v2007.09.13 - Convex Optimization

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642 APPENDIX E. PROJECTIONE 3K ⊥ E 3 (11 T ) + 11 TFigure 129: A few renderings (next page) of normal cone K ⊥ Eto elliptope3E 3 (Figure 86) at point 11 T , projected on R 3 . In [172, fig.2], normalcone is claimed circular in this dimension (severe numerical artifacts corruptboundary and make interior corporeal, drawn truncated).
E.10. ALTERNATING PROJECTION 643When set S is a convex cone K , then the normal cone to K at the originK ⊥ K(0) = −K ∗ (1853)is the negative dual cone. Any point belonging to −K ∗ , projected on K ,projects on the origin. More generally, [73,4.5]K ⊥ K(a) = −(K − a) ∗ (1854)KK(a∈ ⊥ K) = −K ∗ ∩ a ⊥ (1855)⋂The normal cone to Ck at Pb in Figure 122 is the ray{ξ(b −Pb) | ξ ≥0} illustrated in Figure 128. Applying Dykstra’s algorithm tothat example, convergence to the desired result is achieved in two iterations asillustrated in Figure 127. Yet applying Dykstra’s algorithm to the examplein Figure 121 does not improve rate of convergence, unfortunately, becausethe given point b and all the alternating projections already belong to thetranslated normal cone at the vertex of intersection.E.10.3.3speculationFrom these few examples we surmise, unique minimum-distance projection onblunt polyhedral cones having nonempty interior may be found by Dykstra’salgorithm in few iterations.
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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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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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Page 602 and 603: 602 APPENDIX E. PROJECTIONis a nono
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Page 606 and 607: 606 APPENDIX E. PROJECTIONq i q T i
Page 608 and 609: 608 APPENDIX E. PROJECTIONThe test
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Page 612 and 613: 612 APPENDIX E. PROJECTIONE.8 Range
Page 614 and 615: 614 APPENDIX E. PROJECTIONAs for su
Page 616 and 617: 616 APPENDIX E. PROJECTIONWith refe
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Page 622 and 623: 622 APPENDIX E. PROJECTIONThe foreg
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Page 626 and 627: 626 APPENDIX E. PROJECTIONE.10 Alte
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Page 630 and 631: 630 APPENDIX E. PROJECTIONa(a){y |
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Page 634 and 635: 634 APPENDIX E. PROJECTIONwhile, th
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Page 638 and 639: 638 APPENDIX E. PROJECTION10 0dist(
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Page 644 and 645: 644 APPENDIX E. PROJECTION
Page 646 and 647: 646 APPENDIX F. MATLAB PROGRAMSif n
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Page 650 and 651: 650 APPENDIX F. MATLAB PROGRAMSF.1.
Page 652 and 653: 652 APPENDIX F. MATLAB PROGRAMScoun
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Page 658 and 659: 658 APPENDIX F. MATLAB PROGRAMS% so
Page 660 and 661: 660 APPENDIX F. MATLAB PROGRAMS% tr
Page 664 and 665: 664 APPENDIX F. MATLAB PROGRAMSbrea
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Page 668 and 669: 668 APPENDIX F. MATLAB PROGRAMSF.7
Page 670 and 671: 670 APPENDIX F. MATLAB PROGRAMS
Page 672 and 673: 672 APPENDIX G. NOTATION AND A FEW
Page 688 and 689: 688 BIBLIOGRAPHY[7] Abdo Y. Alfakih
Page 690 and 691: 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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