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v2007.11.26 - Convex Optimization
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594 APPENDIX D. MATRIX CALCULUS
594 APPENDIX D. MATRIX CALCULUS
Appendix EProjectionFor any A∈ R m×n , the pseudoinverse [150,7.3, prob.9] [183,6.12, prob.19][110,5.5.4] [251, App.A]A †∆ = limt→0 +(AT A + t I) −1 A T = limt→0 +AT (AA T + t I) −1 ∈ R n×m (1664)is a unique matrix solving minimize ‖AX − I‖ 2 F . For any t > 0XI − A(A T A + t I) −1 A T = t(AA T + t I) −1 (1665)Equivalently, pseudoinverse A † is that unique matrix satisfying theMoore-Penrose conditions: [152,1.3] [284]1. AA † A = A 3. (AA † ) T = AA †2. A † AA † = A † 4. (A † A) T = A † Awhich are necessary and sufficient to establish the pseudoinverse whoseprincipal action is to injectively map R(A) onto R(A T ). Taken rowwise,these conditions are respectively necessary and sufficient for AA † to be theorthogonal projector on R(A) , and for A † A to be the orthogonal projectoron R(A T ).Range and nullspace of the pseudoinverse [197] [248,III.1, exer.1]R(A † ) = R(A T ), R(A †T ) = R(A) (1666)N(A † ) = N(A T ), N(A †T ) = N(A) (1667)can be derived by singular value decomposition (A.6).2001 Jon Dattorro. CO&EDG version 2007.11.26. All rights reserved.Citation: Jon Dattorro, <strong>Convex</strong> <strong>Optimization</strong> & Euclidean Distance Geometry,Meboo Publishing USA, 2005.595
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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 Epigraph . . .
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LIST OF FIGURES 17120 Four fundamen
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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 11: 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 14: 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 73nor
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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 26: 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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2.8. CONE BOUNDARY 95For a proper c
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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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2.9. 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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2.9. 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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2.9. 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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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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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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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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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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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 189n∑π(x) i
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3.1. CONVEX FUNCTION 191This means
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3.1. CONVEX FUNCTION 1933.1.5.3 pos
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3.1. CONVEX FUNCTION 1953.1.6.0.1 E
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3.1. CONVEX FUNCTION 1973.1.7 epigr
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3.1. CONVEX FUNCTION 1993.1.7.0.3 E
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3.1. CONVEX FUNCTION 201Although th
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3.1. CONVEX FUNCTION 203minimize tX
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3.1. CONVEX FUNCTION 2053.1.8 gradi
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3.1. CONVEX FUNCTION 2073.1.8.0.2 E
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3.1. CONVEX FUNCTION 209f(Y )[ ∇f
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3.1. CONVEX FUNCTION 211g is convex
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3.1. CONVEX FUNCTION 213αβα ≥
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3.1. CONVEX FUNCTION 2153.1.11 seco
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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 221matrices in the
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3.3. QUASICONVEX 223on R M + . Alth
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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 2414.2.3.1.1 Example
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4.2. FRAMEWORK 243where δ is the m
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4.2. FRAMEWORK 2454.2.3.1.2 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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4.4. RANK-CONSTRAINED SEMIDEFINITE
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4.4. RANK-CONSTRAINED SEMIDEFINITE
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4.4. RANK-CONSTRAINED SEMIDEFINITE
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4.4. RANK-CONSTRAINED SEMIDEFINITE
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4.4. RANK-CONSTRAINED SEMIDEFINITE
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4.4. RANK-CONSTRAINED SEMIDEFINITE
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4.5. CONSTRAINING CARDINALITY 2734.
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4.5. CONSTRAINING CARDINALITY 275Th
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4.5. CONSTRAINING CARDINALITY 277wh
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4.5. CONSTRAINING CARDINALITY 279f
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4.5. CONSTRAINING CARDINALITY 281al
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4.6. CARDINALITY AND RANK CONSTRAIN
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4.6. CARDINALITY AND RANK CONSTRAIN
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4.6. CARDINALITY AND RANK CONSTRAIN
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4.6. CARDINALITY AND RANK CONSTRAIN
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4.6. CARDINALITY AND RANK CONSTRAIN
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4.6. CARDINALITY AND RANK CONSTRAIN
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4.6. CARDINALITY AND RANK CONSTRAIN
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4.6. CARDINALITY AND RANK CONSTRAIN
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4.6. CARDINALITY AND RANK CONSTRAIN
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4.7. CONVEX ITERATION RANK-1 301whi
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4.7. CONVEX ITERATION RANK-1 303the
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Chapter 5Euclidean Distance MatrixT
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5.2. FIRST METRIC PROPERTIES 307cor
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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 313The collecti
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5.4. EDM DEFINITION 3155.4.2 Gram-f
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5.4. EDM DEFINITION 317D ∈ EDM N
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5.4. EDM DEFINITION 3195.4.2.2.1 Ex
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5.4. EDM DEFINITION 321ten affine e
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5.4. EDM DEFINITION 323spheres:Then
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5.4. EDM DEFINITION 325By eliminati
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5.4. EDM DEFINITION 327whereΦ ij =
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5.4. EDM DEFINITION 3295.4.2.2.5 De
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5.4. EDM DEFINITION 331105ˇx 4ˇx
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5.4. EDM DEFINITION 333corrected by
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5.4. EDM DEFINITION 335by translate
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5.4. EDM DEFINITION 337Crippen & Ha
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5.4. EDM DEFINITION 339where ([√t
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5.4. EDM DEFINITION 341because (A.3
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5.5. INVARIANCE 3435.5.1.0.1 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 3515.
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5.7. EMBEDDING IN AFFINE HULL 353Fo
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5.7. EMBEDDING IN AFFINE HULL 3555.
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5.8. EUCLIDEAN METRIC VERSUS MATRIX
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5.8. EUCLIDEAN METRIC VERSUS MATRIX
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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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5.9. BRIDGE: CONVEX POLYHEDRA TO ED
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5.9. BRIDGE: CONVEX POLYHEDRA TO ED
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5.9. BRIDGE: CONVEX POLYHEDRA TO ED
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5.10. EDM-ENTRY COMPOSITION 373(ii)
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5.11. EDM INDEFINITENESS 3755.11.1
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5.11. EDM INDEFINITENESS 377(confer
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5.11. EDM INDEFINITENESS 379we have
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5.11. EDM INDEFINITENESS 381For pre
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5.12. LIST RECONSTRUCTION 383where
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5.12. LIST RECONSTRUCTION 385(a)(c)
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5.13. RECONSTRUCTION EXAMPLES 387D
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5.13. RECONSTRUCTION EXAMPLES 389Th
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5.13. RECONSTRUCTION EXAMPLES 391wh
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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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5.14. FIFTH PROPERTY OF EUCLIDEAN M
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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 6Cone of distance matricesF
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6.1. DEFINING EDM CONE 4056.1 Defin
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6.2. POLYHEDRAL BOUNDS 407This cone
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6.3.√EDM CONE IS NOT CONVEX 409N
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6.4. A GEOMETRY OF COMPLETION 4116.
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6.4. A GEOMETRY OF COMPLETION 413(a
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6.4. A GEOMETRY OF COMPLETION 415Fi
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6.5. EDM DEFINITION IN 11 T 417and
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6.5. EDM DEFINITION IN 11 T 419then
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6.5. EDM DEFINITION IN 11 T 4216.5.
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6.5. EDM DEFINITION IN 11 T 423D =
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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 435When the Fins
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6.8. DUAL EDM CONE 437Proof. First,
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6.8. DUAL EDM CONE 439EDM 2 = S 2 h
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6.8. DUAL EDM CONE 441whose veracit
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6.8. DUAL EDM CONE 4436.8.1.3.1 Exe
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6.8. DUAL EDM CONE 445has dual affi
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6.8. DUAL EDM CONE 4476.8.1.7 Schoe
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6.9. THEOREM OF THE ALTERNATIVE 449
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6.10. POSTSCRIPT 451When 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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457HS N h0EDM NK = S N h ∩ R N×N
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4597.0.3 Problem approachProblems t
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7.1. FIRST PREVALENT PROBLEM: 461fi
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7.1. FIRST PREVALENT PROBLEM: 4637.
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7.1. FIRST PREVALENT PROBLEM: 465di
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7.1. FIRST PREVALENT PROBLEM: 4677.
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7.1. FIRST PREVALENT PROBLEM: 469wh
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7.1. FIRST PREVALENT PROBLEM: 471Th
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7.2. SECOND PREVALENT PROBLEM: 473O
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7.2. SECOND PREVALENT PROBLEM: 475S
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7.2. SECOND PREVALENT PROBLEM: 477r
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7.2. SECOND PREVALENT PROBLEM: 479c
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7.2. SECOND PREVALENT PROBLEM: 4817
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7.3. THIRD PREVALENT PROBLEM: 483fo
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7.3. THIRD PREVALENT PROBLEM: 485a
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7.3. THIRD PREVALENT PROBLEM: 4877.
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7.3. THIRD PREVALENT PROBLEM: 4897.
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7.3. THIRD PREVALENT PROBLEM: 491Ou
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7.4. CONCLUSION 493The rank constra
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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.2. SEMIDEFINITENESS: DOMAIN OF TE
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A.2. SEMIDEFINITENESS: DOMAIN OF TE
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A.3. PROPER STATEMENTS 503A.3.0.0.1
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A.3. PROPER STATEMENTS 505By simila
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A.3. PROPER STATEMENTS 507Because R
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For A,B ∈ R n×n x T Ax ≥ x T B
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A.3. PROPER STATEMENTS 511A.3.1.0.2
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A.3. PROPER STATEMENTS 513We can de
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A.4. SCHUR COMPLEMENT 515Origin of
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A.4. SCHUR COMPLEMENT 517Schur-form
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A.5. EIGEN DECOMPOSITION 519A.5.0.1
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A.6. SINGULAR VALUE DECOMPOSITION,
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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 527For diagonalizable ma
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A.7. ZEROS 529A.7.4For X,A∈ S M +
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A.7. ZEROS 531A.7.5.0.1 Proposition
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Appendix BSimple matricesMathematic
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B.1. RANK-ONE MATRIX (DYAD) 535R(v)
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B.1. RANK-ONE MATRIX (DYAD) 537rang
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B.2. DOUBLET 539R([u v ])R(Π)= R([
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B.3. ELEMENTARY MATRIX 541If λ ≠
- Page 543 and 544: B.4. AUXILIARY V -MATRICES 543the n
- Page 545 and 546: B.4. AUXILIARY V -MATRICES 54518. V
- Page 547 and 548: B.5. ORTHOGONAL MATRIX 547B.5 Ortho
- Page 549 and 550: B.5. ORTHOGONAL MATRIX 549B.5.3.0.1
- Page 551 and 552: Appendix CSome analytical optimal r
- Page 553 and 554: C.2. DIAGONAL, TRACE, SINGULAR AND
- Page 555 and 556: C.2. DIAGONAL, TRACE, SINGULAR AND
- Page 557 and 558: C.2. DIAGONAL, TRACE, SINGULAR AND
- Page 559 and 560: C.3. ORTHOGONAL PROCRUSTES PROBLEM
- Page 561 and 562: C.4. TWO-SIDED ORTHOGONAL PROCRUSTE
- Page 563 and 564: C.4. TWO-SIDED ORTHOGONAL PROCRUSTE
- Page 565 and 566: Appendix DMatrix calculusFrom too m
- Page 567 and 568: D.1. DIRECTIONAL DERIVATIVE, TAYLOR
- Page 569 and 570: D.1. DIRECTIONAL DERIVATIVE, TAYLOR
- Page 571 and 572: D.1. DIRECTIONAL DERIVATIVE, TAYLOR
- Page 573 and 574: D.1. DIRECTIONAL DERIVATIVE, TAYLOR
- Page 575 and 576: D.1. DIRECTIONAL DERIVATIVE, TAYLOR
- Page 577 and 578: D.1. DIRECTIONAL DERIVATIVE, TAYLOR
- Page 579 and 580: D.1. DIRECTIONAL DERIVATIVE, TAYLOR
- Page 581 and 582: D.1. DIRECTIONAL DERIVATIVE, TAYLOR
- Page 583 and 584: D.1. DIRECTIONAL DERIVATIVE, TAYLOR
- Page 585 and 586: D.1. DIRECTIONAL DERIVATIVE, TAYLOR
- Page 587 and 588: D.2. TABLES OF GRADIENTS AND DERIVA
- Page 589 and 590: D.2. TABLES OF GRADIENTS AND DERIVA
- Page 591 and 592: D.2. TABLES OF GRADIENTS AND DERIVA
- Page 593: D.2. TABLES OF GRADIENTS AND DERIVA
- Page 597 and 598: 597Equivalent to the corresponding
- Page 599 and 600: E.1. IDEMPOTENT MATRICES 599where R
- Page 601 and 602: E.1. IDEMPOTENT MATRICES 601TxT ⊥
- Page 603 and 604: E.2. I − P , PROJECTION ON ALGEBR
- Page 605 and 606: E.3. SYMMETRIC IDEMPOTENT MATRICES
- Page 607 and 608: E.3. SYMMETRIC IDEMPOTENT MATRICES
- Page 609 and 610: E.3. SYMMETRIC IDEMPOTENT MATRICES
- Page 611 and 612: E.5. PROJECTION EXAMPLES 611E.5.0.0
- Page 613 and 614: E.5. PROJECTION EXAMPLES 613of rela
- Page 615 and 616: E.5. PROJECTION EXAMPLES 615E.5.0.0
- Page 617 and 618: E.6. VECTORIZATION INTERPRETATION,
- Page 619 and 620: E.6. VECTORIZATION INTERPRETATION,
- Page 621 and 622: E.6. VECTORIZATION INTERPRETATION,
- Page 623 and 624: E.6. VECTORIZATION INTERPRETATION,
- Page 625 and 626: E.7. ON VECTORIZED MATRICES OF HIGH
- Page 627 and 628: E.7. ON VECTORIZED MATRICES OF HIGH
- Page 629 and 630: E.9. PROJECTION ON CONVEX SET 629E.
- Page 631 and 632: E.9. PROJECTION ON CONVEX SET 631
- Page 633 and 634: E.9. PROJECTION ON CONVEX SET 633Pr
- Page 635 and 636: E.9. PROJECTION ON CONVEX SET 635E.
- Page 637 and 638: E.9. PROJECTION ON CONVEX SET 637wh
- Page 639 and 640: E.9. PROJECTION ON CONVEX SET 639Un
- Page 641 and 642: E.9. PROJECTION ON CONVEX SET 641E.
- Page 643 and 644: E.10. ALTERNATING PROJECTION 643bC
- Page 645 and 646:
E.10. ALTERNATING PROJECTION 6450
- Page 647 and 648:
E.10. ALTERNATING PROJECTION 647E.1
- Page 649 and 650:
E.10. ALTERNATING PROJECTION 649y 2
- Page 651 and 652:
E.10. ALTERNATING PROJECTION 651By
- Page 653 and 654:
E.10. ALTERNATING PROJECTION 653Bar
- Page 655 and 656:
E.10. ALTERNATING PROJECTION 655bH
- Page 657 and 658:
E.10. ALTERNATING PROJECTION 657K
- Page 659 and 660:
E.10. ALTERNATING PROJECTION 659Whe
- Page 661 and 662:
Appendix FMatlab programsMade by Th
- Page 663 and 664:
F.1. ISEDM() 663% is nonnegativeif
- Page 665 and 666:
F.1. ISEDM() 665F.1.1Subroutines fo
- Page 667 and 668:
F.2. CONIC INDEPENDENCE, CONICI() 6
- Page 669 and 670:
F.2. CONIC INDEPENDENCE, CONICI() 6
- Page 671 and 672:
F.3. MAP OF THE USA 671% plot origi
- Page 673 and 674:
F.3. MAP OF THE USA 673statelat = d
- Page 675 and 676:
F.4. RANK REDUCTION SUBROUTINE, RRF
- Page 677 and 678:
F.4. RANK REDUCTION SUBROUTINE, RRF
- Page 679 and 680:
F.5. STURM’S PROCEDURE 679F.5 Stu
- Page 681 and 682:
F.6. CONVEX ITERATION DEMONSTRATION
- Page 683 and 684:
F.6. CONVEX ITERATION DEMONSTRATION
- Page 685 and 686:
F.7. FAST MAX CUT 685endoldtrace =
- Page 687 and 688:
F.8. SIGNAL DROPOUT PROBLEM 687whil
- Page 689 and 690:
Appendix GNotation and a few defini
- Page 691 and 692:
691l.i.w.r.tlinearly independentwit
- Page 693 and 694:
693t → 0 +t goes to 0 from above;
- Page 695 and 696:
695ψ(Z)DDD T (X)D(X) TD −1 (X)D(
- Page 697 and 698:
697R n −or R n×n−S nS n⊥S n
- Page 699 and 700:
699∂H −∂H +da supporting hype
- Page 701 and 702:
701maximizexargsup Xarg supf(x)subj
- Page 703 and 704:
703⊁≥not positive definitescala
- Page 705 and 706:
Bibliography[1] Suliman Al-Homidan
- Page 707 and 708:
BIBLIOGRAPHY 707[16] Keith Ball. An
- Page 709 and 710:
BIBLIOGRAPHY 709[38] A. W. Bojanczy
- Page 711 and 712:
BIBLIOGRAPHY 711[57] Steven Chu. Au
- Page 713 and 714:
BIBLIOGRAPHY 713[76] Frank R. Deuts
- Page 715 and 716:
BIBLIOGRAPHY 715(AACC), June 2004.h
- Page 717 and 718:
BIBLIOGRAPHY 717[114] John Clifford
- Page 719 and 720:
BIBLIOGRAPHY 719[135] Uwe Helmke an
- Page 721 and 722:
BIBLIOGRAPHY 721[157] Joakim Jaldé
- Page 723 and 724:
BIBLIOGRAPHY 723[178] Jung Rye Lee.
- Page 725 and 726:
BIBLIOGRAPHY 725[200] T. S. Motzkin
- Page 727 and 728:
BIBLIOGRAPHY 727[220] Teemu Pennane
- Page 729 and 730:
BIBLIOGRAPHY 729[240] Joshua A. Sin
- Page 731 and 732:
BIBLIOGRAPHY 731[264] Warren S. Tor
- Page 733 and 734:
BIBLIOGRAPHY 733[285] Bernard Widro
- Page 735 and 736:
Index0-norm, 241, 273, 275, 2941-no
- Page 737 and 738:
INDEX 737clipping, 190, 454, 638, 6
- Page 739 and 740:
INDEX 739hull, 53, 56, 306of outer
- Page 741 and 742:
INDEX 741unique, 519elbow, 457, 458
- Page 743 and 744:
INDEX 743polarity, 62handoff, 335ov
- Page 745 and 746:
INDEX 745manifold, 23, 24, 105, 410
- Page 747 and 748:
INDEX 747objectivebilinear, 256, 27
- Page 749 and 750:
INDEX 749Hadamard, 46, 472, 571, 69
- Page 751 and 752:
INDEX 751trace, 477, 478heuristic,
- Page 753 and 754:
INDEX 753symmetric hollow, 51tangen
- Page 755 and 756:
INDEX 755matrix, 65slack, 227vec, 4
- Page 757 and 758:
757
- Page 760:
Convex Optimization & Euclidean Dis
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