v2006.03.09 - Convex Optimization

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220 CHAPTER 4. EUCLIDEAN DISTANCE MATRIX4.3.1 LookaheadThere must exist at least one requirement more than the four properties ofthe Euclidean metric that makes them altogether necessary and sufficient tocertify realizability of bounded convex polyhedra. Indeed, there are infinitelymany more; there are precisely N + 1 necessary and sufficient Euclideanmetric requirements for N points constituting a generating list (2.3.2). Hereis the fifth requirement:4.3.1.0.1 Fifth Euclidean metric property. Relative-angle inequality.(confer4.14.2.1.1) Augmenting the four fundamental properties of theEuclidean metric in R n , for all i,j,l ≠ k ∈{1... N}, i
4.3. ∃ FIFTH EUCLIDEAN METRIC PROPERTY 221kθ iklθ ikjθ jkliljFigure 58: Nomenclature for fifth Euclidean metric property. Each angle θis made by a vector pair at vertex k while i,j,k,l index four points. Thefifth property is necessary for realization of four or more points reckoned bythree angles in any dimension.
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DATTORROCONVEXOPTIMIZATION&EUCLIDEA
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Mεβoo Publishing
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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 DIS
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List of Figures1 Overview 331 Cocoo
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LIST OF FIGURES 294 Euclidean Dista
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List of Tables2 Convex GeometryTabl
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34 CHAPTER 1. OVERVIEWFigure 1: Coc
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36 CHAPTER 1. OVERVIEWAbsolute posi
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38 CHAPTER 1. OVERVIEWFigure 4: Thi
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40 CHAPTER 1. OVERVIEWoriginalrecon
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42 CHAPTER 1. OVERVIEWWe characteri
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44 CHAPTER 1. OVERVIEWEight appendi
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46 CHAPTER 2. CONVEX GEOMETRY2.1 Co
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48 CHAPTER 2. CONVEX GEOMETRY2.1.3
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50 CHAPTER 2. CONVEX GEOMETRY(a)R(b
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52 CHAPTER 2. CONVEX GEOMETRYwhere
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54 CHAPTER 2. CONVEX GEOMETRYNow le
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58 CHAPTER 2. CONVEX GEOMETRYand wh
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60 CHAPTER 2. CONVEX GEOMETRY2.2.1.
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64 CHAPTER 2. CONVEX GEOMETRYAny ma
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66 CHAPTER 2. CONVEX GEOMETRYIn par
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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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168 CHAPTER 2. CONVEX GEOMETRYΓ 4
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270 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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