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308 CHAPTER 5. EUCLIDEAN DISTANCE MATRIXColumns 1 through 6X = 0 -0.1983 -0.4584 0.1657 0.9399 0.74160 0.6863 0.2936 0.6239 -0.2936 0.39270 -0.4835 0.8146 -0.6448 0.0611 -0.42240 0.5059 0.2004 -0.4093 -0.1632 0.3427Columns 7 through 12-0.4815 -0.9399 -0.7416 0.1983 0.4584 -0.28320 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 -0.5431Columns 13 through 180.2832 -0.2926 -0.6473 0.0943 0.3640 -0.36400.6863 0.9176 -0.6239 -0.2313 -0.0624 0.06240.3922 0.1698 -0.2309 -0.6533 -0.1613 0.16130.5431 -0.2088 0.3721 0.7147 -0.9152 0.9152Columns 19 through 25-0.0943 0.6473 -0.1657 0.2926 -0.5759 0.5759 0.48150.2313 0.6239 -0.6239 -0.9176 0.2313 -0.2313 00.6533 0.2309 0.6448 -0.1698 -0.2224 0.2224 0.8756-0.7147 -0.3721 0.4093 0.2088 -0.7520 0.7520 0.0372This particular optimal solution was found by solving a problem sequencein increasing number of spheres. Numerical problems begin to arise withmatrices 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 5is the densest lattice in five dimensions. So they proved that if a kissing arrangement infive dimensions can be extended to some lattice, then k(5)= 40. Of course, the conjecturein 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 twoA and three 0 with any choice of signs and any ordering of the coordinates; the samecoordinates-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 indimension 24, that is the so-called Leech lattice.Actually, people in coding theory have conjectures on the kissing numbers for dimensionsup to 32 (or even greater?). However, sometimes they found better lower bounds. I knowthat Ericson & Zinoviev a few years ago discovered (by hand, no computer) in dimensions13 and 14 better kissing arrangements than were known before. −Oleg Musin
5.4. EDM DEFINITION 309By 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(745)I 2(which does not generally hold) where identity matrix I ∈ S N−1 has one lessdimension than EDM D . By defining an EDM principal submatrix[ ]ˆD = ∆ 0T[0 I ] D ∈ S N−1Ih(746)we get a convex problem equivalent to (741)minimize − tr(W ˆD)ˆD∈S N−1subject to ˆDij ≥ 1 , 1 ≤ i < j = 2... N −111 T − ˆD 1 2 ≽ 0δ( ˆD) = 0(747)Any feasible matrix 11 T − ˆD 1 2belongs to an elliptope (5.9.1.0.1). This next example shows how finding the common point of intersectionfor 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 absoluteposition is determined from its noiseless measured distance-square ď i1to each of three anchors (Figure 2, Figure 78(a)). This trilaterationcan be expressed as a convex optimization problem in terms of listX = ∆ [x 1 ˇx 2 ˇx 3 ˇx 4 ]∈ R 2×4 and Gram matrix G∈ S 4 (714):minimize trGG∈S 4 , X∈R2×4 subject to tr(GΦ i1 ) = ďi1 , i = 2, 3, 4tr ( )Ge i e T i = ‖ˇx i ‖ 2 , i = 2, 3, 4tr(G(e i e T j + e j e T i )/2) = ˇx T i ˇx j , 2 ≤ i < j = 3, 4X(:, 2:4) = [ ˇx 2 ˇx 3 ˇx 4 ][ ] I XX T≽ 0G(748)
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DATTORROCONVEXOPTIMIZATION&EUCLIDEA
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Convex Optimization&Euclidean Dista
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for Jennie Columba♦Antonio♦♦&
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PreludeThe constant demands of my d
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Convex Optimization&Euclidean Dista
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CONVEX OPTIMIZATION & EUCLIDEAN DIS
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List of Figures1 Overview 191 Orion
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LIST OF FIGURES 1559 Quadratic func
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LIST OF FIGURES 17E Projection 5791
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Chapter 1OverviewConvex Optimizatio
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ˇx 4ˇx 3ˇx 2Figure 2: Applicatio
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23Figure 4: This coarsely discretiz
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ases (biorthogonal expansion). We e
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27Figure 7: These bees construct a
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its membership to the EDM cone. The
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31appendicesProvided so as to be mo
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Chapter 2Convex geometryConvexity h
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2.1. CONVEX SET 35Figure 9: A slab
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2.1. CONVEX SET 372.1.6 empty set v
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2.1. CONVEX SET 392.1.7.1 Line inte
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2.1. CONVEX SET 41(a)R 2(b)R 3(c)(d
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2.1. CONVEX SET 43This theorem in c
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2.2. VECTORIZED-MATRIX INNER PRODUC
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2.2. VECTORIZED-MATRIX INNER PRODUC
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2.2. VECTORIZED-MATRIX INNER PRODUC
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2.2. VECTORIZED-MATRIX INNER PRODUC
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2.3. HULLS 53Figure 12: Convex hull
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2.3. HULLS 55Aaffine hull (drawn tr
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2.3. HULLS 57The union of relative
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2.4. HALFSPACE, HYPERPLANE 59of dim
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2.4. HALFSPACE, HYPERPLANE 61H +ay
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2.4. HALFSPACE, HYPERPLANE 63Inters
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2.4. HALFSPACE, HYPERPLANE 65Conver
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2.4. HALFSPACE, HYPERPLANE 67A 1A 2
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2.4. HALFSPACE, HYPERPLANE 69tradit
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2.4. HALFSPACE, HYPERPLANE 71There
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2.5. SUBSPACE REPRESENTATIONS 732.5
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2.5. SUBSPACE REPRESENTATIONS 752.5
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2.6. EXTREME, EXPOSED 77In other wo
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2.6. EXTREME, EXPOSED 792.6.1 Expos
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2.7. CONES 812.6.1.3.1 Definition.
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2.7. CONES 830Figure 24: Boundary o
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2.7. CONES 852.7.2 Convex coneWe ca
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2.7. CONES 87Thus the simplest and
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2.7. CONES 89nomenclature generaliz
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2.8. CONE BOUNDARY 91Proper cone {0
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2.8. CONE BOUNDARY 93the same extre
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96 CHAPTER 2. CONVEX GEOMETRYBCADFi
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98 CHAPTER 2. CONVEX GEOMETRYThe po
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100 CHAPTER 2. CONVEX GEOMETRY2.9.0
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102 CHAPTER 2. CONVEX GEOMETRYwhere
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104 CHAPTER 2. CONVEX GEOMETRY√2
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106 CHAPTER 2. CONVEX GEOMETRYwhich
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108 CHAPTER 2. CONVEX GEOMETRY2.9.2
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110 CHAPTER 2. CONVEX GEOMETRYA con
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2.9. POSITIVE SEMIDEFINITE (PSD) CO
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2.9. POSITIVE SEMIDEFINITE (PSD) CO
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2.9. POSITIVE SEMIDEFINITE (PSD) CO
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2.10. CONIC INDEPENDENCE (C.I.) 121
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2.10. CONIC INDEPENDENCE (C.I.) 123
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2.10. CONIC INDEPENDENCE (C.I.) 125
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2.12. CONVEX POLYHEDRA 127It follow
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2.12. CONVEX POLYHEDRA 129Coefficie
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2.12. CONVEX POLYHEDRA 1312.12.3 Un
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2.12. CONVEX POLYHEDRA 133
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2.13. DUAL CONE & GENERALIZED INEQU
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2.13. DUAL CONE & GENERALIZED INEQU
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2.13. DUAL CONE & GENERALIZED INEQU
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2.13. DUAL CONE & GENERALIZED INEQU
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Chapter 3Geometry of convex functio
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3.1. CONVEX FUNCTION 185f 1 (x)f 2
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3.1. CONVEX FUNCTION 1873.1.3 norm
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3.1. CONVEX FUNCTION 189where the n
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3.1. CONVEX FUNCTION 191where k ∈
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3.1. CONVEX FUNCTION 193f(z)Az 2z 1
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3.1. CONVEX FUNCTION 195{a T z 1 +
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3.1. CONVEX FUNCTION 197When an epi
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3.1. CONVEX FUNCTION 199orthonormal
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3.1. CONVEX FUNCTION 201[30,1.1] Ex
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3.1. CONVEX FUNCTION 20321.510.5Y 2
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3.1. CONVEX FUNCTION 2053.1.8.0.1 E
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3.1. CONVEX FUNCTION 207This equiva
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3.1. CONVEX FUNCTION 2093.1.8.1 mon
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3.1. CONVEX FUNCTION 211[ Yt]∈ ep
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3.1. CONVEX FUNCTION 213→Y −Xwh
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3.2. MATRIX-VALUED CONVEX FUNCTION
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3.2. MATRIX-VALUED CONVEX FUNCTION
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3.2. MATRIX-VALUED CONVEX FUNCTION
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3.3. QUASICONVEX 221A quasiconcave
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3.4. SALIENT PROPERTIES 2236.A nonn
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Chapter 4Semidefinite programmingPr
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4.1. CONIC PROBLEM 227where K is a
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4.1. CONIC PROBLEM 229C0PΓ 1Γ 2S+
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4.1. CONIC PROBLEM 231faces of S 3
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4.1. CONIC PROBLEM 2334.1.1.3 Previ
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4.2. FRAMEWORK 235Equivalently, pri
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4.2. FRAMEWORK 237is positive semid
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4.2. FRAMEWORK 239Optimal value of
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4.2. FRAMEWORK 2414.2.3.0.2 Example
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4.2. FRAMEWORK 243where δ is the m
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4.2. FRAMEWORK 2454.2.3.0.3 Example
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4.3. RANK REDUCTION 2474.3 Rank red
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4.3. RANK REDUCTION 249A rank-reduc
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4.3. RANK REDUCTION 251(t ⋆ i)
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4.3. RANK REDUCTION 2534.3.3.0.1 Ex
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4.3. RANK REDUCTION 2554.3.3.0.2 Ex
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4.4. RANK-CONSTRAINED SEMIDEFINITE
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4.4. RANK-CONSTRAINED SEMIDEFINITE
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Chapter 5Euclidean Distance MatrixT
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5.2. FIRST METRIC PROPERTIES 291cor
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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 297The collecti
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5.4. EDM DEFINITION 2995.4.2 Gram-f
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5.4. EDM DEFINITION 301D ∈ EDM N
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5.4. EDM DEFINITION 3035.4.2.2.1 Ex
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5.4. EDM DEFINITION 305ten affine e
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5.4. EDM DEFINITION 307spheres:Then
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5.4. EDM DEFINITION 311whereΦ ij =
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5.4. EDM DEFINITION 3135.4.2.2.5 De
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5.4. EDM DEFINITION 315105ˇx 4ˇx
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5.4. EDM DEFINITION 317corrected by
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5.4. EDM DEFINITION 319aptly be app
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5.4. EDM DEFINITION 321As before, a
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5.4. EDM DEFINITION 323where ([√t
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5.4. EDM DEFINITION 325because (A.3
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5.5. INVARIANCE 3275.5.1.0.1 Exampl
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5.6. INJECTIVITY OF D & UNIQUE RECO
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5.6. INJECTIVITY OF D & UNIQUE RECO
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5.6. INJECTIVITY OF D & UNIQUE RECO
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5.7. EMBEDDING IN AFFINE HULL 3355.
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5.7. EMBEDDING IN AFFINE HULL 337Fo
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5.7. EMBEDDING IN AFFINE HULL 3395.
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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 357(ii)
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5.11. EDM INDEFINITENESS 3595.11.1
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5.11. EDM INDEFINITENESS 361(confer
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5.11. EDM INDEFINITENESS 363we have
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5.11. EDM INDEFINITENESS 365For pre
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5.12. LIST RECONSTRUCTION 367where
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5.12. LIST RECONSTRUCTION 369(a)(c)
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5.13. RECONSTRUCTION EXAMPLES 371D
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5.13. RECONSTRUCTION EXAMPLES 373Th
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5.13. RECONSTRUCTION EXAMPLES 375wh
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5.14. FIFTH PROPERTY OF EUCLIDEAN M
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5.14. FIFTH PROPERTY OF EUCLIDEAN M
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Chapter 6EDM coneFor N > 3, the con
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6.1. DEFINING EDM CONE 3896.1 Defin
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6.2. POLYHEDRAL BOUNDS 391This cone
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6.3.√EDM CONE IS NOT CONVEX 393N
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6.4. A GEOMETRY OF COMPLETION 3956.
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6.4. A GEOMETRY OF COMPLETION 397(a
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6.4. A GEOMETRY OF COMPLETION 399Fi
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6.5. EDM DEFINITION IN 11 T 401and
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6.5. EDM DEFINITION IN 11 T 403then
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6.5. EDM DEFINITION IN 11 T 4056.5.
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6.5. EDM DEFINITION IN 11 T 407D =
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6.6. CORRESPONDENCE TO PSD CONE S N
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6.6. CORRESPONDENCE TO PSD CONE S N
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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 419When the Fins
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6.8. DUAL EDM CONE 421Proof. First,
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6.8. DUAL EDM CONE 423EDM 2 = S 2 h
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6.8. DUAL EDM CONE 425whose veracit
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6.8. DUAL EDM CONE 4276.8.1.3.1 Exe
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6.8. DUAL EDM CONE 429has dual affi
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6.8. DUAL EDM CONE 4316.8.1.7 Schoe
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6.9. THEOREM OF THE ALTERNATIVE 433
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6.10. POSTSCRIPT 435When D is an ED
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Chapter 7Proximity problemsIn summa
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In contrast, order of projection on
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441HS N h0EDM NK = S N h ∩ R N×N
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4437.0.3 Problem approachProblems t
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7.1. FIRST PREVALENT PROBLEM: 445fi
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7.1. FIRST PREVALENT PROBLEM: 4477.
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7.1. FIRST PREVALENT PROBLEM: 449di
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7.1. FIRST PREVALENT PROBLEM: 4517.
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7.1. FIRST PREVALENT PROBLEM: 453wh
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7.1. FIRST PREVALENT PROBLEM: 455Th
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7.2. SECOND PREVALENT PROBLEM: 457O
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7.2. SECOND PREVALENT PROBLEM: 459S
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7.2. SECOND PREVALENT PROBLEM: 461r
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7.2. SECOND PREVALENT PROBLEM: 463c
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7.2. SECOND PREVALENT PROBLEM: 4657
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7.3. THIRD PREVALENT PROBLEM: 467fo
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7.3. THIRD PREVALENT PROBLEM: 469a
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7.3. THIRD PREVALENT PROBLEM: 4717.
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7.3. THIRD PREVALENT PROBLEM: 4737.
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7.3. THIRD PREVALENT PROBLEM: 475Ou
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478 CHAPTER 7. PROXIMITY PROBLEMSth
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480 APPENDIX A. LINEAR ALGEBRAA.1.1
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482 APPENDIX A. LINEAR ALGEBRAA.1.2
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484 APPENDIX A. LINEAR ALGEBRAonly
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486 APPENDIX A. LINEAR ALGEBRA(AB)
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488 APPENDIX A. LINEAR ALGEBRAA.3.1
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490 APPENDIX A. LINEAR ALGEBRAFor A
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492 APPENDIX A. LINEAR ALGEBRADiago
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494 APPENDIX A. LINEAR ALGEBRAFor A
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496 APPENDIX A. LINEAR ALGEBRAA.3.1
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498 APPENDIX A. LINEAR ALGEBRAA.4 S
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500 APPENDIX A. LINEAR ALGEBRAA.4.0
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502 APPENDIX A. LINEAR ALGEBRAA.5 e
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504 APPENDIX A. LINEAR ALGEBRAs i w
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506 APPENDIX A. LINEAR ALGEBRAA.6.2
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508 APPENDIX A. LINEAR ALGEBRAΣq 2
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Page 510 and 511:
510 APPENDIX A. LINEAR ALGEBRAA.7 Z
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Page 512 and 513:
512 APPENDIX A. LINEAR ALGEBRAThere
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Page 514 and 515:
514 APPENDIX A. LINEAR ALGEBRAA.7.5
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Page 516 and 517:
516 APPENDIX A. LINEAR ALGEBRA
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Page 518 and 519:
518 APPENDIX B. SIMPLE MATRICESB.1
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Page 520 and 521:
520 APPENDIX B. SIMPLE MATRICESProo
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Page 522 and 523:
522 APPENDIX B. SIMPLE MATRICESB.1.
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Page 524 and 525:
524 APPENDIX B. SIMPLE MATRICESN(u
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Page 526 and 527:
526 APPENDIX B. SIMPLE MATRICESDue
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Page 528 and 529:
528 APPENDIX B. SIMPLE MATRICESB.4.
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Page 530 and 531:
530 APPENDIX B. SIMPLE MATRICEShas
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Page 532 and 533:
532 APPENDIX B. SIMPLE MATRICESFigu
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Page 534 and 535:
534 APPENDIX B. SIMPLE MATRICESB.5.
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Page 536 and 537:
536 APPENDIX C. SOME ANALYTICAL OPT
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Page 538 and 539:
538 APPENDIX C. SOME ANALYTICAL OPT
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Page 540 and 541:
540 APPENDIX C. SOME ANALYTICAL OPT
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Page 542 and 543:
542 APPENDIX C. SOME ANALYTICAL OPT
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Page 544 and 545:
544 APPENDIX C. SOME ANALYTICAL OPT
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Page 546 and 547:
546 APPENDIX C. SOME ANALYTICAL OPT
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Page 548 and 549:
548 APPENDIX C. SOME ANALYTICAL OPT
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Page 550 and 551:
550 APPENDIX D. MATRIX CALCULUSThe
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Page 552 and 553:
552 APPENDIX D. MATRIX CALCULUSGrad
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Page 554 and 555:
554 APPENDIX D. MATRIX CALCULUSBeca
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Page 556 and 557:
556 APPENDIX D. MATRIX CALCULUSwhic
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Page 558 and 559:
558 APPENDIX D. MATRIX CALCULUS⎡
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Page 560 and 561:
560 APPENDIX D. MATRIX CALCULUS→Y
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Page 562 and 563:
562 APPENDIX D. MATRIX CALCULUSD.1.
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Page 564 and 565:
564 APPENDIX D. MATRIX CALCULUSwhic
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Page 566 and 567:
566 APPENDIX D. MATRIX CALCULUSIn t
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Page 568 and 569:
568 APPENDIX D. MATRIX CALCULUSD.1.
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Page 570 and 571:
570 APPENDIX D. MATRIX CALCULUSD.2
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Page 572 and 573:
572 APPENDIX D. MATRIX CALCULUSalge
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Page 574 and 575:
574 APPENDIX D. MATRIX CALCULUStrac
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Page 576 and 577:
576 APPENDIX D. MATRIX CALCULUSD.2.
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Page 578 and 579:
578 APPENDIX D. MATRIX CALCULUS
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Page 580 and 581:
580 APPENDIX E. PROJECTIONThe follo
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Page 582 and 583:
582 APPENDIX E. PROJECTIONFor matri
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Page 584 and 585:
584 APPENDIX E. PROJECTION(⇐) To
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Page 586 and 587:
586 APPENDIX E. PROJECTIONNonorthog
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Page 588 and 589:
588 APPENDIX E. PROJECTIONE.2.0.0.1
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Page 590 and 591:
590 APPENDIX E. PROJECTIONE.3.2Orth
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Page 592 and 593:
592 APPENDIX E. PROJECTIONE.3.5Unif
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Page 594 and 595:
594 APPENDIX E. PROJECTIONE.4 Algeb
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Page 596 and 597:
596 APPENDIX E. PROJECTIONa ∗ 2K
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Page 598 and 599:
598 APPENDIX E. PROJECTIONwhere Y =
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Page 600 and 601:
600 APPENDIX E. PROJECTION(B.4.2).
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Page 602 and 603:
602 APPENDIX E. PROJECTIONis a nono
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Page 604 and 605:
604 APPENDIX E. PROJECTIONE.6.4.1Or
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Page 606 and 607:
606 APPENDIX E. PROJECTIONq i q T i
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Page 608 and 609:
608 APPENDIX E. PROJECTIONThe test
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Page 610 and 611:
610 APPENDIX E. PROJECTIONPerpendic
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Page 612 and 613:
612 APPENDIX E. PROJECTIONE.8 Range
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Page 614 and 615:
614 APPENDIX E. PROJECTIONAs for su
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Page 616 and 617:
616 APPENDIX E. PROJECTIONWith refe
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Page 618 and 619:
618 APPENDIX E. PROJECTIONProjectio
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Page 620 and 621:
620 APPENDIX E. PROJECTIONE.9.2.2.2
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Page 622 and 623:
622 APPENDIX E. PROJECTIONThe foreg
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Page 624 and 625:
624 APPENDIX E. PROJECTION❇❇❇
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Page 626 and 627:
626 APPENDIX E. PROJECTIONE.10 Alte
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Page 628 and 629:
628 APPENDIX E. PROJECTIONbH 1H 2P
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Page 630 and 631:
630 APPENDIX E. PROJECTIONa(a){y |
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Page 632 and 633:
632 APPENDIX E. PROJECTION(a feasib
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Page 634 and 635:
634 APPENDIX E. PROJECTIONwhile, th
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Page 636 and 637:
636 APPENDIX E. PROJECTIONE.10.2.1.
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Page 638 and 639:
638 APPENDIX E. PROJECTION10 0dist(
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Page 640 and 641:
640 APPENDIX E. PROJECTIONE.10.3.1D
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Page 642 and 643:
642 APPENDIX E. PROJECTIONE 3K ⊥
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Page 644 and 645:
644 APPENDIX E. PROJECTION
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Page 646 and 647:
646 APPENDIX F. MATLAB PROGRAMSif n
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Page 648 and 649:
648 APPENDIX F. MATLAB PROGRAMSend%
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Page 650 and 651:
650 APPENDIX F. MATLAB PROGRAMSF.1.
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Page 652 and 653:
652 APPENDIX F. MATLAB PROGRAMScoun
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Page 654 and 655:
654 APPENDIX F. MATLAB PROGRAMSF.3
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Page 656 and 657:
656 APPENDIX F. MATLAB PROGRAMSF.3.
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Page 658 and 659:
658 APPENDIX F. MATLAB PROGRAMS% so
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Page 660 and 661:
660 APPENDIX F. MATLAB PROGRAMS% tr
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Page 662 and 663:
662 APPENDIX F. MATLAB PROGRAMSF.4.
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Page 664 and 665:
664 APPENDIX F. MATLAB PROGRAMSbrea
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Page 666 and 667:
666 APPENDIX F. MATLAB PROGRAMSwhil
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Page 668 and 669:
668 APPENDIX F. MATLAB PROGRAMSF.7
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Page 670 and 671:
670 APPENDIX F. MATLAB PROGRAMS
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Page 672 and 673:
672 APPENDIX G. NOTATION AND A FEW
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Page 674 and 675:
674 APPENDIX G. NOTATION AND A FEW
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Page 676 and 677:
676 APPENDIX G. NOTATION AND A FEW
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Page 678 and 679:
678 APPENDIX G. NOTATION AND A FEW
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Page 680 and 681:
680 APPENDIX G. NOTATION AND A FEW
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Page 682 and 683:
682 APPENDIX G. NOTATION AND A FEW
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Page 684 and 685:
684 APPENDIX G. NOTATION AND A FEW
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Page 686 and 687:
686 APPENDIX G. NOTATION AND A FEW
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Page 688 and 689:
688 BIBLIOGRAPHY[7] Abdo Y. Alfakih
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Page 690 and 691:
690 BIBLIOGRAPHY[27] Aharon Ben-Tal
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Page 692 and 693:
692 BIBLIOGRAPHY[48] Lev M. Brègma
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Page 694 and 695:
694 BIBLIOGRAPHY[67] Joel Dawson, S
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Page 696 and 697:
696 BIBLIOGRAPHY[85] Alan Edelman,
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Page 698 and 699:
698 BIBLIOGRAPHY[102] Philip E. Gil
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Page 700 and 701:
700 BIBLIOGRAPHYWeiss, editors, Pol
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Page 702 and 703:
702 BIBLIOGRAPHY[146] Jean-Baptiste
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Page 704 and 705:
704 BIBLIOGRAPHY[168] Jean B. Lasse
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Page 706 and 707:
706 BIBLIOGRAPHY[189] Rudolf Mathar
-
Page 708 and 709:
708 BIBLIOGRAPHY[211] M. L. Overton
-
Page 710 and 711:
710 BIBLIOGRAPHY[229] C. K. Rushfor
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Page 712 and 713:
712 BIBLIOGRAPHY[252] Jos F. Sturm
-
Page 714 and 715:
714 BIBLIOGRAPHY[274] È. B. Vinber
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Page 716 and 717:
[294] Yinyu Ye. Semidefinite progra
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Page 718 and 719:
718 INDEXobtuse, 62positive semidef
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Page 720 and 721:
720 INDEXelliptope, 642orthant, 177
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Page 722 and 723:
722 INDEXdistancegeometry, 20, 317m
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Page 724 and 725:
724 INDEX-dimensional, 37, 89rank,
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Page 726 and 727:
726 INDEXGram form, 331is, 674isedm
-
Page 728 and 729:
728 INDEXdiscretized, 152, 431in su
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Page 730 and 731:
730 INDEXboundary, 115dimension, 10
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Page 732 and 733:
732 INDEXlinear operator, 587, 591,
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Page 734 and 735:
734 INDEXlargest entries, 188monoto
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Page 736:
736 INDEXsimilarity, 606translation
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