v2007.11.26 - Convex Optimization

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10 CONVEX OPTIMIZATION & EUCLIDEAN DISTANCE GEOMETRY4 Semidefinite programming 2254.1 Conic problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 2264.2 Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2344.3 Rank reduction . . . . . . . . . . . . . . . . . . . . . . . . . . 2474.4 Rank-constrained semidefinite program . . . . . . . . . . . . . 2564.5 Constraining cardinality . . . . . . . . . . . . . . . . . . . . . 2734.6 Cardinality and rank constraint examples . . . . . . . . . . . . 2824.7 <strong>Convex</strong> Iteration rank-1 . . . . . . . . . . . . . . . . . . . . . 3005 Euclidean Distance Matrix 3055.1 EDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3065.2 First metric properties . . . . . . . . . . . . . . . . . . . . . . 3075.3 ∃ fifth Euclidean metric property . . . . . . . . . . . . . . . . 3075.4 EDM definition . . . . . . . . . . . . . . . . . . . . . . . . . . 3125.5 Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3425.6 Injectivity of D & unique reconstruction . . . . . . . . . . . . 3455.7 Embedding in affine hull . . . . . . . . . . . . . . . . . . . . . 3515.8 Euclidean metric versus matrix criteria . . . . . . . . . . . . . 3565.9 Bridge: <strong>Convex</strong> polyhedra to EDMs . . . . . . . . . . . . . . . 3645.10 EDM-entry composition . . . . . . . . . . . . . . . . . . . . . 3725.11 EDM indefiniteness . . . . . . . . . . . . . . . . . . . . . . . . 3745.12 List reconstruction . . . . . . . . . . . . . . . . . . . . . . . . 3825.13 Reconstruction examples . . . . . . . . . . . . . . . . . . . . . 3865.14 Fifth property of Euclidean metric . . . . . . . . . . . . . . . 3936 Cone of distance matrices 4036.1 Defining EDM cone . . . . . . . . . . . . . . . . . . . . . . . . 4056.2√Polyhedral bounds . . . . . . . . . . . . . . . . . . . . . . . . 4076.3 EDM cone is not convex . . . . . . . . . . . . . . . . . . . . 4096.4 a geometry of completion . . . . . . . . . . . . . . . . . . . . . 4116.5 EDM definition in 11 T . . . . . . . . . . . . . . . . . . . . . . 4166.6 Correspondence to PSD cone S N−1+ . . . . . . . . . . . . . . . 4256.7 Vectorization & projection interpretation . . . . . . . . . . . . 4306.8 Dual EDM cone . . . . . . . . . . . . . . . . . . . . . . . . . . 4356.9 Theorem of the alternative . . . . . . . . . . . . . . . . . . . . 4496.10 postscript . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451
CONVEX OPTIMIZATION & EUCLIDEAN DISTANCE GEOMETRY 117 Proximity problems 4537.1 First prevalent problem: . . . . . . . . . . . . . . . . . . . . . 4617.2 Second prevalent problem: . . . . . . . . . . . . . . . . . . . . 4727.3 Third prevalent problem: . . . . . . . . . . . . . . . . . . . . . 4837.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493A Linear algebra 495A.1 Main-diagonal δ operator, λ , trace, vec . . . . . . . . . . . . 495A.2 Semidefiniteness: domain of test . . . . . . . . . . . . . . . . . 499A.3 Proper statements . . . . . . . . . . . . . . . . . . . . . . . . . 502A.4 Schur complement . . . . . . . . . . . . . . . . . . . . . . . . 514A.5 eigen decomposition . . . . . . . . . . . . . . . . . . . . . . . . 518A.6 Singular value decomposition, SVD . . . . . . . . . . . . . . . 521A.7 Zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526B Simple matrices 533B.1 Rank-one matrix (dyad) . . . . . . . . . . . . . . . . . . . . . 534B.2 Doublet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539B.3 Elementary matrix . . . . . . . . . . . . . . . . . . . . . . . . 540B.4 Auxiliary V -matrices . . . . . . . . . . . . . . . . . . . . . . . 542B.5 Orthogonal matrix . . . . . . . . . . . . . . . . . . . . . . . . 547C Some analytical optimal results 551C.1 properties of infima . . . . . . . . . . . . . . . . . . . . . . . . 551C.2 diagonal, trace, singular and eigen values . . . . . . . . . . . . 552C.3 Orthogonal Procrustes problem . . . . . . . . . . . . . . . . . 558C.4 Two-sided orthogonal Procrustes . . . . . . . . . . . . . . . . 560D Matrix calculus 565D.1 Directional derivative, Taylor series . . . . . . . . . . . . . . . 565D.2 Tables of gradients and derivatives . . . . . . . . . . . . . . . 586E Projection 595E.1 Idempotent matrices . . . . . . . . . . . . . . . . . . . . . . . 598E.2 I − P , Projection on algebraic complement . . . . . . . . . . . 603E.3 Symmetric idempotent matrices . . . . . . . . . . . . . . . . . 604E.4 Algebra of projection on affine subsets . . . . . . . . . . . . . 610E.5 Projection examples . . . . . . . . . . . . . . . . . . . . . . . 610
Page 1 and 2: DATTORROCONVEXOPTIMIZATION&EUCLIDEA
Page 3 and 4: Convex Optimization&Euclidean Dista
Page 5 and 6: for Jennie Columba♦Antonio♦♦&
Page 7 and 8: PreludeThe constant demands of my d
Page 9: Convex Optimization&Euclidean Dista
Page 13 and 14: List of Figures1 Overview 191 Orion
Page 15 and 16: LIST OF FIGURES 1559 Epigraph . . .
Page 17 and 18: LIST OF FIGURES 17120 Four fundamen
Page 19 and 20: Chapter 1OverviewConvex Optimizatio
Page 21 and 22: ˇx 4ˇx 3ˇx 2Figure 2: Applicatio
Page 23 and 24: 23Figure 4: This coarsely discretiz
Page 25 and 26: ases (biorthogonal expansion). We e
Page 27 and 28: 27Figure 7: These bees construct a
Page 29 and 30: its membership to the EDM cone. The
Page 31 and 32: 31appendicesProvided so as to be mo
Page 33 and 34: Chapter 2Convex geometryConvexity h
Page 35 and 36: 2.1. CONVEX SET 35Figure 11: A slab
Page 37 and 38: 2.1. CONVEX SET 372.1.6 empty set v
Page 39 and 40: 2.1. CONVEX SET 392.1.7.1 Line inte
Page 41 and 42: 2.1. CONVEX SET 41(a)R 2(b)R 3(c)(d
Page 43 and 44: 2.1. CONVEX SET 43This theorem in c
Page 45 and 46: 2.2. VECTORIZED-MATRIX INNER PRODUC
Page 53 and 54: 2.3. HULLS 53Figure 14: Convex hull
Page 55 and 56: 2.3. HULLS 55Aaffine hull (drawn tr
Page 57 and 58: 2.3. HULLS 57The union of relative
Page 59 and 60: 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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2.10. CONIC INDEPENDENCE (C.I.) 121
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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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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 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 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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4.5. 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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4.7. 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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5.7. 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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5.9. 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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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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6.7. 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.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 ≠
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B.4. AUXILIARY V -MATRICES 543the n
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B.4. AUXILIARY V -MATRICES 54518. V
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B.5. ORTHOGONAL MATRIX 547B.5 Ortho
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B.5. ORTHOGONAL MATRIX 549B.5.3.0.1
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Appendix CSome analytical optimal r
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C.2. DIAGONAL, TRACE, SINGULAR AND
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C.3. ORTHOGONAL PROCRUSTES PROBLEM
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C.4. TWO-SIDED ORTHOGONAL PROCRUSTE
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C.4. TWO-SIDED ORTHOGONAL PROCRUSTE
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Appendix 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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597Equivalent to the corresponding
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E.1. IDEMPOTENT MATRICES 599where R
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E.1. IDEMPOTENT MATRICES 601TxT ⊥
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E.2. I − P , PROJECTION ON ALGEBR
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E.3. SYMMETRIC IDEMPOTENT MATRICES
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E.5. PROJECTION EXAMPLES 611E.5.0.0
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E.5. PROJECTION EXAMPLES 613of rela
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E.5. PROJECTION EXAMPLES 615E.5.0.0
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E.6. VECTORIZATION INTERPRETATION,
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E.7. ON VECTORIZED MATRICES OF HIGH
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E.7. ON VECTORIZED MATRICES OF HIGH
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E.9. PROJECTION ON CONVEX SET 629E.
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E.9. PROJECTION ON CONVEX SET 631
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E.9. PROJECTION ON CONVEX SET 633Pr
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E.9. PROJECTION ON CONVEX SET 637wh
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E.9. PROJECTION ON CONVEX SET 639Un
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E.10. ALTERNATING PROJECTION 643bC
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E.10. ALTERNATING PROJECTION 6450
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E.10. ALTERNATING PROJECTION 647E.1
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E.10. ALTERNATING PROJECTION 649y 2
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E.10. ALTERNATING PROJECTION 651By
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E.10. ALTERNATING PROJECTION 653Bar
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E.10. ALTERNATING PROJECTION 655bH
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E.10. ALTERNATING PROJECTION 657K
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E.10. ALTERNATING PROJECTION 659Whe
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Appendix FMatlab programsMade by Th
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F.1. ISEDM() 663% is nonnegativeif
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F.1. ISEDM() 665F.1.1Subroutines fo
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F.2. CONIC INDEPENDENCE, CONICI() 6
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F.2. CONIC INDEPENDENCE, CONICI() 6
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F.3. MAP OF THE USA 671% plot origi
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F.3. MAP OF THE USA 673statelat = d
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F.4. RANK REDUCTION SUBROUTINE, RRF
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F.4. RANK REDUCTION SUBROUTINE, RRF
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F.5. STURM’S PROCEDURE 679F.5 Stu
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F.6. CONVEX ITERATION DEMONSTRATION
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F.6. CONVEX ITERATION DEMONSTRATION
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F.7. FAST MAX CUT 685endoldtrace =
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F.8. SIGNAL DROPOUT PROBLEM 687whil
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Appendix GNotation and a few defini
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691l.i.w.r.tlinearly independentwit
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693t → 0 +t goes to 0 from above;
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695ψ(Z)DDD T (X)D(X) TD −1 (X)D(
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697R n −or R n×n−S nS n⊥S n
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699∂H −∂H +da supporting hype
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701maximizexargsup Xarg supf(x)subj
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703⊁≥not positive definitescala
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Bibliography[1] Suliman Al-Homidan
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BIBLIOGRAPHY 707[16] Keith Ball. An
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BIBLIOGRAPHY 709[38] A. W. Bojanczy
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BIBLIOGRAPHY 711[57] Steven Chu. Au
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BIBLIOGRAPHY 713[76] Frank R. Deuts
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BIBLIOGRAPHY 715(AACC), June 2004.h
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BIBLIOGRAPHY 717[114] John Clifford
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BIBLIOGRAPHY 719[135] Uwe Helmke an
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BIBLIOGRAPHY 721[157] Joakim Jaldé
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BIBLIOGRAPHY 723[178] Jung Rye Lee.
Page 725 and 726:
BIBLIOGRAPHY 725[200] T. S. Motzkin
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BIBLIOGRAPHY 727[220] Teemu Pennane
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BIBLIOGRAPHY 729[240] Joshua A. Sin
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BIBLIOGRAPHY 731[264] Warren S. Tor
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BIBLIOGRAPHY 733[285] Bernard Widro
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Index0-norm, 241, 273, 275, 2941-no
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INDEX 737clipping, 190, 454, 638, 6
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INDEX 739hull, 53, 56, 306of outer
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INDEX 741unique, 519elbow, 457, 458
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INDEX 743polarity, 62handoff, 335ov
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INDEX 745manifold, 23, 24, 105, 410
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INDEX 747objectivebilinear, 256, 27
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INDEX 749Hadamard, 46, 472, 571, 69
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INDEX 751trace, 477, 478heuristic,
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INDEX 753symmetric hollow, 51tangen
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INDEX 755matrix, 65slack, 227vec, 4
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757
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Convex Optimization & Euclidean Dis
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