D = ∆ √d12 √ ⎣ 0 d23 ⎦ √d13 √ (1086) d23 0 illustrated in Figure 114(b). 6.2 Polyhedral bounds The convex cone of EDMs is nonpolyhedral in d ij for N > 2 ; e.g., Figure 114(a). Still we found necessary and sufficient bounding polyhedral relations consistent with EDM cones for cardinality N = 1, 2, 3, 4: N = 3. Transforming distance-square coordinates d ij by taking their positive square root provides polyhedral cone in Figure 114(b); polyhedral because an intersection of three halfspaces in natural coordinates √ dij is provided by triangle inequalities (967). This polyhedral cone implicitly encompasses necessary and sufficient metric properties: nonnegativity, self-distance, symmetry, and triangle inequality.
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448 CHAPTER 6. CONE OF DISTANCE MATRICES<br />
dvec rel∂EDM 3<br />
d 0 13 0.2<br />
0.2<br />
d 12<br />
0.4<br />
0.4<br />
0.6<br />
0.6<br />
0.8<br />
0.8<br />
1<br />
1<br />
1<br />
d 23<br />
0.8<br />
0.6<br />
0.4<br />
d 23<br />
(a)<br />
0.2<br />
0<br />
(d)<br />
d 12<br />
d 13<br />
0.8<br />
0.8<br />
√<br />
d23 0.6<br />
0.6<br />
√ √ 0 0 d13 0.2<br />
0.2 d12<br />
0.2<br />
0.2<br />
0.4<br />
0.4 0.4<br />
0.4<br />
0.6<br />
0.6 0.6<br />
0.6<br />
0.8<br />
0.8 0.8<br />
0.8<br />
1<br />
1 1<br />
1<br />
1<br />
1<br />
0.4<br />
0.4<br />
0.2<br />
0.2<br />
(b)<br />
0<br />
0<br />
(c)<br />
Figure 114: Relative boundary (tiled) of EDM cone EDM 3 drawn truncated<br />
in isometrically isomorphic subspace R 3 . (a) EDM cone drawn in usual<br />
distance-square coordinates d ij . View is from interior toward origin. Unlike<br />
positive semidefinite cone, EDM cone is not self-dual, neither is it proper<br />
in ambient symmetric subspace (dual EDM cone for this example belongs<br />
to isomorphic R 6 ). (b) Drawn in its natural coordinates √ d ij (absolute<br />
distance), cone remains convex (confer5.10); intersection of three halfspaces<br />
(967) whose partial boundaries each contain origin. Cone geometry becomes<br />
“complicated” (nonpolyhedral) in higher dimension. [159,3] (c) Two<br />
coordinate systems artificially superimposed. Coordinate transformation<br />
from d ij to √ d ij appears a topological contraction. (d) Sitting on<br />
its vertex 0, pointed EDM 3 is a circular cone having axis of revolution<br />
dvec(−E)= dvec(11 T − I) (1000) (66). Rounded vertex is plot artifact.
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