v2007.09.17 - Convex Optimization

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22 CHAPTER 1. OVERVIEWFigure 3: [137] [80] Distance data collected via nuclear magnetic resonance(NMR) helped render this 3-dimensional depiction of a protein molecule.At the beginning of the 1980s, Kurt Wüthrich [Nobel laureate], developedan idea about how NMR could be extended to cover biological moleculessuch as proteins. He invented a systematic method of pairing each NMRsignal with the right hydrogen nucleus (proton) in the macromolecule. Themethod is called sequential assignment and is today a cornerstone of all NMRstructural investigations. He also showed how it was subsequently possible todetermine pairwise distances between a large number of hydrogen nuclei anduse this information with a mathematical method based on distance-geometryto calculate a three-dimensional structure for the molecule. [207] [286] [132]Geometric problems involving distance between points can sometimes bereduced to convex optimization problems. Mathematics of this combinedstudy of geometry and optimization is rich and deep. Its application hasalready proven invaluable discerning organic molecular conformation bymeasuring interatomic distance along covalent bonds; e.g., Figure 3. [60][265] [132] [286] [95] Many disciplines have already benefitted and simplifiedconsequent to this theory; e.g., distance-based pattern recognition (Figure 4),localization in wireless sensor networks by measurement of intersensordistance along channels of communication, [35,5] [297] [33] wireless locationof a radio-signal source such as a cell phone by multiple measurements ofsignal strength, the global positioning system (GPS), and multidimensionalscaling which is a numerical representation of qualitative data by finding alow-dimensional scale.
23Figure 4: This coarsely discretized triangulated algorithmically flattenedhuman face (made by Kimmel & the Bronsteins [165]) represents a stagein machine recognition of human identity; called face recognition. Distancegeometry is applied to determine discriminating-features.Euclidean distance geometry together with convex optimization have alsofound application in artificial intelligence:to machine learning by discerning naturally occurring manifolds in:– Euclidean bodies (Figure 5,6.4.0.0.1),– Fourier spectra of kindred utterances [156],– and image sequences [281],to robotics; e.g., automatic control of vehicles maneuvering information. (Figure 8)by chapterWe study the pervasive convex Euclidean bodies and their variousrepresentations. In particular, we make convex polyhedra, cones, and dualcones more visceral through illustration in chapter 2, <strong>Convex</strong> geometry,and we study the geometric relation of polyhedral cones to nonorthogonal
Page 2 and 3: DattorroCONVEXOPTIMIZATION&EUCLIDEA
Page 4 and 5: Meboo Publishing USA345 Stanford Sh
Page 6 and 7: EDM = S h ∩ ( S ⊥ c − S +)
Page 8 and 9: There is a great race under way to
Page 10 and 11: 10 CONVEX OPTIMIZATION & EUCLIDEAN
Page 12 and 13: 12 CONVEX OPTIMIZATION & EUCLIDEAN
Page 14 and 15: 14 LIST OF FIGURES24 Nonconvex cone
Page 16 and 17: 16 LIST OF FIGURES92 Largest ten ei
Page 18 and 19: List of Tables2 Convex geometryTabl
Page 20 and 21: 20 CHAPTER 1. OVERVIEWFigure 1: Ori
Page 24 and 25: 24 CHAPTER 1. OVERVIEWFigure 5: Swi
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Page 28 and 29: 28 CHAPTER 1. OVERVIEWFigure 8: Rob
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Page 32 and 33: 32 CHAPTER 1. OVERVIEW
Page 34 and 35: 34 CHAPTER 2. CONVEX GEOMETRY2.1 Co
Page 36 and 37: 36 CHAPTER 2. CONVEX GEOMETRY2.1.3
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Page 40 and 41: 40 CHAPTER 2. CONVEX GEOMETRYwhere
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Page 48 and 49: 48 CHAPTER 2. CONVEX GEOMETRY2.2.1.
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72 CHAPTER 2. CONVEX GEOMETRYto vec
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74 CHAPTER 2. CONVEX GEOMETRY2.5.1
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78 CHAPTER 2. CONVEX GEOMETRYABCDFi
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80 CHAPTER 2. CONVEX GEOMETRY2.6.1.
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82 CHAPTER 2. CONVEX GEOMETRYX(a)00
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84 CHAPTER 2. CONVEX GEOMETRYXXFigu
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86 CHAPTER 2. CONVEX GEOMETRYFamili
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88 CHAPTER 2. CONVEX GEOMETRYC 1C 2
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90 CHAPTER 2. CONVEX GEOMETRYA prop
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92 CHAPTER 2. CONVEX GEOMETRY∂K
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94 CHAPTER 2. CONVEX GEOMETRYWhen t
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96 CHAPTER 2. CONVEX GEOMETRYBCADFi
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98 CHAPTER 2. CONVEX GEOMETRYThe po
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102 CHAPTER 2. CONVEX GEOMETRYwhere
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104 CHAPTER 2. CONVEX GEOMETRY√2
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106 CHAPTER 2. CONVEX GEOMETRYwhich
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110 CHAPTER 2. CONVEX GEOMETRYA con
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112 CHAPTER 2. CONVEX GEOMETRY0-1-0
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114 CHAPTER 2. CONVEX GEOMETRYposit
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118 CHAPTER 2. CONVEX GEOMETRYThe c
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120 CHAPTER 2. CONVEX GEOMETRYWhen
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122 CHAPTER 2. CONVEX GEOMETRY2.10.
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124 CHAPTER 2. CONVEX GEOMETRY{extr
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126 CHAPTER 2. CONVEX GEOMETRY2.11
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128 CHAPTER 2. CONVEX GEOMETRYFrom
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130 CHAPTER 2. CONVEX GEOMETRYS = {
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132 CHAPTER 2. CONVEX GEOMETRYFigur
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136 CHAPTER 2. CONVEX GEOMETRYK ∗
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138 CHAPTER 2. CONVEX GEOMETRYKK
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140 CHAPTER 2. CONVEX GEOMETRYthe p
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142 CHAPTER 2. CONVEX GEOMETRY(dual
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146 CHAPTER 2. CONVEX GEOMETRYfor w
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148 CHAPTER 2. CONVEX GEOMETRYBy al
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150 CHAPTER 2. CONVEX GEOMETRYb −
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154 CHAPTER 2. CONVEX GEOMETRYDual
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160 CHAPTER 2. CONVEX GEOMETRYΓ 4
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162 CHAPTER 2. CONVEX GEOMETRYEigen
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164 CHAPTER 2. CONVEX GEOMETRYunder
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166 CHAPTER 2. CONVEX GEOMETRYWhen
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168 CHAPTER 2. CONVEX GEOMETRYFor e
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170 CHAPTER 2. CONVEX GEOMETRY10.80
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172 CHAPTER 2. CONVEX GEOMETRYx 210
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174 CHAPTER 2. CONVEX GEOMETRYwhile
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176 CHAPTER 2. CONVEX GEOMETRYαα
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178 CHAPTER 2. CONVEX GEOMETRYFrom
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182 CHAPTER 2. CONVEX GEOMETRYhavin
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184 CHAPTER 3. GEOMETRY OF CONVEX F
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226 CHAPTER 4. SEMIDEFINITE PROGRAM
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292 CHAPTER 5. EUCLIDEAN DISTANCE M
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390 CHAPTER 6. EDM CONEa resemblanc
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392 CHAPTER 6. EDM CONEdvec rel∂E
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394 CHAPTER 6. EDM CONE(b)(c)dvec r
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396 CHAPTER 6. EDM CONE(a)2 nearest
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398 CHAPTER 6. EDM CONEthe graph. T
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400 CHAPTER 6. EDM CONEwhere e i
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402 CHAPTER 6. EDM CONE6.5 EDM defi
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404 CHAPTER 6. EDM CONEN(1 T )δ(V
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406 CHAPTER 6. EDM CONE10(a)-110-1V
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408 CHAPTER 6. EDM CONE6.5.3 Faces
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410 CHAPTER 6. EDM CONE6.5.3.2 Smal
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412 CHAPTER 6. EDM CONE6.6.0.0.1 Pr
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414 CHAPTER 6. EDM CONEdvec rel∂E
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416 CHAPTER 6. EDM CONE6.7 Vectoriz
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418 CHAPTER 6. EDM CONEsvec ∂ S 2
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420 CHAPTER 6. EDM CONEIn fact, the
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422 CHAPTER 6. EDM CONEThe ordinary
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424 CHAPTER 6. EDM CONEEDM 2 = S 2
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426 CHAPTER 6. EDM CONEFrom the res
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428 CHAPTER 6. EDM CONE6.8.1.3 Dual
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430 CHAPTER 6. EDM CONED ◦ = δ(D
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432 CHAPTER 6. EDM CONE6.8.1.6 EDM
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434 CHAPTER 6. EDM CONEBecause 〈
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436 CHAPTER 6. EDM CONE0dvec rel∂
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438 CHAPTER 6. EDM CONE
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440 CHAPTER 7. PROXIMITY PROBLEMS7.
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442 CHAPTER 7. PROXIMITY PROBLEMS..
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444 CHAPTER 7. PROXIMITY PROBLEMSTh
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446 CHAPTER 7. PROXIMITY PROBLEMSwh
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448 CHAPTER 7. PROXIMITY PROBLEMSpo
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452 CHAPTER 7. PROXIMITY PROBLEMSof
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456 CHAPTER 7. PROXIMITY PROBLEMSR
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458 CHAPTER 7. PROXIMITY PROBLEMSwh
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462 CHAPTER 7. PROXIMITY PROBLEMSCo
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466 CHAPTER 7. PROXIMITY PROBLEMSTo
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468 CHAPTER 7. PROXIMITY PROBLEMSth
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470 CHAPTER 7. PROXIMITY PROBLEMSco
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472 CHAPTER 7. PROXIMITY PROBLEMSto
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476 CHAPTER 7. PROXIMITY PROBLEMSdi
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478 CHAPTER 7. PROXIMITY PROBLEMSve
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480 CHAPTER 7. PROXIMITY PROBLEMSth
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482 APPENDIX A. LINEAR ALGEBRAA.1.1
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486 APPENDIX A. LINEAR ALGEBRAonly
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488 APPENDIX A. LINEAR ALGEBRA(AB)
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492 APPENDIX A. LINEAR ALGEBRAFor A
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494 APPENDIX A. LINEAR ALGEBRADiago
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496 APPENDIX A. LINEAR ALGEBRAFor A
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500 APPENDIX A. LINEAR ALGEBRAA.4 S
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504 APPENDIX A. LINEAR ALGEBRAA.5 e
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506 APPENDIX A. LINEAR ALGEBRAs i w
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510 APPENDIX A. LINEAR ALGEBRAΣq 2
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512 APPENDIX A. LINEAR ALGEBRAA.7 Z
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514 APPENDIX A. LINEAR ALGEBRAThere
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518 APPENDIX A. LINEAR ALGEBRA
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520 APPENDIX B. SIMPLE MATRICESB.1
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522 APPENDIX B. SIMPLE MATRICESProo
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524 APPENDIX B. SIMPLE MATRICESB.1.
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526 APPENDIX B. SIMPLE MATRICESN(u
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528 APPENDIX B. SIMPLE MATRICESDue
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532 APPENDIX B. SIMPLE MATRICEShas
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534 APPENDIX B. SIMPLE MATRICESFigu
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538 APPENDIX C. SOME ANALYTICAL OPT
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552 APPENDIX D. MATRIX CALCULUSThe
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554 APPENDIX D. MATRIX CALCULUSGrad
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556 APPENDIX D. MATRIX CALCULUSBeca
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558 APPENDIX D. MATRIX CALCULUSwhic
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560 APPENDIX D. MATRIX CALCULUS⎡
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562 APPENDIX D. MATRIX CALCULUS→Y
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564 APPENDIX D. MATRIX CALCULUSD.1.
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566 APPENDIX D. MATRIX CALCULUSwhic
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568 APPENDIX D. MATRIX CALCULUSIn t
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572 APPENDIX D. MATRIX CALCULUSD.2
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574 APPENDIX D. MATRIX CALCULUSalge
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576 APPENDIX D. MATRIX CALCULUStrac
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580 APPENDIX D. MATRIX CALCULUS
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582 APPENDIX E. PROJECTIONThe follo
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584 APPENDIX E. PROJECTIONFor matri
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586 APPENDIX E. PROJECTION(⇐) To
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588 APPENDIX E. PROJECTIONNonorthog
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590 APPENDIX E. PROJECTIONE.2.0.0.1
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592 APPENDIX E. PROJECTIONE.3.2Orth
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594 APPENDIX E. PROJECTIONE.3.5Unif
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596 APPENDIX E. PROJECTIONE.4 Algeb
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598 APPENDIX E. PROJECTIONa ∗ 2K
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600 APPENDIX E. PROJECTIONwhere Y =
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602 APPENDIX E. PROJECTION(B.4.2).
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604 APPENDIX E. PROJECTIONis a nono
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606 APPENDIX E. PROJECTIONE.6.4.1Or
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608 APPENDIX E. PROJECTIONq i q T i
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610 APPENDIX E. PROJECTIONThe test
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612 APPENDIX E. PROJECTIONPerpendic
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614 APPENDIX E. PROJECTIONE.8 Range
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616 APPENDIX E. PROJECTIONAs for su
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618 APPENDIX E. PROJECTIONWith refe
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620 APPENDIX E. PROJECTIONProjectio
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622 APPENDIX E. PROJECTIONE.9.2.2.2
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624 APPENDIX E. PROJECTIONThe foreg
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626 APPENDIX E. PROJECTION❇❇❇
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628 APPENDIX E. PROJECTIONE.10 Alte
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630 APPENDIX E. PROJECTIONbH 1H 2P
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632 APPENDIX E. PROJECTIONa(a){y |
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634 APPENDIX E. PROJECTION(a feasib
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636 APPENDIX E. PROJECTIONwhile, th
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638 APPENDIX E. PROJECTIONE.10.2.1.
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640 APPENDIX E. PROJECTION10 0dist(
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642 APPENDIX E. PROJECTIONE.10.3.1D
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644 APPENDIX E. PROJECTIONE 3K ⊥
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646 APPENDIX E. PROJECTION
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648 APPENDIX F. MATLAB PROGRAMSif n
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650 APPENDIX F. MATLAB PROGRAMSend%
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652 APPENDIX F. MATLAB PROGRAMSF.1.
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654 APPENDIX F. MATLAB PROGRAMScoun
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656 APPENDIX F. MATLAB PROGRAMSF.3
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660 APPENDIX F. MATLAB PROGRAMS% so
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662 APPENDIX F. MATLAB PROGRAMS% tr
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666 APPENDIX F. MATLAB PROGRAMSbrea
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668 APPENDIX F. MATLAB PROGRAMSwhil
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670 APPENDIX F. MATLAB PROGRAMSF.7
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672 APPENDIX F. MATLAB PROGRAMS
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674 APPENDIX G. NOTATION AND A FEW
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690 BIBLIOGRAPHY[7] Abdo Y. Alfakih
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692 BIBLIOGRAPHY[27] Aharon Ben-Tal
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694 BIBLIOGRAPHY[48] Lev M. Brègma
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696 BIBLIOGRAPHY[67] Joel Dawson, S
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698 BIBLIOGRAPHY[85] Carl Eckart an
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700 BIBLIOGRAPHY[102] Laurent El Gh
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702 BIBLIOGRAPHY[124] Peter Gritzma
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704 BIBLIOGRAPHY[146] Jean-Baptiste
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706 BIBLIOGRAPHY[168] Jean B. Lasse
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708 BIBLIOGRAPHY[190] Rudolf Mathar
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710 BIBLIOGRAPHY[212] M. L. Overton
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712 BIBLIOGRAPHY[230] R. Tyrrell Ro
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714 BIBLIOGRAPHY[254] Jos F. Sturm
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716 BIBLIOGRAPHY[276] È. B. Vinber
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[295] Naoki Yamamoto and Maryam Faz
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720 INDEXobtuse, 62positive semidef
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722 INDEXnormal, 175, 415, 622, 642
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724 INDEXdiscretization, 152, 185,
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726 INDEXminimization, 206, 611full
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728 INDEXinvariant set, 359inversio
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730 INDEXunitary, 533maximalcomplem
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732 INDEXdifference, 118Farkas’ l
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734 INDEXcommutative, 426, 628direc
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736 INDEXset, 683unique, 42, 185, 1
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738 INDEXaffine, 44, 68, 86, 101, 2
show all

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