v2006.03.09 - Convex Optimization

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140 CHAPTER 2. CONVEX GEOMETRY2.12.2.2 Pointedness[210,2.10] Assuming all generators constituting the columns of X ∈ R n×Nare nonzero, K is pointed (2.7.2.1.2) if and only if there is no nonzero a ≽ 0that solves Xa=0; id est, iff N(X) ∩ R N + = 0. (If rankX = n , then thedual cone K ∗ is pointed. (253))A polyhedral proper cone in R n must have at least n linearly independentgenerators, or be the intersection of at least n halfspaces whose partialboundaries have normals that are linearly independent. Otherwise, the conewill contain at least one line and there can be no vertex; id est, the conecannot otherwise be pointed.For any pointed polyhedral cone, there is a one-to-one correspondence ofone-dimensional faces with extreme directions.Examples of pointed closed convex cones K are not limited to polyhedralcones: the origin, any 0-based ray in a subspace, any two-dimensionalV-shaped cone in a subspace, the Lorentz (ice-cream) cone and its polyhedralflavors, the cone of Euclidean distance matrices EDM N in S N h , the propercones: S M + in ambient S M , any orthant in R n or R m×n ; e.g., the nonnegativereal line R + in vector space R .2.12.3 Unit simplexA peculiar convex subset of the nonnegative orthant havinghalfspace-descriptionS ∆ = {s | s ≽ 0, 1 T s ≤ 1} ⊆ R n + (241)is a unique bounded convex polyhedron called unit simplex (Figure 38)having nonempty interior, n + 1 vertices, and dimension [38,2.2.4]dim S = n (242)The origin supplies one vertex while heads of the standard basis [125][212] {e i , i=1... n} in R n constitute those remaining; 2.41 thus itsvertex-description:2.41 In R 0 the unit simplex is the point at the origin, in R the unit simplex is the linesegment [0,1], in R 2 it is a triangle and its relative interior, in R 3 it is the convex hull ofa tetrahedron (Figure 38), in R 4 it is the convex hull of a pentatope [244], and so on.
2.12. CONVEX POLYHEDRA 141S = {s | s ≽ 0, 1 T s ≤ 1}1Figure 38: Unit simplex S in R 3 is a unique solid tetrahedron, but is notregular.
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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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62 CHAPTER 2. CONVEX GEOMETRYwhere
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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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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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