v2007.09.17 - 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 . . . . . . . . . . . . . 2565 Euclidean Distance Matrix 2915.1 EDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2925.2 First metric properties . . . . . . . . . . . . . . . . . . . . . . 2935.3 ∃ fifth Euclidean metric property . . . . . . . . . . . . . . . . 2935.4 EDM definition . . . . . . . . . . . . . . . . . . . . . . . . . . 2985.5 Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3285.6 Injectivity of D & unique reconstruction . . . . . . . . . . . . 3315.7 Embedding in affine hull . . . . . . . . . . . . . . . . . . . . . 3375.8 Euclidean metric versus matrix criteria . . . . . . . . . . . . . 3425.9 Bridge: <strong>Convex</strong> polyhedra to EDMs . . . . . . . . . . . . . . . 3505.10 EDM-entry composition . . . . . . . . . . . . . . . . . . . . . 3585.11 EDM indefiniteness . . . . . . . . . . . . . . . . . . . . . . . . 3605.12 List reconstruction . . . . . . . . . . . . . . . . . . . . . . . . 3685.13 Reconstruction examples . . . . . . . . . . . . . . . . . . . . . 3725.14 Fifth property of Euclidean metric . . . . . . . . . . . . . . . 3796 EDM cone 3896.1 Defining EDM cone . . . . . . . . . . . . . . . . . . . . . . . . 3916.2 Polyhedral bounds . . . . . . . . . . . . . . . . . . . . . . . . 3936.3√EDM cone is not convex . . . . . . . . . . . . . . . . . . . . 3956.4 a geometry of completion . . . . . . . . . . . . . . . . . . . . . 3976.5 EDM definition in 11 T . . . . . . . . . . . . . . . . . . . . . . 4026.6 Correspondence to PSD cone S N−1+ . . . . . . . . . . . . . . . 4116.7 Vectorization & projection interpretation . . . . . . . . . . . . 4166.8 Dual EDM cone . . . . . . . . . . . . . . . . . . . . . . . . . . 4216.9 Theorem of the alternative . . . . . . . . . . . . . . . . . . . . 4356.10 postscript . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4377 Proximity problems 4397.1 First prevalent problem: . . . . . . . . . . . . . . . . . . . . . 4477.2 Second prevalent problem: . . . . . . . . . . . . . . . . . . . . 4587.3 Third prevalent problem: . . . . . . . . . . . . . . . . . . . . . 469
CONVEX OPTIMIZATION & EUCLIDEAN DISTANCE GEOMETRY 117.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479A Linear algebra 481A.1 Main-diagonal δ operator, λ , trace, vec . . . . . . . . . . . . 481A.2 Semidefiniteness: domain of test . . . . . . . . . . . . . . . . . 485A.3 Proper statements . . . . . . . . . . . . . . . . . . . . . . . . . 488A.4 Schur complement . . . . . . . . . . . . . . . . . . . . . . . . 500A.5 eigen decomposition . . . . . . . . . . . . . . . . . . . . . . . . 504A.6 Singular value decomposition, SVD . . . . . . . . . . . . . . . 507A.7 Zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512B Simple matrices 519B.1 Rank-one matrix (dyad) . . . . . . . . . . . . . . . . . . . . . 520B.2 Doublet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525B.3 Elementary matrix . . . . . . . . . . . . . . . . . . . . . . . . 526B.4 Auxiliary V -matrices . . . . . . . . . . . . . . . . . . . . . . . 528B.5 Orthogonal matrix . . . . . . . . . . . . . . . . . . . . . . . . 533C Some analytical optimal results 537C.1 properties of infima . . . . . . . . . . . . . . . . . . . . . . . . 537C.2 diagonal, trace, singular and eigen values . . . . . . . . . . . . 538C.3 Orthogonal Procrustes problem . . . . . . . . . . . . . . . . . 544C.4 Two-sided orthogonal Procrustes . . . . . . . . . . . . . . . . 546D Matrix calculus 551D.1 Directional derivative, Taylor series . . . . . . . . . . . . . . . 551D.2 Tables of gradients and derivatives . . . . . . . . . . . . . . . 572E Projection 581E.1 Idempotent matrices . . . . . . . . . . . . . . . . . . . . . . . 584E.2 I − P , Projection on algebraic complement . . . . . . . . . . . 589E.3 Symmetric idempotent matrices . . . . . . . . . . . . . . . . . 590E.4 Algebra of projection on affine subsets . . . . . . . . . . . . . 596E.5 Projection examples . . . . . . . . . . . . . . . . . . . . . . . 596E.6 Vectorization interpretation, . . . . . . . . . . . . . . . . . . . 603E.7 on vectorized matrices of higher rank . . . . . . . . . . . . . . 610E.8 Range/Rowspace interpretation . . . . . . . . . . . . . . . . . 614E.9 Projection on convex set . . . . . . . . . . . . . . . . . . . . . 614
Page 2 and 3: DattorroCONVEXOPTIMIZATION&EUCLIDEA
Page 4 and 5: Meboo Publishing USA345 Stanford Sh
Page 6 and 7: EDM = S h ∩ ( S ⊥ c − S +)
Page 8 and 9: There is a great race under way to
Page 12 and 13: 12 CONVEX OPTIMIZATION & EUCLIDEAN
Page 14 and 15: 14 LIST OF FIGURES24 Nonconvex cone
Page 16 and 17: 16 LIST OF FIGURES92 Largest ten ei
Page 18 and 19: List of Tables2 Convex geometryTabl
Page 20 and 21: 20 CHAPTER 1. OVERVIEWFigure 1: Ori
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Page 32 and 33: 32 CHAPTER 1. OVERVIEW
Page 34 and 35: 34 CHAPTER 2. CONVEX GEOMETRY2.1 Co
Page 36 and 37: 36 CHAPTER 2. CONVEX GEOMETRY2.1.3
Page 38 and 39: 38 CHAPTER 2. CONVEX GEOMETRY(a)R(b
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60 CHAPTER 2. CONVEX GEOMETRYFigure
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62 CHAPTER 2. CONVEX GEOMETRY2.4.1
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64 CHAPTER 2. CONVEX GEOMETRY11−1
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66 CHAPTER 2. CONVEX GEOMETRYHyperp
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68 CHAPTER 2. CONVEX GEOMETRYCH −
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70 CHAPTER 2. CONVEX GEOMETRYnonemp
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72 CHAPTER 2. CONVEX GEOMETRYto vec
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78 CHAPTER 2. CONVEX GEOMETRYABCDFi
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80 CHAPTER 2. CONVEX GEOMETRY2.6.1.
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82 CHAPTER 2. CONVEX GEOMETRYX(a)00
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84 CHAPTER 2. CONVEX GEOMETRYXXFigu
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86 CHAPTER 2. CONVEX GEOMETRYFamili
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88 CHAPTER 2. CONVEX GEOMETRYC 1C 2
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90 CHAPTER 2. CONVEX GEOMETRYA prop
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92 CHAPTER 2. CONVEX GEOMETRY∂K
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94 CHAPTER 2. CONVEX GEOMETRYWhen t
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96 CHAPTER 2. CONVEX GEOMETRYBCADFi
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98 CHAPTER 2. CONVEX GEOMETRYThe po
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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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110 CHAPTER 2. CONVEX GEOMETRYA con
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112 CHAPTER 2. CONVEX GEOMETRY0-1-0
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114 CHAPTER 2. CONVEX GEOMETRYposit
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118 CHAPTER 2. CONVEX GEOMETRYThe c
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120 CHAPTER 2. CONVEX GEOMETRYWhen
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122 CHAPTER 2. CONVEX GEOMETRY2.10.
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124 CHAPTER 2. CONVEX GEOMETRY{extr
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126 CHAPTER 2. CONVEX GEOMETRY2.11
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128 CHAPTER 2. CONVEX GEOMETRYFrom
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130 CHAPTER 2. CONVEX GEOMETRYS = {
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132 CHAPTER 2. CONVEX GEOMETRYFigur
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136 CHAPTER 2. CONVEX GEOMETRYK ∗
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138 CHAPTER 2. CONVEX GEOMETRYKK
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140 CHAPTER 2. CONVEX GEOMETRYthe p
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142 CHAPTER 2. CONVEX GEOMETRY(dual
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146 CHAPTER 2. CONVEX GEOMETRYfor w
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148 CHAPTER 2. CONVEX GEOMETRYBy al
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150 CHAPTER 2. CONVEX GEOMETRYb −
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154 CHAPTER 2. CONVEX GEOMETRYDual
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160 CHAPTER 2. CONVEX GEOMETRYΓ 4
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162 CHAPTER 2. CONVEX GEOMETRYEigen
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164 CHAPTER 2. CONVEX GEOMETRYunder
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166 CHAPTER 2. CONVEX GEOMETRYWhen
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168 CHAPTER 2. CONVEX GEOMETRYFor e
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170 CHAPTER 2. CONVEX GEOMETRY10.80
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172 CHAPTER 2. CONVEX GEOMETRYx 210
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174 CHAPTER 2. CONVEX GEOMETRYwhile
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176 CHAPTER 2. CONVEX GEOMETRYαα
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178 CHAPTER 2. CONVEX GEOMETRYFrom
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182 CHAPTER 2. CONVEX GEOMETRYhavin
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184 CHAPTER 3. GEOMETRY OF CONVEX F
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226 CHAPTER 4. SEMIDEFINITE PROGRAM
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292 CHAPTER 5. EUCLIDEAN DISTANCE M
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390 CHAPTER 6. EDM CONEa resemblanc
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392 CHAPTER 6. EDM CONEdvec rel∂E
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394 CHAPTER 6. EDM CONE(b)(c)dvec r
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396 CHAPTER 6. EDM CONE(a)2 nearest
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398 CHAPTER 6. EDM CONEthe graph. T
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400 CHAPTER 6. EDM CONEwhere e i
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402 CHAPTER 6. EDM CONE6.5 EDM defi
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404 CHAPTER 6. EDM CONEN(1 T )δ(V
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406 CHAPTER 6. EDM CONE10(a)-110-1V
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408 CHAPTER 6. EDM CONE6.5.3 Faces
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410 CHAPTER 6. EDM CONE6.5.3.2 Smal
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412 CHAPTER 6. EDM CONE6.6.0.0.1 Pr
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414 CHAPTER 6. EDM CONEdvec rel∂E
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416 CHAPTER 6. EDM CONE6.7 Vectoriz
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418 CHAPTER 6. EDM CONEsvec ∂ S 2
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420 CHAPTER 6. EDM CONEIn fact, the
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422 CHAPTER 6. EDM CONEThe ordinary
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424 CHAPTER 6. EDM CONEEDM 2 = S 2
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426 CHAPTER 6. EDM CONEFrom the res
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428 CHAPTER 6. EDM CONE6.8.1.3 Dual
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430 CHAPTER 6. EDM CONED ◦ = δ(D
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432 CHAPTER 6. EDM CONE6.8.1.6 EDM
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434 CHAPTER 6. EDM CONEBecause 〈
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436 CHAPTER 6. EDM CONE0dvec rel∂
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438 CHAPTER 6. EDM CONE
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440 CHAPTER 7. PROXIMITY PROBLEMS7.
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442 CHAPTER 7. PROXIMITY PROBLEMS..
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444 CHAPTER 7. PROXIMITY PROBLEMSTh
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446 CHAPTER 7. PROXIMITY PROBLEMSwh
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448 CHAPTER 7. PROXIMITY PROBLEMSpo
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452 CHAPTER 7. PROXIMITY PROBLEMSof
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456 CHAPTER 7. PROXIMITY PROBLEMSR
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458 CHAPTER 7. PROXIMITY PROBLEMSwh
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462 CHAPTER 7. PROXIMITY PROBLEMSCo
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466 CHAPTER 7. PROXIMITY PROBLEMSTo
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468 CHAPTER 7. PROXIMITY PROBLEMSth
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470 CHAPTER 7. PROXIMITY PROBLEMSco
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472 CHAPTER 7. PROXIMITY PROBLEMSto
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476 CHAPTER 7. PROXIMITY PROBLEMSdi
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478 CHAPTER 7. PROXIMITY PROBLEMSve
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480 CHAPTER 7. PROXIMITY PROBLEMSth
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482 APPENDIX A. LINEAR ALGEBRAA.1.1
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486 APPENDIX A. LINEAR ALGEBRAonly
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488 APPENDIX A. LINEAR ALGEBRA(AB)
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492 APPENDIX A. LINEAR ALGEBRAFor A
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494 APPENDIX A. LINEAR ALGEBRADiago
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496 APPENDIX A. LINEAR ALGEBRAFor A
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500 APPENDIX A. LINEAR ALGEBRAA.4 S
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504 APPENDIX A. LINEAR ALGEBRAA.5 e
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506 APPENDIX A. LINEAR ALGEBRAs i w
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510 APPENDIX A. LINEAR ALGEBRAΣq 2
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512 APPENDIX A. LINEAR ALGEBRAA.7 Z
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514 APPENDIX A. LINEAR ALGEBRAThere
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518 APPENDIX A. LINEAR ALGEBRA
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520 APPENDIX B. SIMPLE MATRICESB.1
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522 APPENDIX B. SIMPLE MATRICESProo
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524 APPENDIX B. SIMPLE MATRICESB.1.
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526 APPENDIX B. SIMPLE MATRICESN(u
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528 APPENDIX B. SIMPLE MATRICESDue
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532 APPENDIX B. SIMPLE MATRICEShas
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534 APPENDIX B. SIMPLE MATRICESFigu
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538 APPENDIX C. SOME ANALYTICAL OPT
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552 APPENDIX D. MATRIX CALCULUSThe
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554 APPENDIX D. MATRIX CALCULUSGrad
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556 APPENDIX D. MATRIX CALCULUSBeca
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558 APPENDIX D. MATRIX CALCULUSwhic
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560 APPENDIX D. MATRIX CALCULUS⎡
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562 APPENDIX D. MATRIX CALCULUS→Y
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564 APPENDIX D. MATRIX CALCULUSD.1.
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566 APPENDIX D. MATRIX CALCULUSwhic
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568 APPENDIX D. MATRIX CALCULUSIn t
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572 APPENDIX D. MATRIX CALCULUSD.2
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574 APPENDIX D. MATRIX CALCULUSalge
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576 APPENDIX D. MATRIX CALCULUStrac
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580 APPENDIX D. MATRIX CALCULUS
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582 APPENDIX E. PROJECTIONThe follo
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584 APPENDIX E. PROJECTIONFor matri
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586 APPENDIX E. PROJECTION(⇐) To
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588 APPENDIX E. PROJECTIONNonorthog
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590 APPENDIX E. PROJECTIONE.2.0.0.1
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592 APPENDIX E. PROJECTIONE.3.2Orth
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594 APPENDIX E. PROJECTIONE.3.5Unif
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596 APPENDIX E. PROJECTIONE.4 Algeb
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598 APPENDIX E. PROJECTIONa ∗ 2K
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600 APPENDIX E. PROJECTIONwhere Y =
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602 APPENDIX E. PROJECTION(B.4.2).
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604 APPENDIX E. PROJECTIONis a nono
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606 APPENDIX E. PROJECTIONE.6.4.1Or
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608 APPENDIX E. PROJECTIONq i q T i
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610 APPENDIX E. PROJECTIONThe test
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612 APPENDIX E. PROJECTIONPerpendic
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614 APPENDIX E. PROJECTIONE.8 Range
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616 APPENDIX E. PROJECTIONAs for su
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618 APPENDIX E. PROJECTIONWith refe
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620 APPENDIX E. PROJECTIONProjectio
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622 APPENDIX E. PROJECTIONE.9.2.2.2
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624 APPENDIX E. PROJECTIONThe foreg
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626 APPENDIX E. PROJECTION❇❇❇
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628 APPENDIX E. PROJECTIONE.10 Alte
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630 APPENDIX E. PROJECTIONbH 1H 2P
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632 APPENDIX E. PROJECTIONa(a){y |
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634 APPENDIX E. PROJECTION(a feasib
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636 APPENDIX E. PROJECTIONwhile, th
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638 APPENDIX E. PROJECTIONE.10.2.1.
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640 APPENDIX E. PROJECTION10 0dist(
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642 APPENDIX E. PROJECTIONE.10.3.1D
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644 APPENDIX E. PROJECTIONE 3K ⊥
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646 APPENDIX E. PROJECTION
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648 APPENDIX F. MATLAB PROGRAMSif n
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650 APPENDIX F. MATLAB PROGRAMSend%
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652 APPENDIX F. MATLAB PROGRAMSF.1.
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654 APPENDIX F. MATLAB PROGRAMScoun
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656 APPENDIX F. MATLAB PROGRAMSF.3
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660 APPENDIX F. MATLAB PROGRAMS% so
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662 APPENDIX F. MATLAB PROGRAMS% tr
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666 APPENDIX F. MATLAB PROGRAMSbrea
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668 APPENDIX F. MATLAB PROGRAMSwhil
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670 APPENDIX F. MATLAB PROGRAMSF.7
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672 APPENDIX F. MATLAB PROGRAMS
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674 APPENDIX G. NOTATION AND A FEW
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690 BIBLIOGRAPHY[7] Abdo Y. Alfakih
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692 BIBLIOGRAPHY[27] Aharon Ben-Tal
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694 BIBLIOGRAPHY[48] Lev M. Brègma
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696 BIBLIOGRAPHY[67] Joel Dawson, S
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698 BIBLIOGRAPHY[85] Carl Eckart an
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700 BIBLIOGRAPHY[102] Laurent El Gh
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702 BIBLIOGRAPHY[124] Peter Gritzma
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704 BIBLIOGRAPHY[146] Jean-Baptiste
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706 BIBLIOGRAPHY[168] Jean B. Lasse
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708 BIBLIOGRAPHY[190] Rudolf Mathar
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710 BIBLIOGRAPHY[212] M. L. Overton
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712 BIBLIOGRAPHY[230] R. Tyrrell Ro
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714 BIBLIOGRAPHY[254] Jos F. Sturm
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716 BIBLIOGRAPHY[276] È. B. Vinber
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[295] Naoki Yamamoto and Maryam Faz
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720 INDEXobtuse, 62positive semidef
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722 INDEXnormal, 175, 415, 622, 642
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724 INDEXdiscretization, 152, 185,
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726 INDEXminimization, 206, 611full
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728 INDEXinvariant set, 359inversio
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730 INDEXunitary, 533maximalcomplem
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732 INDEXdifference, 118Farkas’ l
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734 INDEXcommutative, 426, 628direc
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736 INDEXset, 683unique, 42, 185, 1
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738 INDEXaffine, 44, 68, 86, 101, 2
show all

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