v2004.06.19 - Convex Optimization

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12 CONTENTSB.2 Doublet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331B.3 Elementary matrix . . . . . . . . . . . . . . . . . . . . . . . . 332B.3.1 Householder matrix . . . . . . . . . . . . . . . . . . . . 333B.4 Auxiliary V -matrices . . . . . . . . . . . . . . . . . . . . . . . 334B.4.1 Auxiliary matrix V . . . . . . . . . . . . . . . . . . . . 334B.4.2 Schoenberg auxiliary matrix V N . . . . . . . . . . . . . 336B.4.3 Auxiliary matrix V W . . . . . . . . . . . . . . . . . . . 337B.4.4 Auxiliary V -matrix Table . . . . . . . . . . . . . . . . 337B.4.5 More auxiliary matrices . . . . . . . . . . . . . . . . . 338B.5 Orthogonal matrix . . . . . . . . . . . . . . . . . . . . . . . . 338B.5.1 Reflection . . . . . . . . . . . . . . . . . . . . . . . . . 339B.5.2 Matrix rotation . . . . . . . . . . . . . . . . . . . . . . 339B.6 Circulant matrices... . . . . . . . . . . . . . . . . . . . . . . . 340C Some optimal analytical results 343C.1 involving diagonal, trace, eigenvalues . . . . . . . . . . . . . . 343C.2 Orthogonal Procrustes problem . . . . . . . . . . . . . . . . . 346C.2.1 Two-sided orthogonal Procrustes problems . . . . . . . 347D Matrix calculus 351D.1 Directional derivative, Taylor series . . . . . . . . . . . . . . . 351D.1.1 Gradients . . . . . . . . . . . . . . . . . . . . . . . . . 351D.1.2 Product rules for matrix-functions . . . . . . . . . . . . 355D.1.3 Chain rules for composite matrix-functions . . . . . . . 357D.1.4 First directional derivative . . . . . . . . . . . . . . . . 358D.1.5 Second directional derivative . . . . . . . . . . . . . . . 363D.1.6 Taylor series . . . . . . . . . . . . . . . . . . . . . . . . 366D.1.7 Correspondence of gradient to derivative . . . . . . . . 367D.2 Tables of gradients and derivatives . . . . . . . . . . . . . . . 370D.2.1 Algebraic . . . . . . . . . . . . . . . . . . . . . . . . . 373D.2.2 Trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376D.2.3 Trace Kronecker . . . . . . . . . . . . . . . . . . . . . . 378D.2.4 Log determinant . . . . . . . . . . . . . . . . . . . . . 379D.2.5 Determinant . . . . . . . . . . . . . . . . . . . . . . . . 380D.2.6 Logarithmic . . . . . . . . . . . . . . . . . . . . . . . . 381D.2.7 Exponential . . . . . . . . . . . . . . . . . . . . . . . . 381
CONTENTS 13E Pseudoinverse, Projection 383E.0.1 Logical deductions . . . . . . . . . . . . . . . . . . . . 384E.1 Idempotent matrices . . . . . . . . . . . . . . . . . . . . . . . 385E.1.1 Biorthogonal characterization . . . . . . . . . . . . . . 386E.1.2 Idempotence summary . . . . . . . . . . . . . . . . . . 389E.2 I − P , Projection on algebraic complement . . . . . . . . . . . 389E.3 Symmetric idempotent matrices . . . . . . . . . . . . . . . . . 390E.3.1 Four subspaces . . . . . . . . . . . . . . . . . . . . . . 391E.3.2 Orthogonal characterization . . . . . . . . . . . . . . . 392E.3.3 Symmetric idempotence summary . . . . . . . . . . . . 393E.3.4 Orthonormal decomposition . . . . . . . . . . . . . . . 393E.4 Algebra of projection on affine subsets . . . . . . . . . . . . . 396E.5 Projection examples . . . . . . . . . . . . . . . . . . . . . . . 396E.6 Vectorization interpretation, . . . . . . . . . . . . . . . . . . . 403E.6.1 Nonorthogonal projection on a vector . . . . . . . . . . 403E.6.2 Nonorthogonal projection on vectorized matrix . . . . . 403E.6.3 Orthogonal projection on a vector . . . . . . . . . . . . 405E.6.4 Orthogonal projection on a vectorized matrix . . . . . 405E.7 Range/Rowspace interpretation . . . . . . . . . . . . . . . . . 410E.7.1 Projection on vectorized matrices of higher rank . . . . 411E.8 Projection on convex set . . . . . . . . . . . . . . . . . . . . . 414E.8.1 Projection on cone . . . . . . . . . . . . . . . . . . . . 415E.8.2 Easy projections . . . . . . . . . . . . . . . . . . . . . 418E.8.3 Projection on convex set in subspace . . . . . . . . . . 419E.9 Alternating projection . . . . . . . . . . . . . . . . . . . . . . 422E.9.1 Distance and existence . . . . . . . . . . . . . . . . . . 426E.9.2 Feasibility and convergence . . . . . . . . . . . . . . . 426E.9.3 <strong>Optimization</strong> . . . . . . . . . . . . . . . . . . . . . . . 432F MATLAB programs... 435F.1 isedm() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435F.1.1 Subroutines for isedm() . . . . . . . . . . . . . . . . . 435F.2 conic independence . . . . . . . . . . . . . . . . . . . . . . . . 437F.2.1 lp() . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438F.3 Map of the USA . . . . . . . . . . . . . . . . . . . . . . . . . . 439F.3.1 EDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439F.3.2 EDM using ordinal data . . . . . . . . . . . . . . . . . 441
Page 1 and 2: Euclidean Distance GeometryviaConve
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62 CHAPTER 2. CONVEX GEOMETRY2.6.6.
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66 CHAPTER 2. CONVEX GEOMETRYwhich
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70 CHAPTER 2. CONVEX GEOMETRYAssumi
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72 CHAPTER 2. CONVEX GEOMETRY2.6.8
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74 CHAPTER 2. CONVEX GEOMETRYFrom t
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76 CHAPTER 2. CONVEX GEOMETRYFor an
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78 CHAPTER 2. CONVEX GEOMETRY2.8.1
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80 CHAPTER 2. CONVEX GEOMETRY• (S
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82 CHAPTER 2. CONVEX GEOMETRYmerely
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84 CHAPTER 2. CONVEX GEOMETRYy ∈
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86 CHAPTER 2. CONVEX GEOMETRY2.8.2.
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88 CHAPTER 2. CONVEX GEOMETRYthat f
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90 CHAPTER 2. CONVEX GEOMETRYWhen X
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92 CHAPTER 2. CONVEX GEOMETRY2.9 Fo
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94 CHAPTER 2. CONVEX GEOMETRYFor γ
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96 CHAPTER 2. CONVEX GEOMETRYConver
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98 CHAPTER 2. CONVEX GEOMETRYAssume
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100 CHAPTER 2. CONVEX GEOMETRYwhere
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102 CHAPTER 2. CONVEX GEOMETRYbiort
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X 1 =4X †T1 =⎡⎢⎣⎡⎢⎣10
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106 CHAPTER 2. CONVEX GEOMETRYFigur
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108 CHAPTER 2. CONVEX GEOMETRY∂K
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110 CHAPTER 2. CONVEX GEOMETRYΓ 4
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112 CHAPTER 2. CONVEX GEOMETRY∂K
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114 CHAPTER 3. GEOMETRY OF CONVEX F
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130 CHAPTER 4. EUCLIDEAN DISTANCE M
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204 CHAPTER 5. EDM CONEFor now we i
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206 CHAPTER 5. EDM CONE5.1 Polyhedr
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208 CHAPTER 5. EDM CONE5.2.1 Range
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210 CHAPTER 5. EDM CONEA statement
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212 CHAPTER 5. EDM CONEIn particula
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214 CHAPTER 5. EDM CONE5.4 Correspo
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216 CHAPTER 5. EDM CONE5.4.2 Monoto
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218 CHAPTER 5. EDM CONEsvec ∂ S 2
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220 CHAPTER 5. EDM CONE5.6 Dual EDM
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222 CHAPTER 5. EDM CONE5.6.2.1 Scho
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224 CHAPTER 5. EDM CONE
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226 CHAPTER 6. CONVEX OPTIMIZATIONT
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230 CHAPTER 6. CONVEX OPTIMIZATION6
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232 CHAPTER 6. CONVEX OPTIMIZATIONW
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234 CHAPTER 6. CONVEX OPTIMIZATION6
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236 CHAPTER 6. CONVEX OPTIMIZATIONF
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238 CHAPTER 6. CONVEX OPTIMIZATIONt
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242 CHAPTER 6. CONVEX OPTIMIZATION
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244 CHAPTER 7. EDM PROXIMITY7.0.1 L
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246 CHAPTER 7. EDM PROXIMITY......
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248 CHAPTER 7. EDM PROXIMITYparticu
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250 CHAPTER 7. EDM PROXIMITYmatrice
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252 CHAPTER 7. EDM PROXIMITY7.1.3 C
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256 CHAPTER 7. EDM PROXIMITYFor lac
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258 CHAPTER 7. EDM PROXIMITYH is no
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260 CHAPTER 7. EDM PROXIMITYwhere
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262 CHAPTER 7. EDM PROXIMITYbe zero
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264 CHAPTER 7. EDM PROXIMITYFor the
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266 CHAPTER 7. EDM PROXIMITYK ∗ 2
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K ρ+2λ268 CHAPTER 7. EDM PROXIMIT
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270 CHAPTER 7. EDM PROXIMITYsimply
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272 CHAPTER 7. EDM PROXIMITY
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274 CHAPTER 8. EDM COMPLETION(b)(c)
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276 CHAPTER 8. EDM COMPLETION8.1.0.
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278 CHAPTER 8. EDM COMPLETION8.2 Re
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280 CHAPTER 8. EDM COMPLETION8.2.2
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282 CHAPTER 8. EDM COMPLETIONR 2 ha
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284 CHAPTER 8. EDM COMPLETION0.50.4
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290 CHAPTER 9. PIANO TUNING
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292 APPENDIX A. LINEAR ALGEBRAmatri
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294 APPENDIX A. LINEAR ALGEBRACorol
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296 APPENDIX A. LINEAR ALGEBRA...an
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298 APPENDIX A. LINEAR ALGEBRAλ (
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300 APPENDIX A. LINEAR ALGEBRA• M
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302 APPENDIX A. LINEAR ALGEBRA• F
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304 APPENDIX A. LINEAR ALGEBRA• F
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306 APPENDIX A. LINEAR ALGEBRAA.3.1
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310 APPENDIX A. LINEAR ALGEBRAThe o
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314 APPENDIX A. LINEAR ALGEBRAThe p
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316 APPENDIX A. LINEAR ALGEBRAR{u i
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320 APPENDIX A. LINEAR ALGEBRANow s
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322 APPENDIX A. LINEAR ALGEBRAwhich
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324 APPENDIX A. LINEAR ALGEBRA
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326 APPENDIX B. SIMPLE MATRICESB.1
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328 APPENDIX B. SIMPLE MATRICESB.1.
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330 APPENDIX B. SIMPLE MATRICESby t
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332 APPENDIX B. SIMPLE MATRICESN(u
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334 APPENDIX B. SIMPLE MATRICESthe
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340 APPENDIX B. SIMPLE MATRICESThis
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342 APPENDIX B. SIMPLE MATRICES
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344 APPENDIX C. SOME OPTIMAL ANALYT
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352 APPENDIX D. MATRIX CALCULUSwhil
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354 APPENDIX D. MATRIX CALCULUSa cu
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356 APPENDIX D. MATRIX CALCULUSprod
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358 APPENDIX D. MATRIX CALCULUS∇
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360 APPENDIX D. MATRIX CALCULUSin m
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362 APPENDIX D. MATRIX CALCULUSNoti
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364 APPENDIX D. MATRIX CALCULUS⎡
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366 APPENDIX D. MATRIX CALCULUSD.1.
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368 APPENDIX D. MATRIX CALCULUSExam
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370 APPENDIX D. MATRIX CALCULUSD.2
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372 APPENDIX D. MATRIX CALCULUS
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382 APPENDIX D. MATRIX CALCULUS
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384 APPENDIX E. PSEUDOINVERSE, PROJ
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436 APPENDIX F. MATLAB PROGRAMS...F
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438 APPENDIX F. MATLAB PROGRAMS...F
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440 APPENDIX F. MATLAB PROGRAMS...l
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442 APPENDIX F. MATLAB PROGRAMS...s
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444 APPENDIX F. MATLAB PROGRAMS...
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446 APPENDIX G. NOTATION AND SOME D
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456 BIBLIOGRAPHY[9] Günter M. Zieg
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458 BIBLIOGRAPHY[28] Roger A. Horn
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460 BIBLIOGRAPHY[54] Hong-Xuan Huan
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462 BIBLIOGRAPHY[76] Peter Gritzman
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464 BIBLIOGRAPHYtors, Handbook of S
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466 BIBLIOGRAPHY[117] Shao-Po Wu an
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468 BIBLIOGRAPHY[134] Maryam Fazel,
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470 BIBLIOGRAPHY[154] George P. H.
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472 BIBLIOGRAPHY[178] Jean-Baptiste
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474 BIBLIOGRAPHY
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v2004.06.19 - Convex Optimization

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