v2006.03.09 - Convex Optimization

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44 CHAPTER 1. OVERVIEWEight appendices are provided so as to be more self-contained:linear algebra (appendix A is primarily concerned with properstatements of semidefiniteness for square matrices),simple matrices (dyad, doublet, elementary, Householder, Schoenberg,orthogonal, etcetera, in appendix B),a collection of known analytical solutions to some importantoptimization problems (appendix C),matrix calculus (appendix D concerns matrix-valued functions, theirderivatives and directional derivatives, Taylor series, and tables of firstandsecond-order gradients and derivatives),an elaborate and insightful exposition of orthogonal and nonorthogonalprojection on convex sets (the connection between projection andpositive semidefiniteness, for example, in appendix E),software to discriminate EDMs, conic independence, software to reducerank of an optimal solution to a semidefinite program, and two distinctmethods of reconstructing a map of the United States given onlydistance information (appendix G).
Chapter 2Convex geometryConvexity has an immensely rich structure and numerousapplications. On the other hand, almost every “convex” idea canbe explained by a two-dimensional picture.−Alexander Barvinok [17, p.vii]There is relatively less published pertaining to matrix-valued convex sets andfunctions. [133] [126,6.6] [185] As convex geometry and linear algebra areinextricably bonded, we provide much background material on linear algebra(especially in the appendices) although it is assumed the reader is comfortablewith [212], [214], [125], or any other intermediate-level text. The essentialreferences to convex analysis are [123] [194]. The reader is referred to [210][17] [242] [27] [38] [191] [231] for a comprehensive treatment of convexity.2001 Jon Dattorro. CO&EDG version 03.09.2006. All rights reserved.Citation: Jon Dattorro, Convex Optimization & Euclidean Distance Geometry,Meboo Publishing USA, 2005.45
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 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 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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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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106 CHAPTER 2. CONVEX GEOMETRY2.8.2
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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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