v2010.10.26 - Convex Optimization

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406 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX5.4.2.1 First point at originAssume the first point x 1 in an unknown list X resides at the origin;Xe 1 = 0 ⇔ Ge 1 = 0 (905)Consider the symmetric translation (I − 1e T 1 )D(G)(I − e 1 1 T ) that shiftsthe first row and column of D(G) to the origin; setting Gram-form EDMoperator D(G) = D for convenience,− ( D − (De 1 1 T + 1e T 1D) + 1e T 1De 1 1 T) 12 = G − (Ge 11 T + 1e T 1G) + 1e T 1Ge 1 1 Twheree 1 ⎡⎢⎣10.0⎤(906)⎥⎦ (907)is the first vector from the standard basis. Then it follows for D ∈ S N hwhereG = − ( D − (De 1 1 T + 1e T 1D) ) 12 , x 1 = 0= − [ 0 √ ] TD [ √ ]2V N 0 1 2VN2[ ] 0 0T=0 −VN TDV NV T N GV N = −V T N DV N 1 2I − e 1 1 T =[0 √ ]2V N∀Xis a projector nonorthogonally projecting (E.1) on subspaceS N 1 = {G∈ S N | Ge 1 = 0}{ [0 √ ] T [ √ ]= 2VN Y 0 2VN | Y ∈ SN} (2002)(908)(909)in the Euclidean sense. From (908) we get sufficiency of the first matrixcriterion for an EDM proved by Schoenberg in 1935; [312] 5.7{−VTD ∈ EDM NN DV N ∈ S N−1+⇔(910)D ∈ S N h5.7 From (898) we know R(V N )= N(1 T ) , so (910) is the same as (886). In fact, anymatrix V in place of V N will satisfy (910) whenever R(V )= R(V N )= N(1 T ). But V N isthe matrix implicit in Schoenberg’s seminal exposition.
5.4. EDM DEFINITION 407We provide a rigorous complete more geometric proof of this fundamentalSchoenberg criterion in5.9.1.0.4. [ [isedm(D) ] on Wıκımization]0 0TBy substituting G =0 −VN TDV (908) into D(G) (903),N[ ]0D =δ ( [−VN TDV ) 1 T + 1 0 δ ( ) ] [ ]−VNDV T T 0 0TN − 2N0 −VN TDV (1009)Nassuming x 1 = 0. Details of this bijection are provided in5.6.2.5.4.2.2 0 geometric centerAssume the geometric center (5.5.1.0.1) of an unknown list X is the origin;X1 = 0 ⇔ G1 = 0 (911)Now consider the calculation (I − 1 N 11T )D(G)(I − 1 N 11T ) , a geometriccentering or projection operation. (E.7.2.0.2) Setting D(G) = D forconvenience as in5.4.2.1,G = − ( D − 1 N (D11T + 11 T D) + 1N 2 11 T D11 T) 12 , X1 = 0= −V DV 1 2V GV = −V DV 1 2∀X(912)where more properties of the auxiliary (geometric centering, projection)matrixV I − 1 N 11T ∈ S N (913)are found inB.4. V GV may be regarded as a covariance matrix of means 0.From (912) and the assumption D ∈ S N h we get sufficiency of the more popularform of Schoenberg’s criterion:D ∈ EDM N⇔{−V DV ∈ SN+D ∈ S N h(914)Of particular utility when D ∈ EDM N is the fact, (B.4.2 no.20) (887)tr ( ( )∑−V DV 2) 1 =1∑d2N ij = 12N vec(X)T Φ ij ⊗ I vec Xi,ji,j= tr(V GV ) , G ≽ 0∑= trG = N ‖x l ‖ 2 = ‖X‖ 2 F , X1 = 0l=1(915)
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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 211 Orion
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LIST OF FIGURES 1562 Shrouded polyh
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LIST OF FIGURES 17130 Elliptope E 3
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Chapter 1OverviewConvex Optimizatio
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ˇx 4ˇx 3ˇx 2Figure 2: Applicatio
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25Figure 4: This coarsely discretiz
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(biorthogonal expansion) is examine
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29cardinality Boolean solution to a
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31Figure 8: Robotic vehicles in con
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an elaborate exposition offering in
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Chapter 2Convex geometryConvexity h
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2.1. CONVEX SET 372.1.2 linear inde
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2.1. CONVEX SET 392.1.6 empty set v
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2.1. CONVEX SET 41(a)R(b)R 2(c)R 3F
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2.1. CONVEX SET 43where Q∈ R 3×3
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2.1. CONVEX SET 45Now let’s move
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2.1. CONVEX SET 47By additive inver
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2.1. CONVEX SET 49R nR mR(A T )x px
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2.2. VECTORIZED-MATRIX INNER PRODUC
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2.3. HULLS 61Figure 20: Convex hull
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2.3. HULLS 63Aaffine hull (drawn tr
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2.3. HULLS 65subset of the affine h
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2.3. HULLS 672.3.2.0.2 Example. Nuc
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2.3. HULLS 692.3.2.0.3 Exercise. Co
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2.3. HULLS 71Figure 24: A simplicia
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2.4. HALFSPACE, HYPERPLANE 73H + =
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2.4. HALFSPACE, HYPERPLANE 7511−1
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2.4. HALFSPACE, HYPERPLANE 772.4.2.
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2.4. HALFSPACE, HYPERPLANE 81tradit
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2.5. SUBSPACE REPRESENTATIONS 852.5
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2.5. SUBSPACE REPRESENTATIONS 87(Ex
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2.5. SUBSPACE REPRESENTATIONS 89(a)
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2.5. SUBSPACE REPRESENTATIONS 91are
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2.6. EXTREME, EXPOSED 93A one-dimen
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2.6. EXTREME, EXPOSED 952.6.1.1 Den
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2.7. CONES 97X(a)00(b)XFigure 33: (
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2.7. CONES 99XXFigure 37: Truncated
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2.7. CONES 101Figure 39: Not a cone
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2.7. CONES 103cone that is a halfli
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2.7. CONES 105A pointed closed conv
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2.8. CONE BOUNDARY 107That means th
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2.8. CONE BOUNDARY 1092.8.1.1 extre
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2.8. CONE BOUNDARY 1112.8.2 Exposed
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2.9. POSITIVE SEMIDEFINITE (PSD) CO
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2.10. CONIC INDEPENDENCE (C.I.) 141
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2.12. CONVEX POLYHEDRA 147all dimen
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2.12. CONVEX POLYHEDRA 149convex po
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2.12. CONVEX POLYHEDRA 1512.12.2.2
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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 221f 1 (x)f 2
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3.1. CONVEX FUNCTION 223Rf(b)f(X
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3.2. PRACTICAL NORM FUNCTIONS, ABSO
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3.3. INVERTED FUNCTIONS AND ROOTS 2
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3.4. AFFINE FUNCTION 237rather]x >
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3.4. AFFINE FUNCTION 239f(z)Az 2z 1
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3.5. EPIGRAPH, SUBLEVEL SET 241{a T
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3.5. EPIGRAPH, SUBLEVEL SET 243Subl
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3.5. EPIGRAPH, SUBLEVEL SET 245wher
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3.5. EPIGRAPH, SUBLEVEL SET 247part
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3.5. EPIGRAPH, SUBLEVEL SET 249that
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3.6. GRADIENT 251respect to its vec
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3.6. GRADIENT 253Invertibility is g
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3.6. GRADIENT 2553.6.1.0.2 Theorem.
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3.6. GRADIENT 257f(Y )[ ∇f(X)−1
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3.6. GRADIENT 259αβα ≥ β ≥
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3.6. GRADIENT 2613.6.4 second-order
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3.7. CONVEX MATRIX-VALUED FUNCTION
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3.8. QUASICONVEX 269exponential alw
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3.9. SALIENT PROPERTIES 2713.8.0.0.
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Chapter 4Semidefinite programmingPr
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4.1. CONIC PROBLEM 275(confer p.162
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4.1. CONIC PROBLEM 277PCsemidefinit
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4.1. CONIC PROBLEM 279is the affine
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4.1. CONIC PROBLEM 281faces of S 3
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4.1. CONIC PROBLEM 2834.1.2.3 Previ
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4.2. FRAMEWORK 285Semidefinite Fark
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4.2. FRAMEWORK 287On the other hand
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4.2. FRAMEWORK 2894.2.2.1 Dual prob
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4.2. FRAMEWORK 291For symmetric pos
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4.2. FRAMEWORK 293has norm ‖x ⋆
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4.2. FRAMEWORK 295minimize 1 TˆxX
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4.2. FRAMEWORK 297asminimize ‖ỹ
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4.3. RANK REDUCTION 2994.3 Rank red
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4.3. RANK REDUCTION 301A rank-reduc
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4.3. RANK REDUCTION 303(t ⋆ i)
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4.3. RANK REDUCTION 3054.3.3.0.1 Ex
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4.3. RANK REDUCTION 3074.3.3.0.2 Ex
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4.4. RANK-CONSTRAINED SEMIDEFINITE
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4.5. CONSTRAINING CARDINALITY 333mi
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4.5. CONSTRAINING CARDINALITY 3350R
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4.5. CONSTRAINING CARDINALITY 337it
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4.5. CONSTRAINING CARDINALITY 339m/
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4.5. CONSTRAINING CARDINALITY 341we
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4.5. CONSTRAINING CARDINALITY 343fl
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4.5. CONSTRAINING CARDINALITY 345We
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4.5. CONSTRAINING CARDINALITY 3474.
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4.5. CONSTRAINING CARDINALITY 349R
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4.5. CONSTRAINING CARDINALITY 351pe
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4.6. CARDINALITY AND RANK CONSTRAIN
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Page 395 and 396: Chapter 5Euclidean Distance MatrixT
Page 397 and 398: 5.2. FIRST METRIC PROPERTIES 397to
Page 399 and 400: 5.3. ∃ FIFTH EUCLIDEAN METRIC PRO
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Page 403 and 404: 5.4. EDM DEFINITION 403The collecti
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Page 411 and 412: 5.4. EDM DEFINITION 411is the fact:
Page 413 and 414: 5.4. EDM DEFINITION 413Figure 120:
Page 415 and 416: 5.4. EDM DEFINITION 415is found fro
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Page 425 and 426: 5.4. EDM DEFINITION 425now implicit
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Page 429 and 430: 5.4. EDM DEFINITION 429Crippen & Ha
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Page 433 and 434: 5.4. EDM DEFINITION 433because (A.3
Page 435 and 436: 5.5. INVARIANCE 4355.5.1.0.1 Exampl
Page 437 and 438: 5.5. INVARIANCE 437x 2 x 2x 3 x 1 x
Page 439 and 440: 5.6. INJECTIVITY OF D & UNIQUE RECO
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Page 451 and 452: 5.8. EUCLIDEAN METRIC VERSUS MATRIX
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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 465of a
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5.10. EDM-ENTRY COMPOSITION 467Then
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5.11. EDM INDEFINITENESS 4695.11.1
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5.11. EDM INDEFINITENESS 471we have
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5.11. EDM INDEFINITENESS 473So beca
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5.11. EDM INDEFINITENESS 475holds o
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5.12. LIST RECONSTRUCTION 477where
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5.12. LIST RECONSTRUCTION 479(a)(c)
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5.13. RECONSTRUCTION EXAMPLES 481Wi
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5.13. RECONSTRUCTION EXAMPLES 483d
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5.13. RECONSTRUCTION EXAMPLES 485Th
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5.14. FIFTH PROPERTY OF EUCLIDEAN M
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Chapter 6Cone of distance matricesF
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6.1. DEFINING EDM CONE 4996.1 Defin
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6.2. POLYHEDRAL BOUNDS 501This cone
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6.4. EDM DEFINITION IN 11 T 503That
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6.4. EDM DEFINITION IN 11 T 505N(1
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6.4. EDM DEFINITION IN 11 T 507Then
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6.4. EDM DEFINITION IN 11 T 5096.4.
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6.5. CORRESPONDENCE TO PSD CONE S N
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6.6. VECTORIZATION & PROJECTION INT
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6.7. A GEOMETRY OF COMPLETION 523(b
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6.7. A GEOMETRY OF COMPLETION 525[3
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6.7. A GEOMETRY OF COMPLETION 527wh
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6.8. DUAL EDM CONE 529to the geomet
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6.8. DUAL EDM CONE 531Proof. First,
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6.8. DUAL EDM CONE 533EDM 2 = S 2 h
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6.8. DUAL EDM CONE 535therefore the
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6.8. DUAL EDM CONE 537Elegance of t
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6.8. DUAL EDM CONE 5396.8.1.5 Affin
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6.8. DUAL EDM CONE 5416.8.1.7 Schoe
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6.8. DUAL EDM CONE 5430dvec rel ∂
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6.10. POSTSCRIPT 5456.10 Postscript
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Chapter 7Proximity problemsIn the
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549project on the subspace, then pr
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551HS N h0EDM NK = S N h ∩ R N×N
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5537.0.3 Problem approachProblems t
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7.1. FIRST PREVALENT PROBLEM: 555fi
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7.1. FIRST PREVALENT PROBLEM: 5577.
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7.1. FIRST PREVALENT PROBLEM: 559di
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7.1. FIRST PREVALENT PROBLEM: 5617.
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7.1. FIRST PREVALENT PROBLEM: 563wh
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7.1. FIRST PREVALENT PROBLEM: 565Th
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7.2. SECOND PREVALENT PROBLEM: 567O
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7.2. SECOND PREVALENT PROBLEM: 569S
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7.2. SECOND PREVALENT PROBLEM: 571r
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7.2. SECOND PREVALENT PROBLEM: 573w
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7.2. SECOND PREVALENT PROBLEM: 5757
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7.2. SECOND PREVALENT PROBLEM: 577a
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7.3. THIRD PREVALENT PROBLEM: 579is
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7.3. THIRD PREVALENT PROBLEM: 581We
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7.3. THIRD PREVALENT PROBLEM: 583su
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7.3. THIRD PREVALENT PROBLEM: 585Gi
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7.3. THIRD PREVALENT PROBLEM: 587Op
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7.4. CONCLUSION 589filtering, multi
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Appendix ALinear algebraA.1 Main-di
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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 599(AB) T
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A.3. PROPER STATEMENTS 601A.3.1Semi
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A.3. PROPER STATEMENTS 603For A dia
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A.3. PROPER STATEMENTS 605Diagonali
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A.3. PROPER STATEMENTS 607For A,B
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A.3. PROPER STATEMENTS 609When B is
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A.4. SCHUR COMPLEMENT 611A.4 Schur
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A.4. SCHUR COMPLEMENT 613A.4.0.0.3
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A.4. SCHUR COMPLEMENT 615From Corol
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A.5. EIGENVALUE DECOMPOSITION 617wh
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A.5. EIGENVALUE DECOMPOSITION 619A.
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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 625A.6.5SVD of symmetric
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A.7. ZEROS 627(Transpose.)Likewise,
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A.7. ZEROS 629For X,A∈ S M +[34,2
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A.7. ZEROS 631A.7.5.0.1 Proposition
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Appendix BSimple matricesMathematic
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B.1. RANK-ONE MATRIX (DYAD) 635R(v)
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B.1. RANK-ONE MATRIX (DYAD) 637B.1.
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B.2. DOUBLET 639R([u v ])R(Π)= R([
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B.3. ELEMENTARY MATRIX 641has N −
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B.4. AUXILIARY V -MATRICES 643is an
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B.4. AUXILIARY V -MATRICES 64514. [
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B.5. ORTHOGONAL MATRIX 647Given X
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B.5. ORTHOGONAL MATRIX 649Figure 15
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B.5. ORTHOGONAL MATRIX 651which is
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Appendix CSome analytical optimal r
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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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Appendix DMatrix calculusFrom too m
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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 EProjectionFor any A∈ R
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701U T = U † for orthonormal (inc
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E.1. IDEMPOTENT MATRICES 703where A
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E.1. IDEMPOTENT MATRICES 705order,
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E.1. IDEMPOTENT MATRICES 707are lin
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E.3. SYMMETRIC IDEMPOTENT MATRICES
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E.4. ALGEBRA OF PROJECTION ON AFFIN
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E.5. PROJECTION EXAMPLES 717a ∗ 2
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E.5. PROJECTION EXAMPLES 719where Y
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E.5. PROJECTION EXAMPLES 721(B.4.2)
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E.6. VECTORIZATION INTERPRETATION,
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E.7. PROJECTION ON MATRIX SUBSPACES
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E.7. PROJECTION ON MATRIX SUBSPACES
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E.8. RANGE/ROWSPACE INTERPRETATION
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E.9. PROJECTION ON CONVEX SET 735As
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E.9. PROJECTION ON CONVEX SET 737Wi
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E.9. PROJECTION ON CONVEX SET 739R(
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E.9. PROJECTION ON CONVEX SET 741E.
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E.9. PROJECTION ON CONVEX SET 743E.
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E.9. PROJECTION ON CONVEX SET 745Un
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E.9. PROJECTION ON CONVEX SET 747ac
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E.10. ALTERNATING PROJECTION 749bC
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E.10. ALTERNATING PROJECTION 7510
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E.10. ALTERNATING PROJECTION 753E.1
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E.10. ALTERNATING PROJECTION 755y 2
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E.10. ALTERNATING PROJECTION 757Def
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E.10. ALTERNATING PROJECTION 759Dis
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E.10. ALTERNATING PROJECTION 761mat
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E.10. ALTERNATING PROJECTION 763K
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Appendix FNotation and a few defini
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771A ij or A(i, j) , ij th entry of
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773⊞orthogonal vector sum of sets
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775x +vector x whose negative entri
Page 777 and 778:
777X point list ((76) having cardin
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779SDPSVDSNRdBEDMS n 1S n hS n⊥hS
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781vectorentrycubixquartixfeasible
Page 783 and 784:
783Oorder of magnitude information
Page 785 and 786:
785cofmatrix of cofactors correspon
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Bibliography[1] Edwin A. Abbott. Fl
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BIBLIOGRAPHY 789[23] Dror Baron, Mi
Page 791 and 792:
BIBLIOGRAPHY 791[49] Leonard M. Blu
Page 793 and 794:
BIBLIOGRAPHY 793[74] Yves Chabrilla
Page 795 and 796:
BIBLIOGRAPHY 795[102] Etienne de Kl
Page 797 and 798:
BIBLIOGRAPHY 797[129] Carl Eckart a
Page 799 and 800:
BIBLIOGRAPHY 799[154] James Gleik.
Page 801 and 802:
BIBLIOGRAPHY 801[182] Johan Håstad
Page 803 and 804:
BIBLIOGRAPHY 803[212] Viren Jain an
Page 805 and 806:
BIBLIOGRAPHY 805[237] Monique Laure
Page 807 and 808:
BIBLIOGRAPHY 807[265] Sunderarajan
Page 809 and 810:
BIBLIOGRAPHY 809Notes in Computer S
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BIBLIOGRAPHY 811[319] Anthony Man-C
Page 813 and 814:
BIBLIOGRAPHY 813[346] Pham Dinh Tao
Page 815 and 816:
BIBLIOGRAPHY 815[375] Bernard Widro
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Index∅, see empty set0-norm, 229,
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INDEX 819bees, 30, 413Bellman, 276b
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INDEX 821circular, 130construction,
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INDEX 823measure, 295convex, 21, 36
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INDEX 825duality, 160, 410gap, 161,
Page 827 and 828:
INDEX 827positive semidefinite, 284
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INDEX 829point, 39, 62positive semi
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INDEX 831isomorphic, 52, 56, 59, 77
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INDEX 833nonsymmetric, 703range, 70
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INDEX 835Muller, 624multidimensiona
Page 837 and 838:
INDEX 837projection on, 739, 745tra
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INDEX 839pseudoinverse, 701trace, 5
Page 841 and 842:
INDEX 841coordinate system, 156line
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INDEX 843nonconvex, 40, 97-101nulls
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INDEX 845faces, 106intersection, 10
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INDEX 847projection, 517symmetric,
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849
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Convex Optimization & Euclidean Dis
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