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538 APPENDIX A. LINEAR ALGEBRA A.1.1 Identities This δ notation is efficient and unambiguous as illustrated in the following examples where A ◦ B denotes Hadamard product [176] [134,1.1.4] of matrices of like size, ⊗ Kronecker product [140] (D.1.2.1), y a vector, X a matrix, e i the i th member of the standard basis for R n , S N h the symmetric hollow subspace, σ a vector of (nonincreasingly) ordered singular values, and λ(A) denotes a vector of nonincreasingly ordered eigenvalues of matrix A : 1. δ(A) = δ(A T ) 2. tr(A) = tr(A T ) = δ(A) T 1 = 〈I, A〉 3. δ(cA) = cδ(A), c∈R 4. tr(cA) = c tr(A) = c1 T λ(A), c∈R 5. vec(cA) = c vec(A), c∈R 6. δ(A + B) = δ(A) + δ(B) 7. tr(A + B) = tr(A) + tr(B) 8. vec(A + B) = vec(A) + vec(B) 9. π(δ(A)) = λ(I ◦A) where π is the presorting function. 10. tr(c √ A T A) = c tr √ A T A = c1 T σ(A), c∈R 11. δ(AB) = (A ◦ B T )1 = (B T ◦ A)1 12. δ(AB) T = 1 T (A T ◦ B) = 1 T (B ◦ A T ) ⎡ ⎤ u 1 v 1 13. δ(uv T ⎢ ⎥ ) = ⎣ . ⎦ = u ◦ v , u,v ∈ R N u N v N 14. tr(A T B) = tr(AB T ) = tr(BA T ) = tr(B T A) = 1 T (A ◦ B)1 = 1 T δ(AB T ) = δ(A T B) T 1 = δ(BA T ) T 1 = δ(B T A) T 1 15. D = [d ij ] ∈ S N h , H = [h ij ] ∈ S N h , V = I − 1 N 11T ∈ S N (conferB.4.2 no.20) N tr(−V (D ◦ H)V ) = tr(D T H) = 1 T (D ◦ H)1 = tr(11 T (D ◦ H)) = ∑ d ij h ij i,j
A.1. MAIN-DIAGONAL δ OPERATOR, λ , TRACE, VEC 539 16. tr(ΛA) = δ(Λ) T δ(A), δ 2 (Λ) ∆ = Λ ∈ S N 17. y T Bδ(A) = tr ( Bδ(A)y T) = tr ( δ(B T y)A ) = tr ( Aδ(B T y) ) = δ(A) T B T y = tr ( yδ(A) T B T) = tr ( A T δ(B T y) ) = tr ( δ(B T y)A T) 18. δ 2 (A T A) = ∑ i e i e T iA T Ae i e T i 19. δ ( δ(A)1 T) = δ ( 1δ(A) T) = δ(A) 20. δ(A1)1 = δ(A11 T ) = A1 , δ(y)1 = δ(y1 T ) = y 21. δ(I1) = δ(1) = I 22. δ(e i e T j 1) = δ(e i ) = e i e T i 23. For ζ =[ζ i ]∈ R k and x=[x i ]∈ R k , ∑ ζ i /x i = ζ T δ(x) −1 1 i 24. vec(A ◦ B) = vec(A) ◦ vec(B) = δ(vecA) vec(B) = vec(B) ◦ vec(A) = δ(vecB) vec(A) (35)(1664) 25. vec(AXB) = (B T ⊗ A) vec X ( not H ) 26. vec(BXA) = (A T ⊗ B) vec X 27. tr(AXBX T ) = vec(X) T vec(AXB) = vec(X) T (B T ⊗ A) vec X [140] = δ ( vec(X) vec(X) T (B T ⊗ A) ) T 1 28. tr(AX T BX) = vec(X) T vec(BXA) = vec(X) T (A T ⊗ B) vec X = δ ( vec(X) vec(X) T (A T ⊗ B) ) T 1
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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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Page 493 and 494: 6.10. POSTSCRIPT 493 When D is an E
Page 495 and 496: Chapter 7 Proximity problems In sum
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Page 499 and 500: 499 H S N h 0 EDM N K = S N h ∩ R
Page 501 and 502: 501 7.0.3 Problem approach Problems
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Page 535 and 536: 7.4. CONCLUSION 535 The rank constr
Page 537: Appendix A Linear algebra A.1 Main-
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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 577 and 578: Appendix B Simple matrices Mathemat
Page 579 and 580: B.1. RANK-ONE MATRIX (DYAD) 579 R(v
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Page 583 and 584: B.2. DOUBLET 583 R([u v ]) R(Π)= R
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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
Page 747 and 748:
Index 0-norm, 203, 261, 294, 296, 2
Page 749 and 750:
INDEX 749 product, 43, 92, 147, 254
Page 751 and 752:
INDEX 751 coordinates, 140, 170, 17
Page 753 and 754:
INDEX 753 affine dimension, 485 fea
Page 755 and 756:
INDEX 755 affine, 209 nonlinear, 19
Page 757 and 758:
INDEX 757 of point, 37 ray, 90 rela
Page 759 and 760:
INDEX 759 normal, 47, 548, 563 norm
Page 761 and 762:
INDEX 761 strictly, 515, 520 functi
Page 763 and 764:
INDEX 763 vector, 45, 241, 248, 325
Page 765 and 766:
INDEX 765 convex envelope, see conv
Page 767 and 768:
INDEX 767 cone, 418, 420, 507 dual,
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INDEX 769 trilateration, 21, 42, 36
Page 772:
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v2009.01.01 - Convex Optimization

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