v2009.01.01 - Convex Optimization

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588 APPENDIX B. SIMPLE MATRICES B.4.2 Schoenberg auxiliary matrix V N 1. V N = 1 √ 2 [ −1 T I 2. V T N 1 = 0 3. I − e 1 1 T = [ 0 √ 2V N ] 4. [ 0 √ 2V N ] VN = V N 5. [ 0 √ 2V N ] V = V ] ∈ R N×N−1 6. V [ 0 √ 2V N ] = [ 0 √ 2VN ] 7. [ 0 √ 2V N ] [ 0 √ 2VN ] = [ 0 √ 2VN ] 8. [ 0 √ 2V N ] † = [ 0 0 T 0 I ] V 9. [ 0 √ 2V N ] † V = [ 0 √ 2VN ] † 10. [ 0 √ ] [ √ ] † 2V N 0 2VN = V 11. [ 0 √ [ ] ] † [ √ ] 0 0 T 2V N 0 2VN = 0 I 12. [ 0 √ ] [ ] 0 0 2V T N = [ 0 √ ] 2V 0 I N [ ] [ ] 0 0 T [0 √ ] 0 0 T 13. 2VN = 0 I 0 I 14. [V N 1 √ 2 1 ] −1 = [ ] V † N √ 2 N 1T 15. V † N = √ 2 [ − 1 N 1 I − 1 N 11T] ∈ R N−1×N , ( I − 1 N 11T ∈ S N−1) 16. V † N 1 = 0 17. V † N V N = I
B.4. AUXILIARY V -MATRICES 589 18. V T = V = V N V † N = I − 1 N 11T ∈ S N ( 19. −V † N (11T − I)V N = I , 11 T − I ∈ EDM N) 20. D = [d ij ] ∈ S N h (823) tr(−V DV ) = tr(−V D) = tr(−V † N DV N) = 1 N 1T D 1 = 1 N tr(11T D) = 1 N Any elementary matrix E ∈ S N of the particular form ∑ d ij i,j E = k 1 I − k 2 11 T (1539) where k 1 , k 2 ∈ R , B.7 will make tr(−ED) proportional to ∑ d ij . 21. D = [d ij ] ∈ S N tr(−V DV ) = 1 N 22. D = [d ij ] ∈ S N h ∑ i,j i≠j d ij − N−1 N ∑ d ii = 1 T D1 1 − trD N i tr(−V T N DV N) = ∑ j d 1j 23. For Y ∈ S N B.4.3 V (Y − δ(Y 1))V = Y − δ(Y 1) The skinny matrix ⎡ V ∆ W = ⎢ ⎣ Orthonormal auxiliary matrix V W −1 √ N 1 + −1 N+ √ N −1 N+ √ N . −1 N+ √ N √−1 −1 N · · · √ N −1 N+ √ N ... ... −1 N+ √ N · · · −1 N+ √ N ... −1 N+ √ N ... · · · 1 + −1 N+ √ N . ⎤ ∈ R N×N−1 (1540) ⎥ ⎦ B.7 If k 1 is 1−ρ while k 2 equals −ρ∈R , then all eigenvalues of E for −1/(N −1) < ρ < 1 are guaranteed positive and therefore E is guaranteed positive definite. [261]
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DATTORRO CONVEX OPTIMIZATION & EUCL
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Convex Optimization & Euclidean Dis
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for Jennie Columba ♦ Antonio ♦
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Prelude The constant demands of my
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CONVEX OPTIMIZATION & EUCLIDEAN DIS
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List of Figures 1 Overview 19 1 Ori
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LIST OF FIGURES 15 3 Geometry of co
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LIST OF FIGURES 17 126 Decomposing
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Chapter 1 Overview Convex Optimizat
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ˇx 4 ˇx 3 ˇx 2 Figure 2: Applica
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23 Figure 4: This coarsely discreti
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ases (biorthogonal expansion). We e
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27 Figure 7: These bees construct a
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that establish its membership to th
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31 appendices Provided so as to be
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Chapter 2 Convex geometry Convexity
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2.1. CONVEX SET 35 2.1.2 linear ind
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2.1. CONVEX SET 37 2.1.6 empty set
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2.1. CONVEX SET 39 2.1.7.1 Line int
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2.1. CONVEX SET 41 (a) R 2 (b) R 3
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2.1. CONVEX SET 43 This theorem in
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2.2. VECTORIZED-MATRIX INNER PRODUC
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2.3. HULLS 55 Figure 16: Convex hul
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2.3. HULLS 57 The affine hull of tw
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2.3. HULLS 59 2.3.2 Convex hull The
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2.3. HULLS 61 In case k = N , the F
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2.3. HULLS 63 2.3.2.0.3 Exercise. C
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2.3. HULLS 65 Figure 20: A simplici
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2.4. HALFSPACE, HYPERPLANE 67 H + a
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2.4. HALFSPACE, HYPERPLANE 69 1 1
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2.4. HALFSPACE, HYPERPLANE 71 Recal
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2.4. HALFSPACE, HYPERPLANE 73 C H
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2.4. HALFSPACE, HYPERPLANE 75 2.4.2
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2.4. HALFSPACE, HYPERPLANE 77 (conf
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2.5. SUBSPACE REPRESENTATIONS 79 Ra
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2.5. SUBSPACE REPRESENTATIONS 81 If
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2.6. EXTREME, EXPOSED 83 2.6 Extrem
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2.6. EXTREME, EXPOSED 85 A B C D Fi
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2.7. CONES 87 2.6.1.3.1 Definition.
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2.7. CONES 89 0 Figure 30: Boundary
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2.7. CONES 91 2.7.2 Convex cone We
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2.7. CONES 93 Then a pointed closed
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2.7. CONES 95 A pointed closed conv
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2.8. CONE BOUNDARY 97 So the ray th
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2.8. CONE BOUNDARY 99 2.8.1.1 extre
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2.8. CONE BOUNDARY 101 2.8.2 Expose
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2.8. CONE BOUNDARY 103 From Theorem
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2.9. POSITIVE SEMIDEFINITE (PSD) CO
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2.10. CONIC INDEPENDENCE (C.I.) 127
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2.12. CONVEX POLYHEDRA 133 It follo
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2.12. CONVEX POLYHEDRA 135 Coeffici
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2.12. CONVEX POLYHEDRA 137 2.12.3 U
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2.12. CONVEX POLYHEDRA 139
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2.13. DUAL CONE & GENERALIZED INEQU
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Chapter 3 Geometry of convex functi
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3.1. CONVEX FUNCTION 197 f 1 (x) f
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3.1. CONVEX FUNCTION 199 3.1.3 norm
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3.1. CONVEX FUNCTION 201 A B 1 Figu
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3.1. CONVEX FUNCTION 203 k/m 1 0.9
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k∑ i=1 3.1. CONVEX FUNCTION 205 S
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3.1. CONVEX FUNCTION 207 rather x >
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3.1. CONVEX FUNCTION 209 rather ] x
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3.1. CONVEX FUNCTION 211 3.1.6.0.2
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3.1. CONVEX FUNCTION 213 q(x) f(x)
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3.1. CONVEX FUNCTION 215 3.1.7.0.2
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3.1. CONVEX FUNCTION 217 We learned
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3.1. CONVEX FUNCTION 219 Since opti
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3.1. CONVEX FUNCTION 221 2 1.5 1 0.
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3.1. CONVEX FUNCTION 223 Setting th
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3.1. CONVEX FUNCTION 225 Similarly,
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3.1. CONVEX FUNCTION 227 For vector
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3.1. CONVEX FUNCTION 229 This means
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3.1. CONVEX FUNCTION 231 f(Y ) −
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3.2. MATRIX-VALUED CONVEX FUNCTION
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3.3. QUASICONVEX 239 exponential al
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3.3. QUASICONVEX 241 Unlike convex
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3.4. SALIENT PROPERTIES 243 6. (af
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Chapter 4 Semidefinite programming
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4.1. CONIC PROBLEM 247 where K is a
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4.1. CONIC PROBLEM 249 4.1.1.2 Redu
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4.1. CONIC PROBLEM 251 In any SDP f
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4.1. CONIC PROBLEM 253 Proposition
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4.2. FRAMEWORK 255 sets are closed
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4.2. FRAMEWORK 257 4.2.1.1.3 Exampl
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4.2. FRAMEWORK 259 4.2.2 Duals The
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4.2. FRAMEWORK 261 When equality is
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4.2. FRAMEWORK 263 The pseudoinvers
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4.2. FRAMEWORK 265 For the data giv
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4.2. FRAMEWORK 267 minimizes an aff
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4.3. RANK REDUCTION 269 whose rank
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4.3. RANK REDUCTION 271 and where m
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4.3. RANK REDUCTION 273 4.3.3 Optim
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4.3. RANK REDUCTION 275 Initialize:
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4.4. RANK-CONSTRAINED SEMIDEFINITE
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4.5. CONSTRAINING CARDINALITY 295 m
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4.5. CONSTRAINING CARDINALITY 297 m
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4.5. CONSTRAINING CARDINALITY 299 a
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4.5. CONSTRAINING CARDINALITY 301 f
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4.5. CONSTRAINING CARDINALITY 303 n
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4.5. CONSTRAINING CARDINALITY 305 W
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4.5. CONSTRAINING CARDINALITY 307 t
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4.6. CARDINALITY AND RANK CONSTRAIN
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4.7. CONVEX ITERATION RANK-1 341 fi
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4.7. CONVEX ITERATION RANK-1 343 Gi
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Chapter 5 Euclidean Distance Matrix
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5.2. FIRST METRIC PROPERTIES 347 co
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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 353 The collect
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5.4. EDM DEFINITION 355 5.4.2 Gram-
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5.4. EDM DEFINITION 357 D ∈ EDM N
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5.4. EDM DEFINITION 359 5.4.2.2.1 E
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5.4. EDM DEFINITION 361 ten affine
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5.4. EDM DEFINITION 363 spheres: Th
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5.4. EDM DEFINITION 365 By eliminat
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5.4. EDM DEFINITION 367 where Φ ij
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5.4. EDM DEFINITION 369 5.4.2.2.6 D
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5.4. EDM DEFINITION 371 10 5 ˇx 4
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5.4. EDM DEFINITION 373 corrected b
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5.4. EDM DEFINITION 375 by translat
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5.4. EDM DEFINITION 377 Crippen & H
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5.4. EDM DEFINITION 379 where ([√
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5.4. EDM DEFINITION 381 because (A.
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5.5. INVARIANCE 383 5.5.1.0.1 Examp
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5.5. INVARIANCE 385 x 2 x 2 x 3 x 1
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5.6. INJECTIVITY OF D & UNIQUE RECO
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5.7. EMBEDDING IN AFFINE HULL 393 5
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5.7. EMBEDDING IN AFFINE HULL 395 F
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5.7. EMBEDDING IN AFFINE HULL 397 5
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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 413 of
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5.10. EDM-ENTRY COMPOSITION 415 The
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5.11. EDM INDEFINITENESS 417 5.11.1
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5.11. EDM INDEFINITENESS 419 we hav
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5.11. EDM INDEFINITENESS 421 So bec
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5.11. EDM INDEFINITENESS 423 where
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5.12. LIST RECONSTRUCTION 425 where
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5.12. LIST RECONSTRUCTION 427 (a) (
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5.13. RECONSTRUCTION EXAMPLES 429 D
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5.13. RECONSTRUCTION EXAMPLES 431 T
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5.13. RECONSTRUCTION EXAMPLES 433 w
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5.14. FIFTH PROPERTY OF EUCLIDEAN M
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Chapter 6 Cone of distance matrices
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6.1. DEFINING EDM CONE 447 6.1 Defi
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6.2. POLYHEDRAL BOUNDS 449 This con
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6.3. √ EDM CONE IS NOT CONVEX 451
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6.4. A GEOMETRY OF COMPLETION 453 (
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6.4. A GEOMETRY OF COMPLETION 455 (
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6.4. A GEOMETRY OF COMPLETION 457 F
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6.5. EDM DEFINITION IN 11 T 459 by
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6.5. EDM DEFINITION IN 11 T 461 6.5
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6.5. EDM DEFINITION IN 11 T 463 1 0
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6.5. EDM DEFINITION IN 11 T 465 6.5
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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 477 When the Fin
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6.8. DUAL EDM CONE 479 Proof. First
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6.8. DUAL EDM CONE 481 EDM 2 = S 2
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6.8. DUAL EDM CONE 483 whose veraci
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6.8. DUAL EDM CONE 485 6.8.1.3.1 Ex
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6.8. DUAL EDM CONE 487 has dual aff
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6.8. DUAL EDM CONE 489 6.8.1.7 Scho
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6.9. THEOREM OF THE ALTERNATIVE 491
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6.10. POSTSCRIPT 493 When D is an E
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Chapter 7 Proximity problems In sum
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497 project on the subspace, then p
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499 H S N h 0 EDM N K = S N h ∩ R
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501 7.0.3 Problem approach Problems
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7.1. FIRST PREVALENT PROBLEM: 503 f
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7.1. FIRST PREVALENT PROBLEM: 505 7
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7.1. FIRST PREVALENT PROBLEM: 507 d
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7.1. FIRST PREVALENT PROBLEM: 509 7
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7.1. FIRST PREVALENT PROBLEM: 511 w
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7.1. FIRST PREVALENT PROBLEM: 513 T
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7.2. SECOND PREVALENT PROBLEM: 515
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7.3. THIRD PREVALENT PROBLEM: 525 g
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7.3. THIRD PREVALENT PROBLEM: 527 w
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7.3. THIRD PREVALENT PROBLEM: 529 7
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7.3. THIRD PREVALENT PROBLEM: 531 7
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7.3. THIRD PREVALENT PROBLEM: 533 O
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7.4. CONCLUSION 535 The rank constr
Page 537 and 538: Appendix A Linear algebra A.1 Main-
Page 539 and 540: A.1. MAIN-DIAGONAL δ OPERATOR, λ
Page 541 and 542: A.1. MAIN-DIAGONAL δ OPERATOR, λ
Page 543 and 544: A.2. SEMIDEFINITENESS: DOMAIN OF TE
Page 545 and 546: A.3. PROPER STATEMENTS 545 (AB) T
Page 547 and 548: A.3. PROPER STATEMENTS 547 A.3.1 Se
Page 549 and 550: A.3. PROPER STATEMENTS 549 For A di
Page 551 and 552: A.3. PROPER STATEMENTS 551 Diagonal
Page 553 and 554: A.3. PROPER STATEMENTS 553 For A,B
Page 555 and 556: A.3. PROPER STATEMENTS 555 A.3.1.0.
Page 557 and 558: A.4. SCHUR COMPLEMENT 557 A.4 Schur
Page 559 and 560: A.4. SCHUR COMPLEMENT 559 A.4.0.0.2
Page 561 and 562: A.5. EIGEN DECOMPOSITION 561 When B
Page 563 and 564: A.5. EIGEN DECOMPOSITION 563 dim N(
Page 565 and 566: A.6. SINGULAR VALUE DECOMPOSITION,
Page 567 and 568: A.6. SINGULAR VALUE DECOMPOSITION,
Page 569 and 570: A.7. ZEROS 569 Given symmetric matr
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Page 573 and 574: A.7. ZEROS 573 For X,A∈ S M + [31
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Page 583 and 584: B.2. DOUBLET 583 R([u v ]) R(Π)= R
Page 585 and 586: B.3. ELEMENTARY MATRIX 585 If λ
Page 587: B.4. AUXILIARY V -MATRICES 587 the
Page 591 and 592: B.5. ORTHOGONAL MATRIX 591 B.5 Orth
Page 593 and 594: B.5. ORTHOGONAL MATRIX 593 Figure 1
Page 595 and 596: Appendix C Some analytical optimal
Page 597 and 598: C.2. TRACE, SINGULAR AND EIGEN VALU
Page 603 and 604: C.3. ORTHOGONAL PROCRUSTES PROBLEM
Page 605 and 606: C.4. TWO-SIDED ORTHOGONAL PROCRUSTE
Page 607 and 608: C.4. TWO-SIDED ORTHOGONAL PROCRUSTE
Page 609 and 610: Appendix D Matrix calculus From too
Page 611 and 612: D.1. DIRECTIONAL DERIVATIVE, TAYLOR
Page 631 and 632: D.2. TABLES OF GRADIENTS AND DERIVA
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Appendix E Projection For any A∈
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641 U T = U † for orthonormal (in
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E.1. IDEMPOTENT MATRICES 643 where
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E.1. IDEMPOTENT MATRICES 645 order,
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E.1. IDEMPOTENT MATRICES 647 When t
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E.3. SYMMETRIC IDEMPOTENT MATRICES
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E.5. PROJECTION EXAMPLES 655 E.4.0.
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E.5. PROJECTION EXAMPLES 657 a ∗
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E.5. PROJECTION EXAMPLES 659 E.5.0.
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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.8. RANGE/ROWSPACE INTERPRETATION
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E.9. PROJECTION ON CONVEX SET 675 A
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E.9. PROJECTION ON CONVEX SET 677 W
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E.9. PROJECTION ON CONVEX SET 679 P
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E.9. PROJECTION ON CONVEX SET 681 E
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E.9. PROJECTION ON CONVEX SET 683 T
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E.9. PROJECTION ON CONVEX SET 685
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E.10. ALTERNATING PROJECTION 687 E.
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E.10. ALTERNATING PROJECTION 689 b
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E.10. ALTERNATING PROJECTION 691 a
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E.10. ALTERNATING PROJECTION 693 (a
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E.10. ALTERNATING PROJECTION 695 wh
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E.10. ALTERNATING PROJECTION 699 10
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E.10. ALTERNATING PROJECTION 703 E
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Appendix F Notation and a few defin
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707 a.i. c.i. l.i. w.r.t affinely i
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709 is or ← → t → 0 + as in
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711 ∑ π(γ) Ξ Π ∏ ψ(Z) D D
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713 R m×n Euclidean vector space o
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715 H − H + ∂H ∂H ∂H −
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717 O O sort-index matrix order of
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(x,y) angle between vectors x and y
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Bibliography [1] Suliman Al-Homidan
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BIBLIOGRAPHY 723 [24] Alexander I.
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BIBLIOGRAPHY 725 [52] Stephen Boyd,
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BIBLIOGRAPHY 727 [78] Frank Critchl
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BIBLIOGRAPHY 729 [105] Richard L. D
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BIBLIOGRAPHY 731 [132] Michel X. Go
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BIBLIOGRAPHY 733 [162] T. Herrmann,
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BIBLIOGRAPHY 735 [191] Mark Kahrs a
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BIBLIOGRAPHY 737 [220] K. V. Mardia
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BIBLIOGRAPHY 739 [250] Pythagoras P
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BIBLIOGRAPHY 741 [277] Anthony Man-
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BIBLIOGRAPHY 743 [306] Michael W. T
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[333] Margaret H. Wright. The inter
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Index 0-norm, 203, 261, 294, 296, 2
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INDEX 749 product, 43, 92, 147, 254
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INDEX 751 coordinates, 140, 170, 17
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INDEX 753 affine dimension, 485 fea
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INDEX 755 affine, 209 nonlinear, 19
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INDEX 757 of point, 37 ray, 90 rela
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INDEX 759 normal, 47, 548, 563 norm
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INDEX 761 strictly, 515, 520 functi
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INDEX 763 vector, 45, 241, 248, 325
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INDEX 765 convex envelope, see conv
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INDEX 767 cone, 418, 420, 507 dual,
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INDEX 769 trilateration, 21, 42, 36
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