v2007.11.26 - Convex Optimization

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110 CHAPTER 2. CONVEX GEOMETRYA conic section is the intersection of a cone with any hyperplane. In threedimensions, an intersecting plane perpendicular to a circular cone’s axis ofrevolution produces a section bounded by a circle. (Figure 36) A prominentexample of a circular cone in convex analysis is the Lorentz cone (148). Wealso find that the positive semidefinite cone and cone of Euclidean distancematrices are circular cones, but only in low dimension.The positive semidefinite cone has axis of revolution that is the ray(base 0) through the identity matrix I . Consider the set of normalizedextreme directions of the positive semidefinite cone: for some arbitrarypositive constant a∈ R +{yy T ∈ S M | ‖y‖ = √ a} ⊂ ∂S M + (201)The distance from each extreme direction to the axis of revolution is theradius√R ∆ = infc ‖yyT − cI‖ F = a1 − 1 M(202)which is the distance from yy T to a M I ; the length of vector yyT − a M I .Because distance R (in a particular dimension) from the axis of revolutionto each and every normalized extreme direction is identical, the extremedirections lie on the boundary of a hypersphere in isometrically isomorphicR M(M+1)/2 . From Example 2.9.2.4.1, the convex hull (excluding the vertexat the origin) of the normalized extreme directions is a conic sectionC ∆ = conv{yy T | y ∈ R M , y T y = a} = S M + ∩ {A∈ S M | 〈I , A〉 = a} (203)orthogonal to the identity matrix I ;〈C − a M I , I〉 = tr(C − a I) = 0 (204)MAlthough the positive semidefinite cone possesses some characteristics ofa circular cone, we can prove it is not by demonstrating a shortage of extremedirections; id est, some extreme directions corresponding to each and everyangle of rotation about the axis of revolution are nonexistent: Referring toFigure 37, [290,1-7]
2.9. POSITIVE SEMIDEFINITE (PSD) CONE 111yy TaI θ aMRM 11TFigure 37: Illustrated is a section, perpendicular to axis of revolution, ofcircular cone from Figure 36. Radius R is distance from any extremedirection to axis at a I . Vector aM M 11T is an arbitrary reference by whichto measure angle θ .cos θ =〈 aM 11T − a M I , yyT − a M I〉a 2 (1 − 1 M ) (205)Solving for vector y we geta(1 + (M −1) cosθ) = (1 T y) 2 (206)Because this does not have real solution for every matrix dimension M andfor all 0 ≤ θ ≤ 2π , then we can conclude that the positive semidefinite conemight be circular but only in matrix dimensions 1 and 2 . 2.35 Because of a shortage of extreme directions, conic section (203) cannotbe hyperspherical by the extremes theorem (2.8.1.1.1).2.35 In fact, the positive semidefinite cone is circular in matrix dimensions 1 and 2 whileit is a rotation of the Lorentz cone in matrix dimension 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 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
Page 59 and 60: 2.4. HALFSPACE, HYPERPLANE 59of dim
Page 61 and 62: 2.4. HALFSPACE, HYPERPLANE 61H +ay
Page 63 and 64: 2.4. HALFSPACE, HYPERPLANE 63Inters
Page 65 and 66: 2.4. HALFSPACE, HYPERPLANE 65Conver
Page 67 and 68: 2.4. HALFSPACE, HYPERPLANE 67A 1A 2
Page 69 and 70: 2.4. HALFSPACE, HYPERPLANE 69tradit
Page 71 and 72: 2.4. HALFSPACE, HYPERPLANE 71There
Page 73 and 74: 2.5. SUBSPACE REPRESENTATIONS 73nor
Page 75 and 76: 2.5. SUBSPACE REPRESENTATIONS 752.5
Page 77 and 78: 2.6. EXTREME, EXPOSED 77In other wo
Page 79 and 80: 2.6. EXTREME, EXPOSED 792.6.1 Expos
Page 81 and 82: 2.7. CONES 812.6.1.3.1 Definition.
Page 83 and 84: 2.7. CONES 830Figure 26: Boundary o
Page 85 and 86: 2.7. CONES 852.7.2 Convex coneWe ca
Page 87 and 88: 2.7. CONES 87Thus the simplest and
Page 89 and 90: 2.7. CONES 89nomenclature generaliz
Page 91 and 92: 2.8. CONE BOUNDARY 91Proper cone {0
Page 93 and 94: 2.8. CONE BOUNDARY 93the same extre
Page 95 and 96: 2.8. CONE BOUNDARY 95For a proper c
Page 97 and 98: 2.9. POSITIVE SEMIDEFINITE (PSD) CO
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Page 121 and 122: 2.10. CONIC INDEPENDENCE (C.I.) 121
Page 127 and 128: 2.12. CONVEX POLYHEDRA 127It follow
Page 129 and 130: 2.12. CONVEX POLYHEDRA 129Coefficie
Page 131 and 132: 2.12. CONVEX POLYHEDRA 1312.12.3 Un
Page 133 and 134: 2.12. CONVEX POLYHEDRA 133
Page 135 and 136: 2.13. DUAL CONE & GENERALIZED INEQU
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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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5.4. EDM DEFINITION 341because (A.3
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5.5. INVARIANCE 3435.5.1.0.1 Exampl
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5.6. INJECTIVITY OF D & UNIQUE RECO
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5.7. EMBEDDING IN AFFINE HULL 3515.
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5.7. EMBEDDING IN AFFINE HULL 353Fo
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5.7. EMBEDDING IN AFFINE HULL 3555.
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5.8. EUCLIDEAN METRIC VERSUS MATRIX
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5.9. BRIDGE: CONVEX POLYHEDRA TO ED
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5.10. EDM-ENTRY COMPOSITION 373(ii)
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5.11. EDM INDEFINITENESS 3755.11.1
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5.11. EDM INDEFINITENESS 377(confer
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5.11. EDM INDEFINITENESS 379we have
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5.11. EDM INDEFINITENESS 381For pre
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5.12. LIST RECONSTRUCTION 383where
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5.12. LIST RECONSTRUCTION 385(a)(c)
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5.13. RECONSTRUCTION EXAMPLES 387D
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5.13. RECONSTRUCTION EXAMPLES 389Th
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5.13. RECONSTRUCTION EXAMPLES 391wh
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5.14. FIFTH PROPERTY OF EUCLIDEAN M
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Chapter 6Cone of distance matricesF
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6.1. DEFINING EDM CONE 4056.1 Defin
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6.2. POLYHEDRAL BOUNDS 407This cone
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6.3.√EDM CONE IS NOT CONVEX 409N
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6.4. A GEOMETRY OF COMPLETION 4116.
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6.4. A GEOMETRY OF COMPLETION 413(a
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6.4. A GEOMETRY OF COMPLETION 415Fi
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6.5. EDM DEFINITION IN 11 T 417and
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6.5. EDM DEFINITION IN 11 T 419then
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6.5. EDM DEFINITION IN 11 T 4216.5.
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6.5. EDM DEFINITION IN 11 T 423D =
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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 435When the Fins
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6.8. DUAL EDM CONE 437Proof. First,
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6.8. DUAL EDM CONE 439EDM 2 = S 2 h
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6.8. DUAL EDM CONE 441whose veracit
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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
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BIBLIOGRAPHY 733[285] Bernard Widro
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Index0-norm, 241, 273, 275, 2941-no
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INDEX 737clipping, 190, 454, 638, 6
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INDEX 739hull, 53, 56, 306of outer
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INDEX 741unique, 519elbow, 457, 458
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INDEX 743polarity, 62handoff, 335ov
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INDEX 745manifold, 23, 24, 105, 410
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INDEX 747objectivebilinear, 256, 27
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INDEX 749Hadamard, 46, 472, 571, 69
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INDEX 751trace, 477, 478heuristic,
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INDEX 753symmetric hollow, 51tangen
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INDEX 755matrix, 65slack, 227vec, 4
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Convex Optimization & Euclidean Dis
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v2007.11.26 - Convex Optimization

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