v2007.09.13 - Convex Optimization

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10 CONVEX OPTIMIZATION & EUCLIDEAN DISTANCE GEOMETRY4 Semidefinite programming 2254.1 Conic problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 2264.2 Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2344.3 Rank reduction . . . . . . . . . . . . . . . . . . . . . . . . . . 2474.4 Rank-constrained semidefinite program . . . . . . . . . . . . . 2565 Euclidean Distance Matrix 2895.1 EDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2905.2 First metric properties . . . . . . . . . . . . . . . . . . . . . . 2915.3 ∃ fifth Euclidean metric property . . . . . . . . . . . . . . . . 2915.4 EDM definition . . . . . . . . . . . . . . . . . . . . . . . . . . 2965.5 Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3265.6 Injectivity of D & unique reconstruction . . . . . . . . . . . . 3295.7 Embedding in affine hull . . . . . . . . . . . . . . . . . . . . . 3355.8 Euclidean metric versus matrix criteria . . . . . . . . . . . . . 3405.9 Bridge: <strong>Convex</strong> polyhedra to EDMs . . . . . . . . . . . . . . . 3485.10 EDM-entry composition . . . . . . . . . . . . . . . . . . . . . 3565.11 EDM indefiniteness . . . . . . . . . . . . . . . . . . . . . . . . 3585.12 List reconstruction . . . . . . . . . . . . . . . . . . . . . . . . 3665.13 Reconstruction examples . . . . . . . . . . . . . . . . . . . . . 3705.14 Fifth property of Euclidean metric . . . . . . . . . . . . . . . 3776 EDM cone 3876.1 Defining EDM cone . . . . . . . . . . . . . . . . . . . . . . . . 3896.2 Polyhedral bounds . . . . . . . . . . . . . . . . . . . . . . . . 3916.3√EDM cone is not convex . . . . . . . . . . . . . . . . . . . . 3936.4 a geometry of completion . . . . . . . . . . . . . . . . . . . . . 3956.5 EDM definition in 11 T . . . . . . . . . . . . . . . . . . . . . . 4006.6 Correspondence to PSD cone S N−1+ . . . . . . . . . . . . . . . 4096.7 Vectorization & projection interpretation . . . . . . . . . . . . 4146.8 Dual EDM cone . . . . . . . . . . . . . . . . . . . . . . . . . . 4196.9 Theorem of the alternative . . . . . . . . . . . . . . . . . . . . 4336.10 postscript . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4357 Proximity problems 4377.1 First prevalent problem: . . . . . . . . . . . . . . . . . . . . . 4457.2 Second prevalent problem: . . . . . . . . . . . . . . . . . . . . 4567.3 Third prevalent problem: . . . . . . . . . . . . . . . . . . . . . 467
CONVEX OPTIMIZATION & EUCLIDEAN DISTANCE GEOMETRY 117.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477A Linear algebra 479A.1 Main-diagonal δ operator, λ , trace, vec . . . . . . . . . . . . 479A.2 Semidefiniteness: domain of test . . . . . . . . . . . . . . . . . 483A.3 Proper statements . . . . . . . . . . . . . . . . . . . . . . . . . 486A.4 Schur complement . . . . . . . . . . . . . . . . . . . . . . . . 498A.5 eigen decomposition . . . . . . . . . . . . . . . . . . . . . . . . 502A.6 Singular value decomposition, SVD . . . . . . . . . . . . . . . 505A.7 Zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510B Simple matrices 517B.1 Rank-one matrix (dyad) . . . . . . . . . . . . . . . . . . . . . 518B.2 Doublet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523B.3 Elementary matrix . . . . . . . . . . . . . . . . . . . . . . . . 524B.4 Auxiliary V -matrices . . . . . . . . . . . . . . . . . . . . . . . 526B.5 Orthogonal matrix . . . . . . . . . . . . . . . . . . . . . . . . 531C Some analytical optimal results 535C.1 properties of infima . . . . . . . . . . . . . . . . . . . . . . . . 535C.2 diagonal, trace, singular and eigen values . . . . . . . . . . . . 536C.3 Orthogonal Procrustes problem . . . . . . . . . . . . . . . . . 542C.4 Two-sided orthogonal Procrustes . . . . . . . . . . . . . . . . 544D Matrix calculus 549D.1 Directional derivative, Taylor series . . . . . . . . . . . . . . . 549D.2 Tables of gradients and derivatives . . . . . . . . . . . . . . . 570E Projection 579E.1 Idempotent matrices . . . . . . . . . . . . . . . . . . . . . . . 582E.2 I − P , Projection on algebraic complement . . . . . . . . . . . 587E.3 Symmetric idempotent matrices . . . . . . . . . . . . . . . . . 588E.4 Algebra of projection on affine subsets . . . . . . . . . . . . . 594E.5 Projection examples . . . . . . . . . . . . . . . . . . . . . . . 594E.6 Vectorization interpretation, . . . . . . . . . . . . . . . . . . . 601E.7 on vectorized matrices of higher rank . . . . . . . . . . . . . . 608E.8 Range/Rowspace interpretation . . . . . . . . . . . . . . . . . 612E.9 Projection on convex set . . . . . . . . . . . . . . . . . . . . . 612
Page 1 and 2: DATTORROCONVEXOPTIMIZATION&EUCLIDEA
Page 3 and 4: Convex Optimization&Euclidean Dista
Page 5 and 6: for Jennie Columba♦Antonio♦♦&
Page 7 and 8: PreludeThe constant demands of my d
Page 9: Convex Optimization&Euclidean Dista
Page 13 and 14: List of Figures1 Overview 191 Orion
Page 15 and 16: LIST OF FIGURES 1559 Quadratic func
Page 17 and 18: LIST OF FIGURES 17E Projection 5791
Page 19 and 20: Chapter 1OverviewConvex Optimizatio
Page 21 and 22: ˇx 4ˇx 3ˇx 2Figure 2: Applicatio
Page 23 and 24: 23Figure 4: This coarsely discretiz
Page 25 and 26: ases (biorthogonal expansion). We e
Page 27 and 28: 27Figure 7: These bees construct a
Page 29 and 30: its membership to the EDM cone. The
Page 31 and 32: 31appendicesProvided so as to be mo
Page 33 and 34: Chapter 2Convex geometryConvexity h
Page 35 and 36: 2.1. CONVEX SET 35Figure 9: A slab
Page 37 and 38: 2.1. CONVEX SET 372.1.6 empty set v
Page 39 and 40: 2.1. CONVEX SET 392.1.7.1 Line inte
Page 41 and 42: 2.1. CONVEX SET 41(a)R 2(b)R 3(c)(d
Page 43 and 44: 2.1. CONVEX SET 43This theorem in c
Page 45 and 46: 2.2. VECTORIZED-MATRIX INNER PRODUC
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
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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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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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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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Page 658 and 659:
658 APPENDIX F. MATLAB PROGRAMS% so
Page 660 and 661:
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
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670 APPENDIX F. MATLAB PROGRAMS
Page 672 and 673:
672 APPENDIX G. NOTATION AND A FEW
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Page 676 and 677:
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
Page 708 and 709:
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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v2007.09.13 - Convex Optimization

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