v2007.09.17 - Convex Optimization

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154 CHAPTER 2. CONVEX GEOMETRYDual generalized inequalities with respect to the positive semidefinite conein the ambient space of symmetric matrices can therefore be simply stated:(Fejér)X ≽ 0 ⇔ tr(Y T X) ≥ 0 for all Y ≽ 0 (322)Membership to this cone can be determined in the isometrically isomorphicEuclidean space R M2 via (31). (2.2.1) By the two interpretations in2.13.1,positive semidefinite matrix Y can be interpreted as inward-normal to ahyperplane supporting the positive semidefinite cone.The fundamental statement of positive semidefiniteness, y T Xy ≥0 ∀y(A.3.0.0.1), evokes a particular instance of these dual generalizedinequalities (322):X ≽ 0 ⇔ 〈yy T , X〉 ≥ 0 ∀yy T (≽ 0) (1245)Discretization (2.13.4.2.1) allows replacement of positive semidefinitematrices Y with this minimal set of generators comprising the extremedirections of the positive semidefinite cone (2.9.2.4).2.13.5.1 self-dual conesFrom (110) (a consequence of the halfspaces theorem (2.4.1.1.1)), where theonly finite value of the support function for a convex cone is 0 [148,C.2.3.1],or from discretized definition (319) of the dual cone we get a ratherself-evident characterization of self-duality:K = K ∗ ⇔ K = ⋂ {y | γ T y ≥ 0 } (323)γ∈G(K)In words: Cone K is self-dual iff its own extreme directions areinward-normals to a (minimal) set of hyperplanes bounding halfspaces whoseintersection constructs it. This means each extreme direction of K is normalto a hyperplane exposing one of its own faces; a necessary but insufficientcondition for self-duality (Figure 48, for example).Self-dual cones are of necessarily nonempty interior [25,I] and invariantto rotation about the origin. Their most prominent representatives are theorthants, the positive semidefinite cone S M + in the ambient space of symmetricmatrices (321), and the Lorentz cone (147) [17,II.A] [46, exmp.2.25]. Inthree dimensions, a plane containing the axis of revolution of a self-dual cone(and the origin) will produce a slice whose boundary makes a right angle.
2.13. DUAL CONE & GENERALIZED INEQUALITY 155(a)K0 ∂K ∗∂K ∗K(b)Figure 48: Two (truncated) views of a polyhedral cone K and its dual in R 3 .Each of four extreme directions from K belongs to a face of dual cone K ∗ .Shrouded (inside) cone K is symmetrical about its axis of revolution. Eachpair of diametrically opposed extreme directions from K makes a right angle.An orthant (or any rotation thereof; a simplicial cone) is not the only self-dualpolyhedral cone in three or more dimensions; [17,2.A.21] e.g., consider anequilateral with five extreme directions. In fact, every self-dual polyhedralcone in R 3 has an odd number of extreme directions. [19, thm.3]
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DattorroCONVEXOPTIMIZATION&EUCLIDEA
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Meboo Publishing USA345 Stanford Sh
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EDM = S h ∩ ( S ⊥ c − S +)
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There is a great race under way to
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10 CONVEX OPTIMIZATION & EUCLIDEAN
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12 CONVEX OPTIMIZATION & EUCLIDEAN
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14 LIST OF FIGURES24 Nonconvex cone
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16 LIST OF FIGURES92 Largest ten ei
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List of Tables2 Convex geometryTabl
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20 CHAPTER 1. OVERVIEWFigure 1: Ori
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22 CHAPTER 1. OVERVIEWFigure 3: [13
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24 CHAPTER 1. OVERVIEWFigure 5: Swi
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26 CHAPTER 1. OVERVIEWoriginalrecon
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28 CHAPTER 1. OVERVIEWFigure 8: Rob
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30 CHAPTER 1. OVERVIEWnoveltyp.120
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32 CHAPTER 1. OVERVIEW
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34 CHAPTER 2. CONVEX GEOMETRY2.1 Co
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36 CHAPTER 2. CONVEX GEOMETRY2.1.3
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38 CHAPTER 2. CONVEX GEOMETRY(a)R(b
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40 CHAPTER 2. CONVEX GEOMETRYwhere
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42 CHAPTER 2. CONVEX GEOMETRYNow le
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46 CHAPTER 2. CONVEX GEOMETRYand wh
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48 CHAPTER 2. CONVEX GEOMETRY2.2.1.
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52 CHAPTER 2. CONVEX GEOMETRYAny ma
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54 CHAPTER 2. CONVEX GEOMETRYIn par
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58 CHAPTER 2. CONVEX GEOMETRYsvec
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60 CHAPTER 2. CONVEX GEOMETRYFigure
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64 CHAPTER 2. CONVEX GEOMETRY11−1
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66 CHAPTER 2. CONVEX GEOMETRYHyperp
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68 CHAPTER 2. CONVEX GEOMETRYCH −
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70 CHAPTER 2. CONVEX GEOMETRYnonemp
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72 CHAPTER 2. CONVEX GEOMETRYto vec
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78 CHAPTER 2. CONVEX GEOMETRYABCDFi
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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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204 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
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