v2007.09.13 - Convex Optimization

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354 CHAPTER 5. EUCLIDEAN DISTANCE MATRIXBecause p ∈ P − α may be found by factoring (894), the list Θ p is foundby factoring (893). A unique EDM can be made from that list usinginner-product form definition D(Θ)| Θ=Θp (770). That EDM will be identicalto D if δ(D)=0, by injectivity of D (814).5.9.2 Necessity and sufficiencyFrom (856) we learned that matrix inequality −VN TDV N ≽ 0 is a necessarytest for D to be an EDM. In5.9.1, the connection between convex polyhedraand EDMs was pronounced by the EDM assertion; the matrix inequalitytogether with D ∈ S N h became a sufficient test when the EDM assertiondemanded that every bounded convex polyhedron have a correspondingEDM. For all N >1 (5.8.3), the matrix criteria for the existence of an EDMin (724), (879), and (700) are therefore necessary and sufficient and subsumeall the Euclidean metric properties and further requirements.5.9.3 Example revisitedNow we apply the necessary and sufficient EDM criteria (724) to an earlierproblem.5.9.3.0.1 Example. Small completion problem, III. (confer5.8.3.1.1)Continuing Example 5.3.0.0.2 pertaining to Figure 74 where N = 4,distance-square d 14 is ascertainable from the matrix inequality −VN TDV N ≽0.Because all distances in (697) are known except √ d 14 , we may simplycalculate the smallest eigenvalue of −VN TDV N over a range of d 14 as inFigure 88. We observe a unique value of d 14 satisfying (724) where theabscissa is tangent to the hypograph of the smallest eigenvalue. Sincethe smallest eigenvalue of a symmetric matrix is known to be a concavefunction (5.8.4), we calculate its second partial derivative with respect tod 14 evaluated at 2 and find −1/3. We conclude there are no other satisfyingvalues of d 14 . Further, that value of d 14 does not meet an upper or lowerbound of a triangle inequality like (867), so neither does it cause the collapseof any triangle. Because the smallest eigenvalue is 0, affine dimension r ofany point list corresponding to D cannot exceed N −2. (5.7.1.1)
5.9. BRIDGE: CONVEX POLYHEDRA TO EDMS 355smallest eigenvalue1 2 3 4d 14-0.2-0.4-0.6Figure 88: Smallest eigenvalue of −VN TDV Nvalue of d 14 : 2.makes PSD matrix for only one1.41.2d 1/αijlog 2 (1+d 1/αij )(a)10.80.60.40.21−e −αd ij40.5 1 1.5 2d ij3(b)21d 1/αijlog 2 (1+d 1/αij )1−e −αd ij0.5 1 1.5 2αFigure 89: Some entrywise EDM compositions: (a) α = 2. Concavenondecreasing in d ij . (b) Trajectory convergence in α for d ij = 2.
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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 Quadratic func
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LIST OF FIGURES 17E Projection 5791
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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 9: 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 12: 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 732.5
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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 24: 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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96 CHAPTER 2. CONVEX GEOMETRYBCADFi
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98 CHAPTER 2. CONVEX GEOMETRYThe po
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100 CHAPTER 2. CONVEX GEOMETRY2.9.0
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102 CHAPTER 2. CONVEX GEOMETRYwhere
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104 CHAPTER 2. CONVEX GEOMETRY√2
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106 CHAPTER 2. CONVEX GEOMETRYwhich
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108 CHAPTER 2. CONVEX GEOMETRY2.9.2
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110 CHAPTER 2. CONVEX GEOMETRYA con
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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 189where the n
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3.1. CONVEX FUNCTION 191where k ∈
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3.1. CONVEX FUNCTION 193f(z)Az 2z 1
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3.1. CONVEX FUNCTION 195{a T z 1 +
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3.1. CONVEX FUNCTION 197When an epi
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3.1. CONVEX FUNCTION 199orthonormal
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3.1. CONVEX FUNCTION 201[30,1.1] Ex
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3.1. CONVEX FUNCTION 20321.510.5Y 2
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3.1. CONVEX FUNCTION 2053.1.8.0.1 E
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3.1. CONVEX FUNCTION 207This equiva
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3.1. CONVEX FUNCTION 2093.1.8.1 mon
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3.1. CONVEX FUNCTION 211[ Yt]∈ ep
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3.1. CONVEX FUNCTION 213→Y −Xwh
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3.2. MATRIX-VALUED CONVEX FUNCTION
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3.3. QUASICONVEX 221A quasiconcave
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3.4. SALIENT PROPERTIES 2236.A nonn
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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 2414.2.3.0.2 Example
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4.2. FRAMEWORK 243where δ is the m
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4.2. FRAMEWORK 2454.2.3.0.3 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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Chapter 5Euclidean Distance MatrixT
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5.2. FIRST METRIC PROPERTIES 291cor
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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 297The collecti
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5.4. EDM DEFINITION 2995.4.2 Gram-f
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5.4. EDM DEFINITION 301D ∈ EDM N
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Page 307 and 308: 5.4. EDM DEFINITION 307spheres:Then
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Page 315 and 316: 5.4. EDM DEFINITION 315105ˇx 4ˇx
Page 317 and 318: 5.4. EDM DEFINITION 317corrected by
Page 319 and 320: 5.4. EDM DEFINITION 319aptly be app
Page 321 and 322: 5.4. EDM DEFINITION 321As before, a
Page 323 and 324: 5.4. EDM DEFINITION 323where ([√t
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Page 327 and 328: 5.5. INVARIANCE 3275.5.1.0.1 Exampl
Page 329 and 330: 5.6. INJECTIVITY OF D & UNIQUE RECO
Page 335 and 336: 5.7. EMBEDDING IN AFFINE HULL 3355.
Page 337 and 338: 5.7. EMBEDDING IN AFFINE HULL 337Fo
Page 339 and 340: 5.7. EMBEDDING IN AFFINE HULL 3395.
Page 341 and 342: 5.8. EUCLIDEAN METRIC VERSUS MATRIX
Page 349 and 350: 5.9. BRIDGE: CONVEX POLYHEDRA TO ED
Page 351 and 352: 5.9. BRIDGE: CONVEX POLYHEDRA TO ED
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Page 357 and 358: 5.10. EDM-ENTRY COMPOSITION 357(ii)
Page 359 and 360: 5.11. EDM INDEFINITENESS 3595.11.1
Page 361 and 362: 5.11. EDM INDEFINITENESS 361(confer
Page 363 and 364: 5.11. EDM INDEFINITENESS 363we have
Page 365 and 366: 5.11. EDM INDEFINITENESS 365For pre
Page 367 and 368: 5.12. LIST RECONSTRUCTION 367where
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Page 371 and 372: 5.13. RECONSTRUCTION EXAMPLES 371D
Page 373 and 374: 5.13. RECONSTRUCTION EXAMPLES 373Th
Page 375 and 376: 5.13. RECONSTRUCTION EXAMPLES 375wh
Page 377 and 378: 5.14. FIFTH PROPERTY OF EUCLIDEAN M
Page 387 and 388: Chapter 6EDM coneFor N > 3, the con
Page 389 and 390: 6.1. DEFINING EDM CONE 3896.1 Defin
Page 391 and 392: 6.2. POLYHEDRAL BOUNDS 391This cone
Page 393 and 394: 6.3.√EDM CONE IS NOT CONVEX 393N
Page 395 and 396: 6.4. A GEOMETRY OF COMPLETION 3956.
Page 397 and 398: 6.4. A GEOMETRY OF COMPLETION 397(a
Page 399 and 400: 6.4. A GEOMETRY OF COMPLETION 399Fi
Page 401 and 402: 6.5. EDM DEFINITION IN 11 T 401and
Page 403 and 404: 6.5. EDM DEFINITION IN 11 T 403then
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6.5. EDM DEFINITION IN 11 T 4056.5.
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6.5. EDM DEFINITION IN 11 T 407D =
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6.6. CORRESPONDENCE TO PSD CONE S N
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6.7. VECTORIZATION & PROJECTION INT
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6.7. VECTORIZATION & PROJECTION INT
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6.8. DUAL EDM CONE 419When the Fins
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6.8. DUAL EDM CONE 421Proof. First,
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6.8. DUAL EDM CONE 423EDM 2 = S 2 h
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6.8. DUAL EDM CONE 425whose veracit
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6.8. DUAL EDM CONE 4276.8.1.3.1 Exe
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6.8. DUAL EDM CONE 429has dual affi
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6.8. DUAL EDM CONE 4316.8.1.7 Schoe
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6.9. THEOREM OF THE ALTERNATIVE 433
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6.10. POSTSCRIPT 435When 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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441HS N h0EDM NK = S N h ∩ R N×N
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4437.0.3 Problem approachProblems t
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7.1. FIRST PREVALENT PROBLEM: 445fi
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7.1. FIRST PREVALENT PROBLEM: 4477.
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7.1. FIRST PREVALENT PROBLEM: 449di
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7.1. FIRST PREVALENT PROBLEM: 4517.
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7.1. FIRST PREVALENT PROBLEM: 453wh
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7.1. FIRST PREVALENT PROBLEM: 455Th
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7.2. SECOND PREVALENT PROBLEM: 457O
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7.2. SECOND PREVALENT PROBLEM: 459S
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7.2. SECOND PREVALENT PROBLEM: 461r
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7.2. SECOND PREVALENT PROBLEM: 463c
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7.2. SECOND PREVALENT PROBLEM: 4657
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7.3. THIRD PREVALENT PROBLEM: 467fo
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7.3. THIRD PREVALENT PROBLEM: 469a
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7.3. THIRD PREVALENT PROBLEM: 4717.
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7.3. THIRD PREVALENT PROBLEM: 4737.
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7.3. THIRD PREVALENT PROBLEM: 475Ou
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478 CHAPTER 7. PROXIMITY PROBLEMSth
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480 APPENDIX A. LINEAR ALGEBRAA.1.1
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484 APPENDIX A. LINEAR ALGEBRAonly
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486 APPENDIX A. LINEAR ALGEBRA(AB)
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Page 490 and 491:
490 APPENDIX A. LINEAR ALGEBRAFor A
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492 APPENDIX A. LINEAR ALGEBRADiago
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494 APPENDIX A. LINEAR ALGEBRAFor A
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Page 498 and 499:
498 APPENDIX A. LINEAR ALGEBRAA.4 S
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Page 502 and 503:
502 APPENDIX A. LINEAR ALGEBRAA.5 e
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504 APPENDIX A. LINEAR ALGEBRAs i w
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Page 508 and 509:
508 APPENDIX A. LINEAR ALGEBRAΣq 2
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510 APPENDIX A. LINEAR ALGEBRAA.7 Z
Page 512 and 513:
512 APPENDIX A. LINEAR ALGEBRAThere
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Page 516 and 517:
516 APPENDIX A. LINEAR ALGEBRA
Page 518 and 519:
518 APPENDIX B. SIMPLE MATRICESB.1
Page 520 and 521:
520 APPENDIX B. SIMPLE MATRICESProo
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522 APPENDIX B. SIMPLE MATRICESB.1.
Page 524 and 525:
524 APPENDIX B. SIMPLE MATRICESN(u
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526 APPENDIX B. SIMPLE MATRICESDue
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Page 530 and 531:
530 APPENDIX B. SIMPLE MATRICEShas
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532 APPENDIX B. SIMPLE MATRICESFigu
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Page 536 and 537:
536 APPENDIX C. SOME ANALYTICAL OPT
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Page 548 and 549:
Page 550 and 551:
550 APPENDIX D. MATRIX CALCULUSThe
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552 APPENDIX D. MATRIX CALCULUSGrad
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554 APPENDIX D. MATRIX CALCULUSBeca
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556 APPENDIX D. MATRIX CALCULUSwhic
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558 APPENDIX D. MATRIX CALCULUS⎡
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560 APPENDIX D. MATRIX CALCULUS→Y
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562 APPENDIX D. MATRIX CALCULUSD.1.
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564 APPENDIX D. MATRIX CALCULUSwhic
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566 APPENDIX D. MATRIX CALCULUSIn t
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570 APPENDIX D. MATRIX CALCULUSD.2
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572 APPENDIX D. MATRIX CALCULUSalge
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574 APPENDIX D. MATRIX CALCULUStrac
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Page 578 and 579:
578 APPENDIX D. MATRIX CALCULUS
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580 APPENDIX E. PROJECTIONThe follo
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582 APPENDIX E. PROJECTIONFor matri
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584 APPENDIX E. PROJECTION(⇐) To
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586 APPENDIX E. PROJECTIONNonorthog
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588 APPENDIX E. PROJECTIONE.2.0.0.1
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590 APPENDIX E. PROJECTIONE.3.2Orth
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592 APPENDIX E. PROJECTIONE.3.5Unif
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594 APPENDIX E. PROJECTIONE.4 Algeb
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596 APPENDIX E. PROJECTIONa ∗ 2K
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598 APPENDIX E. PROJECTIONwhere Y =
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600 APPENDIX E. PROJECTION(B.4.2).
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602 APPENDIX E. PROJECTIONis a nono
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604 APPENDIX E. PROJECTIONE.6.4.1Or
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606 APPENDIX E. PROJECTIONq i q T i
Page 608 and 609:
608 APPENDIX E. PROJECTIONThe test
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610 APPENDIX E. PROJECTIONPerpendic
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612 APPENDIX E. PROJECTIONE.8 Range
Page 614 and 615:
614 APPENDIX E. PROJECTIONAs for su
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616 APPENDIX E. PROJECTIONWith refe
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618 APPENDIX E. PROJECTIONProjectio
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620 APPENDIX E. PROJECTIONE.9.2.2.2
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622 APPENDIX E. PROJECTIONThe foreg
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624 APPENDIX E. PROJECTION❇❇❇
Page 626 and 627:
626 APPENDIX E. PROJECTIONE.10 Alte
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628 APPENDIX E. PROJECTIONbH 1H 2P
Page 630 and 631:
630 APPENDIX E. PROJECTIONa(a){y |
Page 632 and 633:
632 APPENDIX E. PROJECTION(a feasib
Page 634 and 635:
634 APPENDIX E. PROJECTIONwhile, th
Page 636 and 637:
636 APPENDIX E. PROJECTIONE.10.2.1.
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638 APPENDIX E. PROJECTION10 0dist(
Page 640 and 641:
640 APPENDIX E. PROJECTIONE.10.3.1D
Page 642 and 643:
642 APPENDIX E. PROJECTIONE 3K ⊥
Page 644 and 645:
644 APPENDIX E. PROJECTION
Page 646 and 647:
646 APPENDIX F. MATLAB PROGRAMSif n
Page 648 and 649:
648 APPENDIX F. MATLAB PROGRAMSend%
Page 650 and 651:
650 APPENDIX F. MATLAB PROGRAMSF.1.
Page 652 and 653:
652 APPENDIX F. MATLAB PROGRAMScoun
Page 654 and 655:
654 APPENDIX F. MATLAB PROGRAMSF.3
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Page 658 and 659:
658 APPENDIX F. MATLAB PROGRAMS% so
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660 APPENDIX F. MATLAB PROGRAMS% tr
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Page 664 and 665:
664 APPENDIX F. MATLAB PROGRAMSbrea
Page 666 and 667:
666 APPENDIX F. MATLAB PROGRAMSwhil
Page 668 and 669:
668 APPENDIX F. MATLAB PROGRAMSF.7
Page 670 and 671:
670 APPENDIX F. MATLAB PROGRAMS
Page 672 and 673:
672 APPENDIX G. NOTATION AND A FEW
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Page 678 and 679:
Page 680 and 681:
Page 682 and 683:
Page 684 and 685:
Page 686 and 687:
Page 688 and 689:
688 BIBLIOGRAPHY[7] Abdo Y. Alfakih
Page 690 and 691:
690 BIBLIOGRAPHY[27] Aharon Ben-Tal
Page 692 and 693:
692 BIBLIOGRAPHY[48] Lev M. Brègma
Page 694 and 695:
694 BIBLIOGRAPHY[67] Joel Dawson, S
Page 696 and 697:
696 BIBLIOGRAPHY[85] Alan Edelman,
Page 698 and 699:
698 BIBLIOGRAPHY[102] Philip E. Gil
Page 700 and 701:
700 BIBLIOGRAPHYWeiss, editors, Pol
Page 702 and 703:
702 BIBLIOGRAPHY[146] Jean-Baptiste
Page 704 and 705:
704 BIBLIOGRAPHY[168] Jean B. Lasse
Page 706 and 707:
706 BIBLIOGRAPHY[189] Rudolf Mathar
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708 BIBLIOGRAPHY[211] M. L. Overton
Page 710 and 711:
710 BIBLIOGRAPHY[229] C. K. Rushfor
Page 712 and 713:
712 BIBLIOGRAPHY[252] Jos F. Sturm
Page 714 and 715:
714 BIBLIOGRAPHY[274] È. B. Vinber
Page 716 and 717:
[294] Yinyu Ye. Semidefinite progra
Page 718 and 719:
718 INDEXobtuse, 62positive semidef
Page 720 and 721:
720 INDEXelliptope, 642orthant, 177
Page 722 and 723:
722 INDEXdistancegeometry, 20, 317m
Page 724 and 725:
724 INDEX-dimensional, 37, 89rank,
Page 726 and 727:
726 INDEXGram form, 331is, 674isedm
Page 728 and 729:
728 INDEXdiscretized, 152, 431in su
Page 730 and 731:
730 INDEXboundary, 115dimension, 10
Page 732 and 733:
732 INDEXlinear operator, 587, 591,
Page 734 and 735:
734 INDEXlargest entries, 188monoto
Page 736:
736 INDEXsimilarity, 606translation
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