v2007.11.26 - Convex Optimization

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392 CHAPTER 5. EUCLIDEAN DISTANCE MATRIXSecondly, the second half (982b) of the alternation takes place in adifferent vector space; S N h (versus S N−1 ). From5.6 we know these twovector spaces are related by an isomorphism, S N−1 =V N (S N h ) (852), but notby an isometry.We have, therefore, no guarantee from theory of alternating projectionthat the alternation (982) converges to a point, in the set of all EDMscorresponding to affine dimension not in excess of 3, belonging todvec EDM N ∩ Π T K M+ .5.13.2.4 InterludeWe have not implemented the second half (985) of alternation (982) forUSA map data because memory-demands exceed the capability of our 32-bitlaptop 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 (974) is available. Given known set of indices Iminimize rankV DVD(988)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 (981). 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 (988) is practically always different from theoptimal solution set for its counterpart, problem (980).
5.14. FIFTH PROPERTY OF EUCLIDEAN METRIC 3935.14 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 toreconstruct 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 (893) and D ∈ S 3 h are necessaryand sufficient conditions for D to be an EDM. By (895), 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(989)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 (895a) 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 (797) corresponds to any one of the(N −1)!/(2!(N −1 − 2)!) principal 2 ×2 submatrices. 5.53 Assuming D ∈ S 4 hand −VN TDV N ∈ S 3 , then by the positive (semi)definite principal submatrices5.53 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)
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DATTORROCONVEXOPTIMIZATION&EUCLIDEA
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Convex Optimization&Euclidean Dista
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for Jennie Columba♦Antonio♦♦&
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PreludeThe constant demands of my d
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Convex Optimization&Euclidean Dista
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CONVEX OPTIMIZATION & EUCLIDEAN DIS
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List of Figures1 Overview 191 Orion
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LIST OF FIGURES 1559 Epigraph . . .
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LIST OF FIGURES 17120 Four fundamen
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Chapter 1OverviewConvex Optimizatio
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ˇx 4ˇx 3ˇx 2Figure 2: Applicatio
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23Figure 4: This coarsely discretiz
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ases (biorthogonal expansion). We e
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27Figure 7: These bees construct a
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its membership to the EDM cone. The
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31appendicesProvided so as to be mo
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Chapter 2Convex geometryConvexity h
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2.1. CONVEX SET 35Figure 11: A slab
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2.1. CONVEX SET 372.1.6 empty set v
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2.1. CONVEX SET 392.1.7.1 Line inte
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2.1. CONVEX SET 41(a)R 2(b)R 3(c)(d
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2.1. CONVEX SET 43This theorem in c
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2.2. VECTORIZED-MATRIX INNER PRODUC
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2.3. HULLS 53Figure 14: Convex hull
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2.3. HULLS 55Aaffine hull (drawn tr
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2.3. HULLS 57The union of relative
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2.4. HALFSPACE, HYPERPLANE 59of dim
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2.4. HALFSPACE, HYPERPLANE 61H +ay
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2.4. HALFSPACE, HYPERPLANE 63Inters
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2.4. HALFSPACE, HYPERPLANE 65Conver
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2.4. HALFSPACE, HYPERPLANE 67A 1A 2
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2.4. HALFSPACE, HYPERPLANE 69tradit
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2.4. HALFSPACE, HYPERPLANE 71There
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2.5. SUBSPACE REPRESENTATIONS 73nor
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2.5. SUBSPACE REPRESENTATIONS 752.5
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2.6. EXTREME, EXPOSED 77In other wo
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2.6. EXTREME, EXPOSED 792.6.1 Expos
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2.7. CONES 812.6.1.3.1 Definition.
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2.7. CONES 830Figure 26: Boundary o
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2.7. CONES 852.7.2 Convex coneWe ca
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2.7. CONES 87Thus the simplest and
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2.7. CONES 89nomenclature generaliz
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2.8. CONE BOUNDARY 91Proper cone {0
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2.8. CONE BOUNDARY 93the same extre
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2.8. CONE BOUNDARY 95For a proper c
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2.9. POSITIVE SEMIDEFINITE (PSD) CO
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2.10. CONIC INDEPENDENCE (C.I.) 121
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2.12. CONVEX POLYHEDRA 127It follow
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2.12. CONVEX POLYHEDRA 129Coefficie
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2.12. CONVEX POLYHEDRA 1312.12.3 Un
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2.12. CONVEX POLYHEDRA 133
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2.13. DUAL CONE & GENERALIZED INEQU
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Chapter 3Geometry of convex functio
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3.1. CONVEX FUNCTION 185f 1 (x)f 2
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3.1. CONVEX FUNCTION 1873.1.3 norm
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3.1. CONVEX FUNCTION 189n∑π(x) i
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3.1. CONVEX FUNCTION 191This means
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3.1. CONVEX FUNCTION 1933.1.5.3 pos
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3.1. CONVEX FUNCTION 1953.1.6.0.1 E
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3.1. CONVEX FUNCTION 1973.1.7 epigr
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3.1. CONVEX FUNCTION 201Although th
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3.1. CONVEX FUNCTION 203minimize tX
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3.1. CONVEX FUNCTION 2053.1.8 gradi
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3.1. CONVEX FUNCTION 209f(Y )[ ∇f
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3.1. CONVEX FUNCTION 211g is convex
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3.1. CONVEX FUNCTION 213αβα ≥
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3.1. CONVEX FUNCTION 2153.1.11 seco
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3.2. MATRIX-VALUED CONVEX FUNCTION
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3.2. MATRIX-VALUED CONVEX FUNCTION
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3.3. QUASICONVEX 221matrices in the
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3.3. QUASICONVEX 223on R M + . Alth
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Chapter 4Semidefinite programmingPr
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4.1. CONIC PROBLEM 227where K is a
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4.1. CONIC PROBLEM 229C0PΓ 1Γ 2S+
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4.1. CONIC PROBLEM 231faces of S 3
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4.1. CONIC PROBLEM 2334.1.1.3 Previ
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4.2. FRAMEWORK 235Equivalently, pri
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4.2. FRAMEWORK 237is positive semid
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4.2. FRAMEWORK 239Optimal value of
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4.2. FRAMEWORK 2414.2.3.1.1 Example
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4.2. FRAMEWORK 243where δ is the m
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4.2. FRAMEWORK 2454.2.3.1.2 Example
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4.3. RANK REDUCTION 2474.3 Rank red
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4.3. RANK REDUCTION 249A rank-reduc
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4.3. RANK REDUCTION 251(t ⋆ i)
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4.3. RANK REDUCTION 2534.3.3.0.1 Ex
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4.3. RANK REDUCTION 2554.3.3.0.2 Ex
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4.4. RANK-CONSTRAINED SEMIDEFINITE
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4.5. CONSTRAINING CARDINALITY 2734.
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4.5. CONSTRAINING CARDINALITY 275Th
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4.5. CONSTRAINING CARDINALITY 277wh
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4.5. CONSTRAINING CARDINALITY 279f
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4.5. CONSTRAINING CARDINALITY 281al
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4.6. CARDINALITY AND RANK CONSTRAIN
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4.7. CONVEX ITERATION RANK-1 301whi
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4.7. CONVEX ITERATION RANK-1 303the
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Chapter 5Euclidean Distance MatrixT
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5.2. FIRST METRIC PROPERTIES 307cor
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5.3. ∃ FIFTH EUCLIDEAN METRIC PRO
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5.3. ∃ FIFTH EUCLIDEAN METRIC PRO
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5.4. EDM DEFINITION 313The collecti
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5.4. EDM DEFINITION 3155.4.2 Gram-f
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5.4. EDM DEFINITION 317D ∈ EDM N
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5.4. EDM DEFINITION 3195.4.2.2.1 Ex
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5.4. EDM DEFINITION 321ten affine e
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5.4. EDM DEFINITION 323spheres:Then
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5.4. EDM DEFINITION 325By eliminati
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5.4. EDM DEFINITION 327whereΦ ij =
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5.4. EDM DEFINITION 3295.4.2.2.5 De
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5.4. EDM DEFINITION 331105ˇx 4ˇx
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5.4. EDM DEFINITION 333corrected by
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5.4. EDM DEFINITION 335by translate
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5.4. EDM DEFINITION 337Crippen & Ha
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5.4. EDM DEFINITION 339where ([√t
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Page 345 and 346: 5.6. INJECTIVITY OF D & UNIQUE RECO
Page 351 and 352: 5.7. EMBEDDING IN AFFINE HULL 3515.
Page 353 and 354: 5.7. EMBEDDING IN AFFINE HULL 353Fo
Page 355 and 356: 5.7. EMBEDDING IN AFFINE HULL 3555.
Page 357 and 358: 5.8. EUCLIDEAN METRIC VERSUS MATRIX
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Page 379 and 380: 5.11. EDM INDEFINITENESS 379we have
Page 381 and 382: 5.11. EDM INDEFINITENESS 381For pre
Page 383 and 384: 5.12. LIST RECONSTRUCTION 383where
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Page 387 and 388: 5.13. RECONSTRUCTION EXAMPLES 387D
Page 389 and 390: 5.13. RECONSTRUCTION EXAMPLES 389Th
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Page 403 and 404: Chapter 6Cone of distance matricesF
Page 405 and 406: 6.1. DEFINING EDM CONE 4056.1 Defin
Page 407 and 408: 6.2. POLYHEDRAL BOUNDS 407This cone
Page 409 and 410: 6.3.√EDM CONE IS NOT CONVEX 409N
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Page 413 and 414: 6.4. A GEOMETRY OF COMPLETION 413(a
Page 415 and 416: 6.4. A GEOMETRY OF COMPLETION 415Fi
Page 417 and 418: 6.5. EDM DEFINITION IN 11 T 417and
Page 419 and 420: 6.5. EDM DEFINITION IN 11 T 419then
Page 421 and 422: 6.5. EDM DEFINITION IN 11 T 4216.5.
Page 423 and 424: 6.5. EDM DEFINITION IN 11 T 423D =
Page 425 and 426: 6.6. CORRESPONDENCE TO PSD CONE S N
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Page 435 and 436: 6.8. DUAL EDM CONE 435When the Fins
Page 437 and 438: 6.8. DUAL EDM CONE 437Proof. First,
Page 439 and 440: 6.8. DUAL EDM CONE 439EDM 2 = S 2 h
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6.8. DUAL EDM CONE 4436.8.1.3.1 Exe
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6.8. DUAL EDM CONE 445has dual affi
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6.8. DUAL EDM CONE 4476.8.1.7 Schoe
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6.9. THEOREM OF THE ALTERNATIVE 449
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6.10. POSTSCRIPT 451When D is an ED
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Chapter 7Proximity problemsIn summa
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In contrast, order of projection on
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457HS N h0EDM NK = S N h ∩ R N×N
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4597.0.3 Problem approachProblems t
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7.1. FIRST PREVALENT PROBLEM: 461fi
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7.1. FIRST PREVALENT PROBLEM: 4637.
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7.1. FIRST PREVALENT PROBLEM: 465di
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7.1. FIRST PREVALENT PROBLEM: 4677.
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7.1. FIRST PREVALENT PROBLEM: 469wh
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7.1. FIRST PREVALENT PROBLEM: 471Th
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7.2. SECOND PREVALENT PROBLEM: 473O
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7.2. SECOND PREVALENT PROBLEM: 475S
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7.2. SECOND PREVALENT PROBLEM: 477r
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7.2. SECOND PREVALENT PROBLEM: 479c
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7.2. SECOND PREVALENT PROBLEM: 4817
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7.3. THIRD PREVALENT PROBLEM: 483fo
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7.3. THIRD PREVALENT PROBLEM: 485a
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7.3. THIRD PREVALENT PROBLEM: 4877.
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7.3. THIRD PREVALENT PROBLEM: 4897.
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7.3. THIRD PREVALENT PROBLEM: 491Ou
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7.4. CONCLUSION 493The rank constra
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Appendix ALinear algebraA.1 Main-di
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A.1. MAIN-DIAGONAL δ OPERATOR, λ
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A.2. SEMIDEFINITENESS: DOMAIN OF TE
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A.2. SEMIDEFINITENESS: DOMAIN OF TE
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A.3. PROPER STATEMENTS 503A.3.0.0.1
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A.3. PROPER STATEMENTS 505By simila
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A.3. PROPER STATEMENTS 507Because R
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For A,B ∈ R n×n x T Ax ≥ x T B
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A.3. PROPER STATEMENTS 511A.3.1.0.2
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A.3. PROPER STATEMENTS 513We can de
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A.4. SCHUR COMPLEMENT 515Origin of
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A.4. SCHUR COMPLEMENT 517Schur-form
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A.5. EIGEN DECOMPOSITION 519A.5.0.1
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A.6. SINGULAR VALUE DECOMPOSITION,
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A.7. ZEROS 527For diagonalizable ma
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A.7. ZEROS 529A.7.4For X,A∈ S M +
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A.7. ZEROS 531A.7.5.0.1 Proposition
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Appendix BSimple matricesMathematic
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B.1. RANK-ONE MATRIX (DYAD) 535R(v)
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B.1. RANK-ONE MATRIX (DYAD) 537rang
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B.2. DOUBLET 539R([u v ])R(Π)= R([
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B.3. ELEMENTARY MATRIX 541If λ ≠
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B.4. AUXILIARY V -MATRICES 543the n
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B.4. AUXILIARY V -MATRICES 54518. V
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B.5. ORTHOGONAL MATRIX 547B.5 Ortho
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B.5. ORTHOGONAL MATRIX 549B.5.3.0.1
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Appendix CSome analytical optimal r
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C.2. DIAGONAL, TRACE, SINGULAR AND
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C.3. ORTHOGONAL PROCRUSTES PROBLEM
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C.4. TWO-SIDED ORTHOGONAL PROCRUSTE
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C.4. TWO-SIDED ORTHOGONAL PROCRUSTE
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Appendix DMatrix calculusFrom too m
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D.1. DIRECTIONAL DERIVATIVE, TAYLOR
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D.2. TABLES OF GRADIENTS AND DERIVA
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Appendix EProjectionFor any A∈ R
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597Equivalent to the corresponding
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E.1. IDEMPOTENT MATRICES 599where R
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E.1. IDEMPOTENT MATRICES 601TxT ⊥
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E.2. I − P , PROJECTION ON ALGEBR
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E.3. SYMMETRIC IDEMPOTENT MATRICES
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E.5. PROJECTION EXAMPLES 611E.5.0.0
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E.5. PROJECTION EXAMPLES 613of rela
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E.5. PROJECTION EXAMPLES 615E.5.0.0
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E.6. VECTORIZATION INTERPRETATION,
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E.7. ON VECTORIZED MATRICES OF HIGH
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E.7. ON VECTORIZED MATRICES OF HIGH
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E.9. PROJECTION ON CONVEX SET 629E.
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E.9. PROJECTION ON CONVEX SET 631
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E.9. PROJECTION ON CONVEX SET 633Pr
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E.9. PROJECTION ON CONVEX SET 637wh
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E.9. PROJECTION ON CONVEX SET 639Un
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E.10. ALTERNATING PROJECTION 643bC
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E.10. ALTERNATING PROJECTION 6450
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E.10. ALTERNATING PROJECTION 647E.1
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E.10. ALTERNATING PROJECTION 649y 2
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E.10. ALTERNATING PROJECTION 651By
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E.10. ALTERNATING PROJECTION 653Bar
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E.10. ALTERNATING PROJECTION 655bH
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E.10. ALTERNATING PROJECTION 657K
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E.10. ALTERNATING PROJECTION 659Whe
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Appendix FMatlab programsMade by Th
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F.1. ISEDM() 663% is nonnegativeif
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F.1. ISEDM() 665F.1.1Subroutines fo
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F.2. CONIC INDEPENDENCE, CONICI() 6
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F.2. CONIC INDEPENDENCE, CONICI() 6
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F.3. MAP OF THE USA 671% plot origi
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F.3. MAP OF THE USA 673statelat = d
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F.4. RANK REDUCTION SUBROUTINE, RRF
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F.4. RANK REDUCTION SUBROUTINE, RRF
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F.5. STURM’S PROCEDURE 679F.5 Stu
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F.6. CONVEX ITERATION DEMONSTRATION
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F.6. CONVEX ITERATION DEMONSTRATION
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F.7. FAST MAX CUT 685endoldtrace =
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F.8. SIGNAL DROPOUT PROBLEM 687whil
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Appendix GNotation and a few defini
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691l.i.w.r.tlinearly independentwit
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693t → 0 +t goes to 0 from above;
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695ψ(Z)DDD T (X)D(X) TD −1 (X)D(
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697R n −or R n×n−S nS n⊥S n
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699∂H −∂H +da supporting hype
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701maximizexargsup Xarg supf(x)subj
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703⊁≥not positive definitescala
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Bibliography[1] Suliman Al-Homidan
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BIBLIOGRAPHY 707[16] Keith Ball. An
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BIBLIOGRAPHY 709[38] A. W. Bojanczy
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BIBLIOGRAPHY 711[57] Steven Chu. Au
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BIBLIOGRAPHY 713[76] Frank R. Deuts
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BIBLIOGRAPHY 715(AACC), June 2004.h
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BIBLIOGRAPHY 717[114] John Clifford
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BIBLIOGRAPHY 719[135] Uwe Helmke an
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BIBLIOGRAPHY 721[157] Joakim Jaldé
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BIBLIOGRAPHY 723[178] Jung Rye Lee.
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BIBLIOGRAPHY 725[200] T. S. Motzkin
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BIBLIOGRAPHY 727[220] Teemu Pennane
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BIBLIOGRAPHY 729[240] Joshua A. Sin
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BIBLIOGRAPHY 731[264] Warren S. Tor
Page 733 and 734:
BIBLIOGRAPHY 733[285] Bernard Widro
Page 735 and 736:
Index0-norm, 241, 273, 275, 2941-no
Page 737 and 738:
INDEX 737clipping, 190, 454, 638, 6
Page 739 and 740:
INDEX 739hull, 53, 56, 306of outer
Page 741 and 742:
INDEX 741unique, 519elbow, 457, 458
Page 743 and 744:
INDEX 743polarity, 62handoff, 335ov
Page 745 and 746:
INDEX 745manifold, 23, 24, 105, 410
Page 747 and 748:
INDEX 747objectivebilinear, 256, 27
Page 749 and 750:
INDEX 749Hadamard, 46, 472, 571, 69
Page 751 and 752:
INDEX 751trace, 477, 478heuristic,
Page 753 and 754:
INDEX 753symmetric hollow, 51tangen
Page 755 and 756:
INDEX 755matrix, 65slack, 227vec, 4
Page 757 and 758:
757
Page 760:
Convex Optimization & Euclidean Dis
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v2007.11.26 - Convex Optimization

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