v2006.03.09 - Convex Optimization

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18 CONVEX OPTIMIZATION & EUCLIDEAN DISTANCE GEOMETRY4.8.3.1 Tightening the triangle inequality . . . . . . . 2684.8.4 Affine dimension reduction in two dimensions . . . . . 2694.9 Bridge: <strong>Convex</strong> polyhedra to EDMs . . . . . . . . . . . . . . . 2704.9.1 Geometric arguments . . . . . . . . . . . . . . . . . . . 2724.9.2 Necessity and sufficiency . . . . . . . . . . . . . . . . . 2754.9.3 Example revisited . . . . . . . . . . . . . . . . . . . . . 2754.10 EDM-entry composition . . . . . . . . . . . . . . . . . . . . . 2764.10.1 EDM by elliptope . . . . . . . . . . . . . . . . . . . . . 2784.11 EDM indefiniteness . . . . . . . . . . . . . . . . . . . . . . . . 2784.11.1 EDM eigenvalues, congruence transformation . . . . . . 2794.11.2 Spectral cones . . . . . . . . . . . . . . . . . . . . . . . 2804.11.2.1 Cayley-Menger versus Schoenberg . . . . . . 2814.11.2.2 Ordered eigenvalues . . . . . . . . . . . . . . 2824.11.2.3 Unordered eigenvalues . . . . . . . . . . . . . 2844.11.2.4 Dual cone versus dual spectral cone . . . . . 2854.12 List reconstruction . . . . . . . . . . . . . . . . . . . . . . . . 2864.12.1 x 1 at the origin. V N . . . . . . . . . . . . . . . . . . . 2864.12.2 0 geometric center. V . . . . . . . . . . . . . . . . . . 2884.13 Reconstruction examples . . . . . . . . . . . . . . . . . . . . . 2904.13.1 Isometric reconstruction . . . . . . . . . . . . . . . . . 2904.13.2 Isotonic reconstruction . . . . . . . . . . . . . . . . . . 2904.13.2.1 Isotonic map of the USA . . . . . . . . . . . . 2924.13.2.2 Isotonic solution with sort constraint . . . . . 2944.13.2.3 Convergence . . . . . . . . . . . . . . . . . . 2954.13.2.4 Interlude . . . . . . . . . . . . . . . . . . . . 2964.14 Fifth property of Euclidean metric . . . . . . . . . . . . . . . 2974.14.1 Recapitulate . . . . . . . . . . . . . . . . . . . . . . . . 2974.14.2 Derivation of the fifth . . . . . . . . . . . . . . . . . . 2974.14.2.1 Nonnegative determinant . . . . . . . . . . . 2984.14.2.2 Beyond the fifth metric property . . . . . . . 3004.14.3 Path not followed . . . . . . . . . . . . . . . . . . . . . 3004.14.3.1 N = 4 . . . . . . . . . . . . . . . . . . . . . . 3014.14.3.2 N = 5 . . . . . . . . . . . . . . . . . . . . . . 3024.14.3.3 Volume of simplices . . . . . . . . . . . . . . 3024.14.4 Affine dimension reduction in three dimensions . . . . . 3044.14.4.1 Exemplum redux . . . . . . . . . . . . . . . . 304
CONVEX OPTIMIZATION & EUCLIDEAN DISTANCE GEOMETRY 195 EDM cone 3055.1 Defining the EDM cone . . . . . . . . . . . . . . . . . . . . . . 3065.2 √Polyhedral bounds . . . . . . . . . . . . . . . . . . . . . . . . 3085.3 EDM cone is not convex . . . . . . . . . . . . . . . . . . . . 3105.4 a geometry of completion . . . . . . . . . . . . . . . . . . . . . 3125.5 EDM definition in 11 T . . . . . . . . . . . . . . . . . . . . . . 3175.5.1 Range of EDM D . . . . . . . . . . . . . . . . . . . . . 3185.5.2 Boundary constituents of EDM cone . . . . . . . . . . 3205.5.3 Faces of EDM cone . . . . . . . . . . . . . . . . . . . . 3235.5.3.1 Isomorphic faces . . . . . . . . . . . . . . . . 3235.5.3.2 Extreme direction of EDM cone . . . . . . . . 3235.5.3.3 Smallest face . . . . . . . . . . . . . . . . . . 3255.5.3.4 Open question . . . . . . . . . . . . . . . . . 3255.6 Correspondence to PSD cone S N−1+ . . . . . . . . . . . . . . . 3265.6.1 Gram-form correspondence to S N−1+ . . . . . . . . . . . 3285.6.2 EDM cone by elliptope . . . . . . . . . . . . . . . . . . 3285.7 Vectorization & projection interpretation . . . . . . . . . . . . 3315.7.1 Face of PSD cone S N + containing V . . . . . . . . . . . 3325.7.2 EDM criteria in 11 T . . . . . . . . . . . . . . . . . . . 3355.8 Dual EDM cone . . . . . . . . . . . . . . . . . . . . . . . . . . 3365.8.1 Ambient S N . . . . . . . . . . . . . . . . . . . . . . . . 3365.8.1.1 EDM cone and its dual in ambient S N . . . . 3375.8.1.2 nonnegative orthant . . . . . . . . . . . . . . 3425.8.1.3 Dual Euclidean distance matrix criterion . . . 3435.8.1.4 Nonorthogonal components of dual EDM . . . 3455.8.1.5 Affine dimension complementarity . . . . . . 3455.8.1.6 EDM cone duality . . . . . . . . . . . . . . . 3475.8.1.7 Schoenberg criterion is membership relation . 3485.8.2 Ambient S N h . . . . . . . . . . . . . . . . . . . . . . . . 3495.9 Theorem of the alternative . . . . . . . . . . . . . . . . . . . . 3505.10 postscript . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3526 Semidefinite programming 3536.1 Conic problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 3546.1.1 Maximal complementarity . . . . . . . . . . . . . . . . 3566.1.1.1 Reduced-rank solution . . . . . . . . . . . . . 3566.1.1.2 Coexistence of low- and high-rank solutions . 3586.1.1.3 Previous work . . . . . . . . . . . . . . . . . . 361
Page 1 and 2: DATTORROCONVEXOPTIMIZATION&EUCLIDEA
Page 3 and 4: Mεβoo Publishing
Page 5 and 6: Convex Optimization&Euclidean Dista
Page 9 and 10: for Jennie Columba♦Antonio♦♦&
Page 11 and 12: PreludeThe constant demands of my d
Page 15 and 16: CONVEX OPTIMIZATION & EUCLIDEAN DIS
Page 17: CONVEX OPTIMIZATION & EUCLIDEAN DIS
Page 27 and 28: List of Figures1 Overview 331 Cocoo
Page 29 and 30: LIST OF FIGURES 294 Euclidean Dista
Page 31: List of Tables2 Convex GeometryTabl
Page 34 and 35: 34 CHAPTER 1. OVERVIEWFigure 1: Coc
Page 36 and 37: 36 CHAPTER 1. OVERVIEWAbsolute posi
Page 38 and 39: 38 CHAPTER 1. OVERVIEWFigure 4: Thi
Page 40 and 41: 40 CHAPTER 1. OVERVIEWoriginalrecon
Page 42 and 43: 42 CHAPTER 1. OVERVIEWWe characteri
Page 44 and 45: 44 CHAPTER 1. OVERVIEWEight appendi
Page 46 and 47: 46 CHAPTER 2. CONVEX GEOMETRY2.1 Co
Page 48 and 49: 48 CHAPTER 2. CONVEX GEOMETRY2.1.3
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Page 52 and 53: 52 CHAPTER 2. CONVEX GEOMETRYwhere
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68 CHAPTER 2. CONVEX GEOMETRY2.3.2
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70 CHAPTER 2. CONVEX GEOMETRY2.3.2.
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72 CHAPTER 2. CONVEX GEOMETRYBy con
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74 CHAPTER 2. CONVEX GEOMETRYH −
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78 CHAPTER 2. CONVEX GEOMETRYA 1A 2
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80 CHAPTER 2. CONVEX GEOMETRYCH −
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84 CHAPTER 2. CONVEX GEOMETRYMany o
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90 CHAPTER 2. CONVEX GEOMETRYABCDFi
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94 CHAPTER 2. CONVEX GEOMETRYX(a)00
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96 CHAPTER 2. CONVEX GEOMETRYXXFigu
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98 CHAPTER 2. CONVEX GEOMETRYFamili
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100 CHAPTER 2. CONVEX GEOMETRYThe v
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102 CHAPTER 2. CONVEX GEOMETRYPrope
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104 CHAPTER 2. CONVEX GEOMETRYthe s
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108 CHAPTER 2. CONVEX GEOMETRYFrom
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110 CHAPTER 2. CONVEX GEOMETRYThe o
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112 CHAPTER 2. CONVEX GEOMETRYCXFig
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114 CHAPTER 2. CONVEX GEOMETRYis a
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118 CHAPTER 2. CONVEX GEOMETRYmatri
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122 CHAPTER 2. CONVEX GEOMETRY0-1-0
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124 CHAPTER 2. CONVEX GEOMETRYposit
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130 CHAPTER 2. CONVEX GEOMETRY000(a
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132 CHAPTER 2. CONVEX GEOMETRYK(a)K
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134 CHAPTER 2. CONVEX GEOMETRYHx 2x
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136 CHAPTER 2. CONVEX GEOMETRY2.12
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138 CHAPTER 2. CONVEX GEOMETRYsubsp
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140 CHAPTER 2. CONVEX GEOMETRY2.12.
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142 CHAPTER 2. CONVEX GEOMETRYS= co
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144 CHAPTER 2. CONVEX GEOMETRY
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146 CHAPTER 2. CONVEX GEOMETRYis a
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148 CHAPTER 2. CONVEX GEOMETRYR 2 R
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150 CHAPTER 2. CONVEX GEOMETRYwhich
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152 CHAPTER 2. CONVEX GEOMETRYK is
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154 CHAPTER 2. CONVEX GEOMETRYKK
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156 CHAPTER 2. CONVEX GEOMETRYWhen
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158 CHAPTER 2. CONVEX GEOMETRYAy
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160 CHAPTER 2. CONVEX GEOMETRYwhere
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168 CHAPTER 2. CONVEX GEOMETRYΓ 4
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170 CHAPTER 2. CONVEX GEOMETRYCoord
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172 CHAPTER 2. CONVEX GEOMETRYunder
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174 CHAPTER 2. CONVEX GEOMETRYWhen
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176 CHAPTER 2. CONVEX GEOMETRYFor e
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188 CHAPTER 2. CONVEX GEOMETRYwhere
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190 CHAPTER 2. CONVEX GEOMETRY
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192 CHAPTER 3. GEOMETRY OF CONVEX F
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216 CHAPTER 4. EUCLIDEAN DISTANCE M
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306 CHAPTER 5. EDM CONE5.1 Defining
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308 CHAPTER 5. EDM CONEThis cone is
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310 CHAPTER 5. EDM CONEN = 4. Relat
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312 CHAPTER 5. EDM CONE5.4 a geomet
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314 CHAPTER 5. EDM CONE(a)(b)Figure
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316 CHAPTER 5. EDM CONEFigure 78: T
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318 CHAPTER 5. EDM CONEProof. Next
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320 CHAPTER 5. EDM CONETo prove tha
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322 CHAPTER 5. EDM CONE{D ∈ EDM N
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324 CHAPTER 5. EDM CONED = δ ( −
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326 CHAPTER 5. EDM CONE5.6 Correspo
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328 CHAPTER 5. EDM CONE5.6.0.0.2 Ex
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330 CHAPTER 5. EDM CONE5.6.2.0.1 Ex
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332 CHAPTER 5. EDM CONEThe set of a
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334 CHAPTER 5. EDM CONEsvec ∂ S 2
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336 CHAPTER 5. EDM CONEWhen the Fin
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338 CHAPTER 5. EDM CONEProof. First
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340 CHAPTER 5. EDM CONEEDM 2 = S 2
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342 CHAPTER 5. EDM CONEwhose veraci
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344 CHAPTER 5. EDM CONED ◦ = δ(D
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346 CHAPTER 5. EDM CONEcone; id est
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348 CHAPTER 5. EDM CONE5.8.1.7 Scho
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350 CHAPTER 5. EDM CONETo find the
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352 CHAPTER 5. EDM CONEWhen D is an
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354 CHAPTER 6. SEMIDEFINITE PROGRAM
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388 CHAPTER 7. EDM PROXIMITY7.0.1 M
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390 CHAPTER 7. EDM PROXIMITY......
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392 CHAPTER 7. EDM PROXIMITYAny pro
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394 CHAPTER 7. EDM PROXIMITYSubstit
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396 CHAPTER 7. EDM PROXIMITY7.1.1 C
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398 CHAPTER 7. EDM PROXIMITYdimensi
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400 CHAPTER 7. EDM PROXIMITYGiven
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402 CHAPTER 7. EDM PROXIMITYProblem
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404 CHAPTER 7. EDM PROXIMITYBecause
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406 CHAPTER 7. EDM PROXIMITY7.2.1 C
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408 CHAPTER 7. EDM PROXIMITY7.2.1.2
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412 CHAPTER 7. EDM PROXIMITYTo see
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414 CHAPTER 7. EDM PROXIMITY∑mini
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416 CHAPTER 7. EDM PROXIMITYcharact
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420 CHAPTER 7. EDM PROXIMITYwhere
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422 CHAPTER 7. EDM PROXIMITY
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424 APPENDIX A. LINEAR ALGEBRAA.1.1
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426 APPENDIX A. LINEAR ALGEBRA27. T
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428 APPENDIX A. LINEAR ALGEBRAonly
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430 APPENDIX A. LINEAR ALGEBRA(AB)
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434 APPENDIX A. LINEAR ALGEBRA(Fan)
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436 APPENDIX A. LINEAR ALGEBRAFor A
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438 APPENDIX A. LINEAR ALGEBRAFor A
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442 APPENDIX A. LINEAR ALGEBRAA.4 S
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448 APPENDIX A. LINEAR ALGEBRAA.5 e
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450 APPENDIX A. LINEAR ALGEBRAs i w
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454 APPENDIX A. LINEAR ALGEBRAΣq 2
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456 APPENDIX A. LINEAR ALGEBRAA.7 Z
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458 APPENDIX A. LINEAR ALGEBRAThe r
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460 APPENDIX A. LINEAR ALGEBRAThe p
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462 APPENDIX B. SIMPLE MATRICESB.1
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464 APPENDIX B. SIMPLE MATRICESB.1.
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468 APPENDIX B. SIMPLE MATRICESN(u
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470 APPENDIX B. SIMPLE MATRICESDue
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474 APPENDIX B. SIMPLE MATRICEShas
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476 APPENDIX B. SIMPLE MATRICESFigu
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480 APPENDIX C. SOME ANALYTICAL OPT
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C.5. TWO-SIDED ORTHOGONAL PROCRUSTE
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496 APPENDIX D. MATRIX CALCULUSThe
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498 APPENDIX D. MATRIX CALCULUSGrad
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500 APPENDIX D. MATRIX CALCULUSBeca
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502 APPENDIX D. MATRIX CALCULUSfrom
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504 APPENDIX D. MATRIX CALCULUSin m
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506 APPENDIX D. MATRIX CALCULUSFigu
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508 APPENDIX D. MATRIX CALCULUS⎡
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510 APPENDIX D. MATRIX CALCULUSD.1.
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514 APPENDIX D. MATRIX CALCULUSD.2
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516 APPENDIX D. MATRIX CALCULUSAlge
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518 APPENDIX D. MATRIX CALCULUSTrac
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522 APPENDIX D. MATRIX CALCULUS
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524 APPENDIX E. PROJECTIONThe follo
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526 APPENDIX E. PROJECTIONE.1.1Bior
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528 APPENDIX E. PROJECTIONTxT ⊥ R
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530 APPENDIX E. PROJECTIONE.2 I−P
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532 APPENDIX E. PROJECTIONIn any ca
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534 APPENDIX E. PROJECTIONProof. We
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536 APPENDIX E. PROJECTIONthe direc
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538 APPENDIX E. PROJECTIONand defin
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540 APPENDIX E. PROJECTIONnonorthog
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542 APPENDIX E. PROJECTIONwhere y p
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544 APPENDIX E. PROJECTIONE.6.2Nono
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546 APPENDIX E. PROJECTIONE.6.3Orth
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548 APPENDIX E. PROJECTIONE.6.4.1.1
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550 APPENDIX E. PROJECTIONBy (1537)
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552 APPENDIX E. PROJECTIONAs for al
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554 APPENDIX E. PROJECTIONcharacter
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558 APPENDIX E. PROJECTION∂H −C
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562 APPENDIX E. PROJECTIONNow assum
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564 APPENDIX E. PROJECTIONProof. [3
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566 APPENDIX E. PROJECTION❇❇❇
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568 APPENDIX E. PROJECTIONE.10 Alte
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570 APPENDIX E. PROJECTIONbH 1H 2P
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572 APPENDIX E. PROJECTIONa(a)C 1C
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574 APPENDIX E. PROJECTION(a feasib
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576 APPENDIX E. PROJECTIONwhile, th
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578 APPENDIX E. PROJECTIONThe dista
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580 APPENDIX E. PROJECTION⎡⎢⎣
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582 APPENDIX E. PROJECTIONK ⊥ H 1
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584 APPENDIX E. PROJECTION
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586 APPENDIX F. PROOF OF EDM COMPOS
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588 APPENDIX F. PROOF OF EDM COMPOS
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590 APPENDIX G. MATLAB PROGRAMSif n
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592 APPENDIX G. MATLAB PROGRAMSend%
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594 APPENDIX G. MATLAB PROGRAMSG.1.
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596 APPENDIX G. MATLAB PROGRAMScoun
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598 APPENDIX G. MATLAB PROGRAMSG.3
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602 APPENDIX G. MATLAB PROGRAMSf(id
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604 APPENDIX G. MATLAB PROGRAMS%try
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608 APPENDIX H. NOTATION AND A FEW
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622 BIBLIOGRAPHY[7] Abdo Y. Alfakih
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624 BIBLIOGRAPHY[29] Pratik Biswas,
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626 BIBLIOGRAPHY[51] Frank Critchle
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628 BIBLIOGRAPHY[70] Ivar Ekeland a
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630 BIBLIOGRAPHY[91] John Clifford
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632 BIBLIOGRAPHYnew software CANDID
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634 BIBLIOGRAPHY[133] Florian Jarre
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636 BIBLIOGRAPHY[155] David G. Luen
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638 BIBLIOGRAPHY[178] Brad Osgood.
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640 BIBLIOGRAPHY[198] Uwe Schäfer.
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642 BIBLIOGRAPHY[222] Akimichi Take
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644 BIBLIOGRAPHY[244] Eric W. Weiss
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646 BIBLIOGRAPHY
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