v2006.03.09 - Convex Optimization

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276 CHAPTER 4. EUCLIDEAN DISTANCE MATRIXfunction (4.8.4), we calculate its second partial derivative with respect tod 14 evaluated at 2 and find −1/3. We conclude there are no other satisfyingvalues of d 14 . Further, that value of d 14 does not meet an upper or lowerbound of a triangle inequality like (611), so neither does it cause the collapseof any triangle. Because the minimum eigenvalue is 0, affine dimension r ofany point list corresponding to D cannot exceed N −2. (4.7.1.1) 4.10 EDM-entry compositionLaurent [145,2.3] applies results from Schoenberg (1938) [200] to showcertain nonlinear compositions of individual EDM entries yield EDMs;in particular,D ∈ EDM N ⇔ [1 − e −αd ij] ∈ EDM N ∀α > 0⇔ [e −αd ij] ∈ E N ∀α > 0(639)where D = [d ij ] and E N is the elliptope (624).Proof.F.Schoenberg’s results [200,6, thm.5] (confer [140, p.108-109]) alsosuggest certain finite positive roots of EDM entries produce EDMs;specifically,D ∈ EDM N ⇔ [d 1/αij ] ∈ EDM N ∀α > 1 (640)The special case α = 2 is of interest because it corresponds to absolutedistance; e.g.,D ∈ EDM N ⇒ ◦√ D ∈ EDM N (641)Assuming that points constituting a corresponding list X are distinct(604), then it follows: for D ∈ S N hlimα→∞ [d1/α ij] = limα→∞[1 − e −αd ij] = −E ∆ = 11 T − I (642)Negative elementary matrix −E (B.3) is relatively interior to the EDM cone(5.6) and terminal to the respective trajectories (639) and (640) as functionsof α . Both trajectories are confined to the EDM cone; in engineering terms,the EDM cone is an invariant set [197] to either trajectory. Further, if D is
4.10. EDM-ENTRY COMPOSITION 277minimum eigenvalue1 2 3 4d 14-0.2-0.4-0.6Figure 68: Minimum eigenvalue of −VN TDV None value of d 14 ; 2.positive semidefinite for only1.41.2d 1/αijlog 2 (1+d 1/αij )(a)10.80.60.40.21−e −αd ij40.5 1 1.5 2d ij3(b)21d 1/αijlog 2 (1+d 1/αij )1−e −αd ij0.5 1 1.5 2αFigure 69: Some entrywise EDM compositions: (a) α = 2. Concavenondecreasing in d ij . (b) Trajectory convergence in α for d ij = 2.
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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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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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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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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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648
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