v2007.09.13 - Convex Optimization

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96 CHAPTER 2. CONVEX GEOMETRYBCADFigure 30: Properties of extreme points carry over to extreme directions.[228,18] Four rays (drawn truncated) on boundary of conic hull oftwo-dimensional closed convex set from Figure 21 lifted to R 3 . Raythrough point A is exposed hence extreme. Extreme direction B on coneboundary is not an exposed direction, although it belongs to the exposedface cone{A,B}. Extreme ray through C is exposed. Point D is neitheran exposed or extreme direction although it belongs to a two-dimensionalexposed face of the conic hull.0
2.9. POSITIVE SEMIDEFINITE (PSD) CONE 97boundary definition (2.6.1.3.1); although it can correctly be inferred: eachand every extreme point and direction belongs to some exposed face.Arbitrary points residing on the relative boundary of a convex set are notnecessarily exposed or extreme points. Similarly, the direction of an arbitraryray, base 0, on the boundary of a convex cone is not necessarily an exposedor extreme direction. For the polyhedral cone illustrated in Figure 15, forexample, there are three two-dimensional exposed faces constituting theentire boundary, each composed of an infinity of rays. Yet there are onlythree exposed directions.Neither is an extreme direction on the boundary of a pointed convex conenecessarily an exposed direction. Lift the two-dimensional set in Figure 21,for example, into three dimensions such that no two points in the set arecollinear with the origin. Then its conic hull can have an extreme directionB on the boundary that is not an exposed direction, illustrated in Figure 30.2.9 Positive semidefinite (PSD) coneThe cone of positive semidefinite matrices studied in this sectionis arguably the most important of all non-polyhedral cones whosefacial structure we completely understand.−Alexander Barvinok [20, p.78]2.9.0.0.1 Definition. Positive semidefinite cone.The set of all symmetric positive semidefinite matrices of particulardimension M is called the positive semidefinite cone:S M ∆+ = { A ∈ S M | A ≽ 0 }= { A ∈ S M | y T Ay ≥0 ∀ ‖y‖ = 1 }= ⋂ {A ∈ S M | 〈yy T , A〉 ≥ 0 } (160)‖y‖=1formed by the intersection of an infinite number of halfspaces (2.4.1.1) invectorized variable A , 2.29 each halfspace having partial boundary containingthe origin in isomorphic R M(M+1)/2 . It is a unique immutable proper conein the ambient space of symmetric matrices S M .2.29 infinite in number when M >1. Because y T A y=y T A T y , matrix A is almost alwaysassumed symmetric. (A.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 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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Page 53 and 54: 2.3. HULLS 53Figure 12: Convex hull
Page 55 and 56: 2.3. HULLS 55Aaffine hull (drawn tr
Page 57 and 58: 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 732.5
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 24: 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 98 and 99: 98 CHAPTER 2. CONVEX GEOMETRYThe po
Page 100 and 101: 100 CHAPTER 2. CONVEX GEOMETRY2.9.0
Page 102 and 103: 102 CHAPTER 2. CONVEX GEOMETRYwhere
Page 104 and 105: 104 CHAPTER 2. CONVEX GEOMETRY√2
Page 106 and 107: 106 CHAPTER 2. CONVEX GEOMETRYwhich
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Page 110: 110 CHAPTER 2. CONVEX GEOMETRYA con
Page 115 and 116: 2.9. POSITIVE SEMIDEFINITE (PSD) CO
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 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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5.7. EMBEDDING IN AFFINE HULL 3355.
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5.7. EMBEDDING IN AFFINE HULL 337Fo
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5.7. EMBEDDING IN AFFINE HULL 3395.
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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 357(ii)
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5.11. EDM INDEFINITENESS 3595.11.1
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5.11. EDM INDEFINITENESS 361(confer
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5.11. EDM INDEFINITENESS 363we have
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5.11. EDM INDEFINITENESS 365For pre
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5.12. LIST RECONSTRUCTION 367where
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5.12. LIST RECONSTRUCTION 369(a)(c)
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5.13. RECONSTRUCTION EXAMPLES 371D
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5.13. RECONSTRUCTION EXAMPLES 373Th
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5.13. RECONSTRUCTION EXAMPLES 375wh
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5.14. FIFTH PROPERTY OF EUCLIDEAN M
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Chapter 6EDM coneFor N > 3, the con
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6.1. DEFINING EDM CONE 3896.1 Defin
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6.2. POLYHEDRAL BOUNDS 391This cone
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6.3.√EDM CONE IS NOT CONVEX 393N
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6.4. A GEOMETRY OF COMPLETION 3956.
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6.4. A GEOMETRY OF COMPLETION 397(a
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6.4. A GEOMETRY OF COMPLETION 399Fi
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6.5. EDM DEFINITION IN 11 T 401and
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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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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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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
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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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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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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,
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698 BIBLIOGRAPHY[102] Philip E. Gil
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700 BIBLIOGRAPHYWeiss, editors, Pol
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702 BIBLIOGRAPHY[146] Jean-Baptiste
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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
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720 INDEXelliptope, 642orthant, 177
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722 INDEXdistancegeometry, 20, 317m
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724 INDEX-dimensional, 37, 89rank,
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726 INDEXGram form, 331is, 674isedm
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728 INDEXdiscretized, 152, 431in su
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730 INDEXboundary, 115dimension, 10
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732 INDEXlinear operator, 587, 591,
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734 INDEXlargest entries, 188monoto
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736 INDEXsimilarity, 606translation
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