v2009.01.01 - Convex Optimization

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430 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX λ(−V T N OV N) j 900 800 700 600 500 400 300 200 100 0 1 2 3 4 5 6 7 8 9 10 j Figure 111: Largest ten eigenvalues, of −V T N OV N for map of USA, sorted by nonincreasing value. Any process of reconstruction that leaves comparative distance information intact is called ordinal multidimensional scaling or isotonic reconstruction. Beyond rotation, reflection, and translation error, (5.5) list reconstruction by isotonic reconstruction is subject to error in absolute scale (dilation) and distance ratio. Yet Borg & Groenen argue: [46,2.2] reconstruction from complete comparative distance information for a large number of points is as highly constrained as reconstruction from an EDM; the larger the number, the better. 5.13.2.1 Isotonic map of the USA To test Borg & Groenen’s conjecture, suppose we make a complete sort-index matrix O ∈ S N h ∩ R N×N + for the map of the USA and then substitute O in place of EDM D in the reconstruction process of5.12. Whereas EDM D returned only three significant eigenvalues (1045), the sort-index matrix O is generally not an EDM (certainly not an EDM with corresponding affine dimension 3) so returns many more. The eigenvalues, calculated with absolute numerical error approximately 5E-7, are plotted in Figure 111: (In the code on Wıκımization, matrix O is normalized by (N(N −1)/2) 2 .) λ(−V T N OV N ) = [880.1 463.9 186.1 46.20 17.12 9.625 8.257 1.701 0.7128 0.6460 · · · ] T (1050)
5.13. RECONSTRUCTION EXAMPLES 431 The extra eigenvalues indicate that affine dimension corresponding to an EDM near O is likely to exceed 3. To realize the map, we must simultaneously reduce that dimensionality and find an EDM D closest to O in some sense (a problem explored more in7) while maintaining the known comparative distance relationship; e.g., given permutation matrix Π expressing the known sorting action on the entries d of unknown D ∈ S N h , (66) d ∆ = 1 √ 2 dvec D = ⎡ ⎢ ⎣ ⎤ d 12 d 13 d 23 d 14 d 24 d 34 ⎥ ⎦ . d N−1,N ∈ R N(N−1)/2 (1051) we can make the sort-index matrix O input to the optimization problem minimize ‖−VN T(D − O)V N ‖ F D subject to rankVN TDV N ≤ 3 (1052) Πd ∈ K M+ D ∈ EDM N that finds the EDM D (corresponding to affine dimension not exceeding 3 in isomorphic dvec EDM N ∩ Π T K M+ ) closest to O in the sense of Schoenberg (817). Analytical solution to this problem, ignoring the sort constraint Πd ∈ K M+ , is known [306]: we get the convex optimization [sic] (7.1) minimize ‖−VN T(D − O)V N ‖ F D subject to rankVN TDV N ≤ 3 (1053) D ∈ EDM N Only the three largest nonnegative eigenvalues in (1050) need be retained to make list (1037); the rest are discarded. The reconstruction from EDM D found in this manner is plotted in Figure 110(e)(f). Matlab code is on Wıκımization. From these plots it becomes obvious that inclusion of the sort constraint is necessary for isotonic reconstruction.
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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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Page 381 and 382: 5.4. EDM DEFINITION 381 because (A.
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Page 385 and 386: 5.5. INVARIANCE 385 x 2 x 2 x 3 x 1
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Page 393 and 394: 5.7. EMBEDDING IN AFFINE HULL 393 5
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Page 399 and 400: 5.8. EUCLIDEAN METRIC VERSUS MATRIX
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Page 413 and 414: 5.10. EDM-ENTRY COMPOSITION 413 of
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Page 419 and 420: 5.11. EDM INDEFINITENESS 419 we hav
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Page 423 and 424: 5.11. EDM INDEFINITENESS 423 where
Page 425 and 426: 5.12. LIST RECONSTRUCTION 425 where
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Page 435 and 436: 5.14. FIFTH PROPERTY OF EUCLIDEAN M
Page 445 and 446: Chapter 6 Cone of distance matrices
Page 447 and 448: 6.1. DEFINING EDM CONE 447 6.1 Defi
Page 449 and 450: 6.2. POLYHEDRAL BOUNDS 449 This con
Page 451 and 452: 6.3. √ EDM CONE IS NOT CONVEX 451
Page 453 and 454: 6.4. A GEOMETRY OF COMPLETION 453 (
Page 455 and 456: 6.4. A GEOMETRY OF COMPLETION 455 (
Page 457 and 458: 6.4. A GEOMETRY OF COMPLETION 457 F
Page 459 and 460: 6.5. EDM DEFINITION IN 11 T 459 by
Page 461 and 462: 6.5. EDM DEFINITION IN 11 T 461 6.5
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Page 465 and 466: 6.5. EDM DEFINITION IN 11 T 465 6.5
Page 467 and 468: 6.6. CORRESPONDENCE TO PSD CONE S N
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Page 477 and 478: 6.8. DUAL EDM CONE 477 When the Fin
Page 479 and 480: 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
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Appendix A Linear algebra A.1 Main-
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A.1. MAIN-DIAGONAL δ OPERATOR, λ
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A.1. MAIN-DIAGONAL δ OPERATOR, λ
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A.2. SEMIDEFINITENESS: DOMAIN OF TE
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A.3. PROPER STATEMENTS 545 (AB) T
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A.3. PROPER STATEMENTS 547 A.3.1 Se
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A.3. PROPER STATEMENTS 549 For A di
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A.3. PROPER STATEMENTS 551 Diagonal
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A.3. PROPER STATEMENTS 553 For A,B
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A.3. PROPER STATEMENTS 555 A.3.1.0.
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A.4. SCHUR COMPLEMENT 557 A.4 Schur
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A.4. SCHUR COMPLEMENT 559 A.4.0.0.2
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A.5. EIGEN DECOMPOSITION 561 When B
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A.5. EIGEN DECOMPOSITION 563 dim N(
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A.6. SINGULAR VALUE DECOMPOSITION,
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A.6. SINGULAR VALUE DECOMPOSITION,
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A.7. ZEROS 569 Given symmetric matr
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A.7. ZEROS 571 (TRANSPOSE.) Likewis
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A.7. ZEROS 573 For X,A∈ S M + [31
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A.7. ZEROS 575 A.7.5.0.1 Propositio
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Appendix B Simple matrices Mathemat
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B.1. RANK-ONE MATRIX (DYAD) 579 R(v
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B.1. RANK-ONE MATRIX (DYAD) 581 ran
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B.2. DOUBLET 583 R([u v ]) R(Π)= R
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B.3. ELEMENTARY MATRIX 585 If λ
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B.4. AUXILIARY V -MATRICES 587 the
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B.4. AUXILIARY V -MATRICES 589 18.
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B.5. ORTHOGONAL MATRIX 591 B.5 Orth
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B.5. ORTHOGONAL MATRIX 593 Figure 1
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Appendix C Some analytical optimal
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C.2. TRACE, SINGULAR AND EIGEN VALU
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C.3. ORTHOGONAL PROCRUSTES PROBLEM
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C.4. TWO-SIDED ORTHOGONAL PROCRUSTE
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C.4. TWO-SIDED ORTHOGONAL PROCRUSTE
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Appendix D Matrix calculus From too
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D.1. DIRECTIONAL DERIVATIVE, TAYLOR
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D.2. TABLES OF GRADIENTS AND DERIVA
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Appendix 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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