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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486 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX5.13.2.4 InterludeMap reconstruction from comparative distance data, isotonic reconstruction,would also prove invaluable to stellar cartography where absolute interstellardistance is difficult to acquire. But we have not yet implemented the secondhalf (1152) of alternation (1149) for USA map data because memory-demandsexceed capability of our computer.5.13.2.4.1 Exercise. Convergence of isotonic solution by alternation.Empirically demonstrate convergence, discussed in5.13.2.3, on a smallerdata set.It would be remiss not to mention another method of solution to thisisotonic reconstruction problem: Once again we assume only comparativedistance data like (1141) is available. Given known set of indices Iminimize rankV DVD(1155)subject to d ij ≤ d kl ≤ d mn ∀(i,j,k,l,m,n)∈ ID ∈ EDM Nthis problem minimizes affine dimension while finding an EDM whoseentries satisfy known comparative relationships. Suitable rank heuristicsare discussed in4.4.1 and7.2.2 that will transform this to a convexoptimization problem.Using contemporary computers, even with a rank heuristic in place of theobjective function, this problem formulation is more difficult to compute thanthe relaxed counterpart problem (1148). That is because there exist efficientalgorithms to compute a selected few eigenvalues and eigenvectors from avery large matrix. Regardless, it is important to recognize: the optimalsolution set for this problem (1155) is practically always different from theoptimal solution set for its counterpart, problem (1147).5.14 Fifth property of Euclidean metricWe continue now with the question raised in5.3 regarding the necessityfor at least one requirement more than the four properties of the Euclideanmetric (5.2) to certify realizability of a bounded convex polyhedron or to

5.14. FIFTH PROPERTY OF EUCLIDEAN METRIC 487reconstruct a generating list for it from incomplete distance information.There we saw that the four Euclidean metric properties are necessary forD ∈ EDM N in the case N = 3, but become insufficient when cardinality Nexceeds 3 (regardless of affine dimension).5.14.1 RecapitulateIn the particular case N = 3, −V T N DV N ≽ 0 (1059) and D ∈ S 3 h are necessaryand sufficient conditions for D to be an EDM. By (1061), triangle inequality isthen the only Euclidean condition bounding the necessarily nonnegative d ij ;and those bounds are tight. That means the first four properties of theEuclidean metric are necessary and sufficient conditions for D to be an EDMin the case N = 3 ; for i,j∈{1, 2, 3}√dij ≥ 0, i ≠ j√dij = 0, i = j√dij = √ d ji √dij≤ √ d ik + √ d kj , i≠j ≠k⇔ −V T N DV N ≽ 0D ∈ S 3 h⇔ D ∈ EDM 3(1156)Yet those four properties become insufficient when N > 3.5.14.2 Derivation of the fifthCorrespondence between the triangle inequality and the EDM was developedin5.8.2 where a triangle inequality (1061a) was revealed within theleading principal 2 ×2 submatrix of −VN TDV N when positive semidefinite.Our choice of the leading principal submatrix was arbitrary; actually,a unique triangle inequality like (956) corresponds to any one of the(N −1)!/(2!(N −1 − 2)!) principal 2 ×2 submatrices. 5.59 Assuming D ∈ S 4 hand −VN TDV N ∈ S 3 , then by the positive (semi)definite principal submatricestheorem (A.3.1.0.4) it is sufficient to prove: all d ij are nonnegative, alltriangle inequalities are satisfied, and det(−VN TDV N) is nonnegative. WhenN = 4, in other words, that nonnegative determinant becomes the fifth andlast Euclidean metric requirement for D ∈ EDM N . We now endeavor toascribe geometric meaning to it.5.59 There are fewer principal 2×2 submatrices in −V T N DV N than there are triangles madeby four or more points because there are N!/(3!(N − 3)!) triangles made by point triples.The triangles corresponding to those submatrices all have vertex x 1 . (confer5.8.2.1)

486 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX5.13.2.4 InterludeMap reconstruction from comparative distance data, isotonic reconstruction,would also prove invaluable to stellar cartography where absolute interstellardistance is difficult to acquire. But we have not yet implemented the secondhalf (1152) of alternation (1149) for USA map data because memory-demandsexceed capability of our computer.5.13.2.4.1 Exercise. Convergence of isotonic solution by alternation.Empirically demonstrate convergence, discussed in5.13.2.3, on a smallerdata set.It would be remiss not to mention another method of solution to thisisotonic reconstruction problem: Once again we assume only comparativedistance data like (1141) is available. Given known set of indices Iminimize rankV DVD(1155)subject to d ij ≤ d kl ≤ d mn ∀(i,j,k,l,m,n)∈ ID ∈ EDM Nthis problem minimizes affine dimension while finding an EDM whoseentries satisfy known comparative relationships. Suitable rank heuristicsare discussed in4.4.1 and7.2.2 that will transform this to a convexoptimization problem.Using contemporary computers, even with a rank heuristic in place of theobjective function, this problem formulation is more difficult to compute thanthe relaxed counterpart problem (1148). That is because there exist efficientalgorithms to compute a selected few eigenvalues and eigenvectors from avery large matrix. Regardless, it is important to recognize: the optimalsolution set for this problem (1155) is practically always different from theoptimal solution set for its counterpart, problem (1147).5.14 Fifth property of Euclidean metricWe continue now with the question raised in5.3 regarding the necessityfor at least one requirement more than the four properties of the Euclideanmetric (5.2) to certify realizability of a bounded convex polyhedron or to

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