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60 CHAPTER 2. CONVEX GEOMETRY svec ∂ S 2 + I [ α β β γ ] γ α √ 2β Figure 18: Two Fantopes. Circle, (radius 1 √ 2 ) shown here on boundary of positive semidefinite cone S 2 + in isometrically isomorphic R 3 from Figure 37, comprises boundary of a Fantope (82) in this dimension (k = 1, N = 2). Lone point illustrated is identity matrix I and is that Fantope corresponding to k = 2, N = 2. (View is from inside PSD cone looking toward origin.)
2.3. HULLS 61 In case k = N , the Fantope is identity matrix I . More generally, the set {UU T | U ∈ R N×k , U T U = I} (84) comprises the extreme points (2.6.0.0.1) of its convex hull. By (1367), each and every extreme point UU T has only k nonzero eigenvalues λ and they all equal 1 ; id est, λ(UU T ) 1:k = λ(U T U) = 1. So the Frobenius norm of each and every extreme point equals the same constant ‖UU T ‖ 2 F = k (85) Each extreme point simultaneously lies on the boundary of the positive semidefinite cone (when k < N ,2.9) and on the boundary of a hypersphere of dimension k(N − √k(1 k + 1) and radius − k ) centered at k I along 2 2 N N the ray (base 0) through the identity matrix I in isomorphic vector space R N(N+1)/2 (2.2.2.1). Figure 18 illustrates extreme points (84) comprising the boundary of a Fantope, the boundary of a disc corresponding to k = 1, N = 2 ; but that circumscription does not hold in higher dimension. (2.9.2.5) 2.3.2.0.2 Example. <strong>Convex</strong> hull of rank-1 matrices. From (83), in Example 2.3.2.0.1, we learn that the convex hull of normalized symmetric rank-1 matrices is a slice of the positive semidefinite cone. In2.9.2.4 we find the convex hull of all symmetric rank-1 matrices to be the entire positive semidefinite cone. In the present example we abandon symmetry; instead posing, what is the convex hull of bounded nonsymmetric rank-1 matrices: conv{uv T | ‖uv T ‖ ≤ 1, u∈ R m , v ∈ R n } = {X ∈ R m×n | ∑ i σ(X) i ≤ 1} (86) where σ(X) is a vector of singular values. Since ‖uv T ‖= ‖u‖‖v‖ (1496), norm of each vector constituting a dyad uv T in the hull is effectively bounded above by 1. Proof. (⇐) Suppose ∑ σ(X) i ≤ 1. As inA.6, define a singular value decomposition: X = UΣV T where U = [u 1 ... u min{m,n} ]∈ R m×min{m,n} , V = [v 1 ... v min{m,n} ]∈ R n×min{m,n} , and whose sum of singular values is ∑ σ(X)i = tr Σ = κ ≤ 1. Then we may write X = ∑ σ i κ √ κui √ κv T i which is a convex combination of dyads each of whose norm does not exceed 1. (Srebro)
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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
Page 9 and 10: Convex Optimization & Euclidean Dis
Page 11 and 12: CONVEX OPTIMIZATION & EUCLIDEAN DIS
Page 13 and 14: List of Figures 1 Overview 19 1 Ori
Page 15 and 16: LIST OF FIGURES 15 3 Geometry of co
Page 17 and 18: LIST OF FIGURES 17 126 Decomposing
Page 19 and 20: Chapter 1 Overview Convex Optimizat
Page 21 and 22: ˇx 4 ˇx 3 ˇx 2 Figure 2: Applica
Page 23 and 24: 23 Figure 4: This coarsely discreti
Page 25 and 26: ases (biorthogonal expansion). We e
Page 27 and 28: 27 Figure 7: These bees construct a
Page 29 and 30: that establish its membership to th
Page 31 and 32: 31 appendices Provided so as to be
Page 33 and 34: Chapter 2 Convex geometry Convexity
Page 35 and 36: 2.1. CONVEX SET 35 2.1.2 linear ind
Page 37 and 38: 2.1. CONVEX SET 37 2.1.6 empty set
Page 39 and 40: 2.1. CONVEX SET 39 2.1.7.1 Line int
Page 41 and 42: 2.1. CONVEX SET 41 (a) R 2 (b) R 3
Page 43 and 44: 2.1. CONVEX SET 43 This theorem in
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Page 55 and 56: 2.3. HULLS 55 Figure 16: Convex hul
Page 57 and 58: 2.3. HULLS 57 The affine hull of tw
Page 59: 2.3. HULLS 59 2.3.2 Convex hull The
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Page 67 and 68: 2.4. HALFSPACE, HYPERPLANE 67 H + a
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Page 71 and 72: 2.4. HALFSPACE, HYPERPLANE 71 Recal
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Page 75 and 76: 2.4. HALFSPACE, HYPERPLANE 75 2.4.2
Page 77 and 78: 2.4. HALFSPACE, HYPERPLANE 77 (conf
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Page 81 and 82: 2.5. SUBSPACE REPRESENTATIONS 81 If
Page 83 and 84: 2.6. EXTREME, EXPOSED 83 2.6 Extrem
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Page 87 and 88: 2.7. CONES 87 2.6.1.3.1 Definition.
Page 89 and 90: 2.7. CONES 89 0 Figure 30: Boundary
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Page 95 and 96: 2.7. CONES 95 A pointed closed conv
Page 97 and 98: 2.8. CONE BOUNDARY 97 So the ray th
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Page 101 and 102: 2.8. CONE BOUNDARY 101 2.8.2 Expose
Page 103 and 104: 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
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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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Convex Optimization & Euclidean Dis
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