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364 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX Columns 1 through 6 X = 0 -0.1983 -0.4584 0.1657 0.9399 0.7416 0 0.6863 0.2936 0.6239 -0.2936 0.3927 0 -0.4835 0.8146 -0.6448 0.0611 -0.4224 0 0.5059 0.2004 -0.4093 -0.1632 0.3427 Columns 7 through 12 -0.4815 -0.9399 -0.7416 0.1983 0.4584 -0.2832 0 0.2936 -0.3927 -0.6863 -0.2936 -0.6863 -0.8756 -0.0611 0.4224 0.4835 -0.8146 -0.3922 -0.0372 0.1632 -0.3427 -0.5059 -0.2004 -0.5431 Columns 13 through 18 0.2832 -0.2926 -0.6473 0.0943 0.3640 -0.3640 0.6863 0.9176 -0.6239 -0.2313 -0.0624 0.0624 0.3922 0.1698 -0.2309 -0.6533 -0.1613 0.1613 0.5431 -0.2088 0.3721 0.7147 -0.9152 0.9152 Columns 19 through 25 -0.0943 0.6473 -0.1657 0.2926 -0.5759 0.5759 0.4815 0.2313 0.6239 -0.6239 -0.9176 0.2313 -0.2313 0 0.6533 0.2309 0.6448 -0.1698 -0.2224 0.2224 0.8756 -0.7147 -0.3721 0.4093 0.2088 -0.7520 0.7520 0.0372 This particular optimal solution was found by solving a problem sequence in increasing number of spheres. Numerical problems begin to arise with matrices of this cardinality due to interior-point methods of solution. years ago. I believe Gauss knew it. Moreover, Korkine & Zolotarev proved in 1882 that D 5 is the densest lattice in five dimensions. So they proved that if a kissing arrangement in five dimensions can be extended to some lattice, then k(5)= 40. Of course, the conjecture in the general case also is: k(5)= 40. You would like to see coordinates? Easily. Let A= √ 2. Then p(1)=(A,A,0,0,0), p(2)=(−A,A,0,0,0), p(3)=(A, −A,0,0,0), ... p(40)=(0,0,0, −A, −A); i.e., we are considering points with coordinates that have two A and three 0 with any choice of signs and any ordering of the coordinates; the same coordinates-expression in dimensions 3 and 4. The first miracle happens in dimension 6. There are better packings than D 6 (Conjecture: k(6)=72). It’s a real miracle how dense the packing is in eight dimensions (E 8 =Korkine & Zolotarev packing that was discovered in 1880s) and especially in dimension 24, that is the so-called Leech lattice. Actually, people in coding theory have conjectures on the kissing numbers for dimensions up to 32 (or even greater?). However, sometimes they found better lower bounds. I know that Ericson & Zinoviev a few years ago discovered (by hand, no computer) in dimensions 13 and 14 better kissing arrangements than were known before. −Oleg Musin
5.4. EDM DEFINITION 365 By eliminating some equality constraints for this particular problem, matrix variable dimension can be reduced. From5.8.3 we have [ ] 0 −VNDV T N = 11 T T 1 − [0 I ] D (838) I 2 (which does not hold more generally) where identity matrix I ∈ S N−1 has one less dimension than EDM D . By defining an EDM principal submatrix [ ] ˆD = ∆ 0 T [0 I ] D ∈ S N−1 I h (839) we get a convex problem equivalent to (834) minimize − tr(W ˆD) ˆD∈S N−1 subject to ˆDij ≥ 1 , 1 ≤ i < j = 2... N −1 11 T − ˆD 1 2 ≽ 0 δ( ˆD) = 0 (840) Any feasible matrix 11 T − ˆD 1 2 belongs to an elliptope (5.9.1.0.1). This next example shows how finding the common point of intersection for three circles in a plane, a nonlinear problem, has convex expression. 5.4.2.2.4 Example. So & Ye trilateration in wireless sensor network. Given three known absolute point positions in R 2 (three anchors ˇx 2 , ˇx 3 , ˇx 4 ) and only one unknown point (one sensor x 1 ∈ R 2 ), the sensor’s absolute position is determined from its noiseless measured distance-square ď i1 to each of three anchors (Figure 2, Figure 97(a)). This trilateration can be expressed as a convex optimization problem in terms of list X = ∆ [x 1 ˇx 2 ˇx 3 ˇx 4 ]∈ R 2×4 and Gram matrix G∈ S 4 (807): minimize trG G∈S 4 , X∈R2×4 subject to tr(GΦ i1 ) = ďi1 , i = 2, 3, 4 tr ( ) Ge i e T i = ‖ˇx i ‖ 2 , i = 2, 3, 4 tr(G(e i e T j + e j e T i )/2) = ˇx T i ˇx j , 2 ≤ i < j = 3, 4 X(:, 2:4) = [ ˇx 2 ˇx 3 ˇx 4 ] [ ] I X X T ≽ 0 G (841)
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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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4.6. CARDINALITY AND RANK CONSTRAIN
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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
Page 733 and 734:
BIBLIOGRAPHY 733 [162] T. Herrmann,
Page 735 and 736:
BIBLIOGRAPHY 735 [191] Mark Kahrs a
Page 737 and 738:
BIBLIOGRAPHY 737 [220] K. V. Mardia
Page 739 and 740:
BIBLIOGRAPHY 739 [250] Pythagoras P
Page 741 and 742:
BIBLIOGRAPHY 741 [277] Anthony Man-
Page 743 and 744:
BIBLIOGRAPHY 743 [306] Michael W. T
Page 745 and 746:
[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,
Page 769 and 770:
INDEX 769 trilateration, 21, 42, 36
Page 772:
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v2009.01.01 - Convex Optimization

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