v2007.09.17 - Convex Optimization

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490 APPENDIX A. LINEAR ALGEBRAA.3.1Semidefiniteness, eigenvalues, nonsymmetricWhen A∈ R n×n , let λ ( 12 (A +AT ) ) ∈ R n denote eigenvalues of thesymmetrized matrix A.8 arranged in nonincreasing order.By positive semidefiniteness of A∈ R n×n we mean, A.9 [200,1.3.1](conferA.3.1.0.1)x T Ax ≥ 0 ∀x∈ R n ⇔ A +A T ≽ 0 ⇔ λ(A +A T ) ≽ 0 (1247)(2.9.0.1)A ≽ 0 ⇒ A T = A (1248)A ≽ B ⇔ A −B ≽ 0 A ≽ 0 or B ≽ 0 (1249)x T Ax≥0 ∀x A T = A (1250)Matrix symmetry is not intrinsic to positive semidefiniteness;A T = A, λ(A) ≽ 0 ⇒ x T Ax ≥ 0 ∀x (1251)If A T = A thenλ(A) ≽ 0 ⇐ A T = A, x T Ax ≥ 0 ∀x (1252)λ(A) ≽ 0 ⇔ A ≽ 0 (1253)meaning, matrix A belongs to the positive semidefinite cone in thesubspace of symmetric matrices if and only if its eigenvalues belong tothe nonnegative orthant.〈A , A〉 = 〈λ(A), λ(A)〉 (1254)For µ∈ R , A∈ R n×n , and vector λ(A)∈ C n holding the orderedeigenvalues of Aλ(µI + A) = µ1 + λ(A) (1255)Proof: A=MJM −1 and µI + A = M(µI + J)M −1 where J isthe Jordan form for A ; [249,5.6, App.B] id est, δ(J) = λ(A) , soλ(µI + A) = δ(µI + J) because µI + J is also a Jordan form. A.8 The symmetrization of A is (A +A T )/2. λ ( 12 (A +AT ) ) = λ(A +A T )/2.A.9 Strang agrees [249, p.334] it is not λ(A) that requires observation. Yet he is mistakenby proposing the Hermitian part alone x H (A+A H )x be tested, because the anti-Hermitianpart does not vanish under complex test unless A is Hermitian. (1238)
A.3. PROPER STATEMENTS 491By similar reasoning,λ(I + µA) = 1 + λ(µA) (1256)For vector σ(A) holding the singular values of any matrix Aσ(I + µA T A) = π(|1 + µσ(A T A)|) (1257)σ(µI + A T A) = π(|µ1 + σ(A T A)|) (1258)where π is the nonlinear permutation operator sorting its vectorargument into nonincreasing order.For A∈ S M and each and every ‖w‖= 1 [150,7.7, prob.9]w T Aw ≤ µ ⇔ A ≼ µI ⇔ λ(A) ≼ µ1 (1259)[150,2.5.4] (confer (36))A is normal ⇔ ‖A‖ 2 F = λ(A) T λ(A) (1260)For A∈ R m×n A T A ≽ 0, AA T ≽ 0 (1261)because, for dimensionally compatible vector x , x T A T Ax = ‖Ax‖ 2 2 ,x T AA T x = ‖A T x‖ 2 2 .For A∈ R n×n and c∈Rtr(cA) = c tr(A) = c1 T λ(A)(A.1.1 no.6)For m a nonnegative integer, (1637)det(A m ) =tr(A m ) =n∏λ(A) m i (1262)i=1n∑λ(A) m i (1263)i=1
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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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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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Page 482 and 483: 482 APPENDIX A. LINEAR ALGEBRAA.1.1
Page 486 and 487: 486 APPENDIX A. LINEAR ALGEBRAonly
Page 488 and 489: 488 APPENDIX A. LINEAR ALGEBRA(AB)
Page 492 and 493: 492 APPENDIX A. LINEAR ALGEBRAFor A
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Page 496 and 497: 496 APPENDIX A. LINEAR ALGEBRAFor A
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Page 506 and 507: 506 APPENDIX A. LINEAR ALGEBRAs i w
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Page 518 and 519: 518 APPENDIX A. LINEAR ALGEBRA
Page 520 and 521: 520 APPENDIX B. SIMPLE MATRICESB.1
Page 522 and 523: 522 APPENDIX B. SIMPLE MATRICESProo
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Page 526 and 527: 526 APPENDIX B. SIMPLE MATRICESN(u
Page 528 and 529: 528 APPENDIX B. SIMPLE MATRICESDue
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Page 534 and 535: 534 APPENDIX B. SIMPLE MATRICESFigu
Page 538 and 539: 538 APPENDIX C. SOME ANALYTICAL OPT
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540 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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