v2007.09.13 - Convex Optimization

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382 CHAPTER 5. EUCLIDEAN DISTANCE MATRIXc i ===[ ]−1 0 12 N−2 (N −2)! 2 det T1 −D i(−1) N2 N−2 (N −2)! 2 detD i(970)(1 T D −1i 1 ) (971)(−1) N2 N−2 (N −2)! 2 1T cof(D i ) T 1 (972)where D i is the i th principal N −1×N −1 submatrix 5.56 of D ∈ EDM N ,and cof(D i ) is the N −1×N −1 matrix of cofactors [247,4] correspondingto D i . The number of principal 3 × 3 submatrices in D is, of course, equalto the number of triangular facets in the tetrahedron; four (N!/(3!(N −3)!))when N = 4.5.14.3.1.1 Exercise. Sufficiency conditions for an EDM of four points.Triangle inequality (property 4) and area inequality (968) are conditionsnecessary for D to be an EDM. Prove their sufficiency in conjunction withthe remaining three Euclidean metric properties.5.14.3.2 N = 5Moving to the next level, we might encounter a Euclidean body calledpolychoron, a bounded polyhedron in four dimensions. 5.57 The polychoronhas five (N!/(4!(N −4)!)) facets, each of them a general tetrahedron whosevolume-squared c i is calculated using the same formula; (970) whereD is the EDM corresponding to the polychoron, and D i is the EDMcorresponding to the i th facet (the principal 4 × 4 submatrix of D ∈ EDM Ncorresponding to the i th tetrahedron). The analogue to triangle & distanceis now polychoron & facet-volume. We could then write another generalized“triangle” inequality like (968) but in terms of facet volume; [284,IV]√ci ≤ √ c j + √ c k + √ c l + √ c m , i≠j ≠k≠l≠m∈{1... 5} (973)5.56 Every principal submatrix of an EDM remains an EDM. [170,4.1]5.57 The simplest polychoron is called a pentatope [279]; a regular simplex hence convex.(A pentahedron is a three-dimensional body having five vertices.)
5.14. FIFTH PROPERTY OF EUCLIDEAN METRIC 383.Figure 93: Length of one-dimensional face a equals height h=a=1 of thisnonsimplicial pyramid in R 3 with square base inscribed in a circle of radius Rcentered at the origin. [279, Pyramid]5.14.3.2.1 Exercise. Sufficiency for an EDM of five points.For N = 5, triangle (distance) inequality (5.2), area inequality (968), andvolume inequality (973) are conditions necessary for D to be an EDM.Prove their sufficiency.5.14.3.3 Volume of simplicesThere is no known formula for the volume of a bounded general convexpolyhedron expressed either by halfspace or vertex-description. [296,2.1][207, p.173] [167] [168] [122] [123] Volume is a concept germane to R 3 ;in higher dimensions it is called content. Applying the EDM assertion(5.9.1.0.3) and a result given in [46,8.3.1], a general nonempty simplex(2.12.3) in R N−1 corresponding to an EDM D ∈ S N h has content√ c = content(S)√det(−V T N DV N) (974)where the content-squared of the unit simplex S ⊂ R N−1 is proportional toits Cayley-Menger determinant;content(S) 2 =[ ]−1 0 12 N−1 (N −1)! 2 det T1 −D([0 e 1 e 2 · · · e N−1 ])where e i ∈ R N−1 and the EDM operator used is D(X) (705).(975)
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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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5.4. EDM DEFINITION 3035.4.2.2.1 Ex
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5.4. EDM DEFINITION 305ten affine e
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5.4. EDM DEFINITION 307spheres:Then
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5.4. EDM DEFINITION 309By eliminati
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5.4. EDM DEFINITION 311whereΦ ij =
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5.4. EDM DEFINITION 3135.4.2.2.5 De
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5.4. EDM DEFINITION 315105ˇx 4ˇx
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5.4. EDM DEFINITION 317corrected by
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5.4. EDM DEFINITION 319aptly be app
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5.4. EDM DEFINITION 321As before, a
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5.4. EDM DEFINITION 323where ([√t
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5.4. EDM DEFINITION 325because (A.3
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5.5. INVARIANCE 3275.5.1.0.1 Exampl
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5.6. INJECTIVITY OF D & UNIQUE RECO
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Page 337 and 338: 5.7. EMBEDDING IN AFFINE HULL 337Fo
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Page 367 and 368: 5.12. LIST RECONSTRUCTION 367where
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Page 371 and 372: 5.13. RECONSTRUCTION EXAMPLES 371D
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Page 377 and 378: 5.14. FIFTH PROPERTY OF EUCLIDEAN M
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Page 387 and 388: Chapter 6EDM coneFor N > 3, the con
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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.
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Page 401 and 402: 6.5. EDM DEFINITION IN 11 T 401and
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Page 407 and 408: 6.5. EDM DEFINITION IN 11 T 407D =
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Page 419 and 420: 6.8. DUAL EDM CONE 419When the Fins
Page 421 and 422: 6.8. DUAL EDM CONE 421Proof. First,
Page 423 and 424: 6.8. DUAL EDM CONE 423EDM 2 = S 2 h
Page 425 and 426: 6.8. DUAL EDM CONE 425whose veracit
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Page 429 and 430: 6.8. DUAL EDM CONE 429has dual affi
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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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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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498 APPENDIX A. LINEAR ALGEBRAA.4 S
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502 APPENDIX A. LINEAR ALGEBRAA.5 e
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504 APPENDIX A. LINEAR ALGEBRAs i w
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508 APPENDIX A. LINEAR ALGEBRAΣq 2
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510 APPENDIX A. LINEAR ALGEBRAA.7 Z
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512 APPENDIX A. LINEAR ALGEBRAThere
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516 APPENDIX A. LINEAR ALGEBRA
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518 APPENDIX B. SIMPLE MATRICESB.1
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520 APPENDIX B. SIMPLE MATRICESProo
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522 APPENDIX B. SIMPLE MATRICESB.1.
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524 APPENDIX B. SIMPLE MATRICESN(u
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526 APPENDIX B. SIMPLE MATRICESDue
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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 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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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
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608 APPENDIX E. PROJECTIONThe test
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610 APPENDIX E. PROJECTIONPerpendic
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612 APPENDIX E. PROJECTIONE.8 Range
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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❇❇❇
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626 APPENDIX E. PROJECTIONE.10 Alte
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628 APPENDIX E. PROJECTIONbH 1H 2P
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630 APPENDIX E. PROJECTIONa(a){y |
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632 APPENDIX E. PROJECTION(a feasib
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634 APPENDIX E. PROJECTIONwhile, th
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636 APPENDIX E. PROJECTIONE.10.2.1.
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638 APPENDIX E. PROJECTION10 0dist(
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640 APPENDIX E. PROJECTIONE.10.3.1D
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642 APPENDIX E. PROJECTIONE 3K ⊥
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644 APPENDIX E. PROJECTION
Page 646 and 647:
646 APPENDIX F. MATLAB PROGRAMSif n
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648 APPENDIX F. MATLAB PROGRAMSend%
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650 APPENDIX F. MATLAB PROGRAMSF.1.
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652 APPENDIX F. MATLAB PROGRAMScoun
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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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664 APPENDIX F. MATLAB PROGRAMSbrea
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666 APPENDIX F. MATLAB PROGRAMSwhil
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668 APPENDIX F. MATLAB PROGRAMSF.7
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670 APPENDIX F. MATLAB PROGRAMS
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672 APPENDIX G. NOTATION AND A FEW
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Page 686 and 687:
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688 BIBLIOGRAPHY[7] Abdo Y. Alfakih
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690 BIBLIOGRAPHY[27] Aharon Ben-Tal
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692 BIBLIOGRAPHY[48] Lev M. Brègma
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694 BIBLIOGRAPHY[67] Joel Dawson, S
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696 BIBLIOGRAPHY[85] Alan Edelman,
Page 698 and 699:
698 BIBLIOGRAPHY[102] Philip E. Gil
Page 700 and 701:
700 BIBLIOGRAPHYWeiss, editors, Pol
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702 BIBLIOGRAPHY[146] Jean-Baptiste
Page 704 and 705:
704 BIBLIOGRAPHY[168] Jean B. Lasse
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706 BIBLIOGRAPHY[189] Rudolf Mathar
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708 BIBLIOGRAPHY[211] M. L. Overton
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710 BIBLIOGRAPHY[229] C. K. Rushfor
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712 BIBLIOGRAPHY[252] Jos F. Sturm
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714 BIBLIOGRAPHY[274] È. B. Vinber
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[294] Yinyu Ye. Semidefinite progra
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718 INDEXobtuse, 62positive semidef
Page 720 and 721:
720 INDEXelliptope, 642orthant, 177
Page 722 and 723:
722 INDEXdistancegeometry, 20, 317m
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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
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732 INDEXlinear operator, 587, 591,
Page 734 and 735:
734 INDEXlargest entries, 188monoto
Page 736:
736 INDEXsimilarity, 606translation
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