v2006.03.09 - Convex Optimization

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346 CHAPTER 5. EDM CONEcone; id est, the optimal solution of 5.8minimize ‖D ◦ − H‖ FD ◦ ∈ S Nsubject to D ◦ − δ(D ◦ 1) ≽ 0(825)has dual affine dimension complementary to affine dimension correspondingto the optimal solution ofminimize ‖D − H‖ FD∈S N hsubject to −VN TDV N ≽ 0(826)Precisely,rank(D ◦⋆ −δ(D ◦⋆ 1)) + rank(V T ND ⋆ V N ) = N −1 (827)and rank(D ◦⋆ −δ(D ◦⋆ 1))≤N−1 because vector 1 is always in the nullspaceof rank’s argument. This is similar to the known result for projection on theself-dual positive semidefinite cone and its polar:rankP −S N+H + rankP S N+H = N (828)When low affine dimension is a desirable result of projection on theEDM cone, projection on the polar EDM cone should be performed instead.<strong>Convex</strong> polar problem (825) can be solved for D ◦⋆ by transforming to anequivalent Schur-form semidefinite program (A.4.1). Interior-point methodsfor numerically solving semidefinite programs tend to produce high-ranksolutions. (6.1.1) Then D ⋆ = H − D ◦⋆ ∈ EDM N by Corollary E.9.2.2.1, andD ⋆ will tend to have low affine dimension. This approach breaks whenattempting projection on a cone subset discriminated by affine dimensionor rank, because then we have no complementarity relation like (827) or(828) (7.1.4.1).5.8 This dual projection can be solved quickly (without semidefinite programming) viaLemma 5.8.1.1.1; rewriting,minimize ‖(D ◦ − δ(D ◦ 1)) − (H − δ(D ◦ 1))‖ FD ◦ ∈ S Nsubject to D ◦ − δ(D ◦ 1) ≽ 0which is the projection of affinely transformed optimal solution H − δ(D ◦⋆ 1) on S N c ∩ S N + ;D ◦⋆ − δ(D ◦⋆ 1) = P S N+P S N c(H − δ(D ◦⋆ 1))Foreknowledge of an optimal solution D ◦⋆ as argument to projection suggests recursion.
5.8. DUAL EDM CONE 3475.8.1.6 EDM cone dualityIn4.6.1.1, via Gram-form EDM operatorD(G) = δ(G)1 T + 1δ(G) T − 2G ∈ EDM N ⇐ G ≽ 0 (472)we established clear connection between the EDM cone and that face (783)of positive semidefinite cone S N + in the geometric center subspace:EDM N = D(S N c ∩ S N +) (556)V(EDM N ) = S N c ∩ S N + (557)whereIn4.6.1 we establishedV(D) = −V DV 1 2(545)S N c ∩ S N + = V N S N−1+ V T N (543)Then from (816), (824), and (790) we can deduceδ(EDM N∗ 1) − EDM N∗ = V N S N−1+ V T N = S N c ∩ S N + (829)which, by (556) and (557), means the EDM cone can be related to the dualEDM cone by an equality:(EDM N = D δ(EDM N∗ 1) − EDM N∗) (830)V(EDM N ) = δ(EDM N∗ 1) − EDM N∗ (831)This means projection −V(EDM N ) of the EDM cone on the geometric centersubspace S N c (E.7.2.0.2) is an affine transformation of the dual EDM cone:EDM N∗ − δ(EDM N∗ 1). Secondarily, it means the EDM cone is not self-dual.
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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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Page 296 and 297: 296 CHAPTER 4. EUCLIDEAN DISTANCE M
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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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