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804 BIBLIOGRAPHYComputing Sciences, Tokyo Institute of Technology, Japan, July 2009.http://math.ewha.ac.kr/~skim/Research/B-457.pdf[225] Yoonsoo Kim and Mehran Mesbahi. On the rank minimization problem. InProceedings of the American Control Conference, volume 3, pages 2015–2020.American Automatic Control Council (AACC), June 2004.[226] Ron Kimmel. Numerical Geometry of Images: Theory, Algorithms, andApplications. Springer-Verlag, 2003.[227] Erwin Kreyszig. Introductory Functional Analysis with Applications. Wiley, 1989.[228] Anthony Kuh, Chaopin Zhu, and Danilo Mandic. Sensor network localization usingleast squares kernel regression. In 10 th International Conference of Knowledge-BasedIntelligent Information and Engineering Systems (KES), volume 4253(III) of LectureNotes in Computer Science, pages 1280–1287, Bournemouth UK, October 2006.Springer-Verlag.http://www.convexoptimization.com/TOOLS/Kuh.pdf[229] Harold W. Kuhn. Nonlinear programming: a historical view. In Richard W. Cottleand Carlton E. Lemke, editors, Nonlinear Programming, pages 1–26. AmericanMathematical Society, 1976.[230] Takahito Kuno, Yasutoshi Yajima, and Hiroshi Konno. An outer approximationmethod for minimizing the product of several convex functions on a convex set.Journal of Global Optimization, 3(3):325–335, September 1993.[231] Amy N. Langville and Carl D. Meyer. Google’s PageRank and Beyond: The Scienceof Search Engine Rankings. Princeton University Press, 2006.[232] Jean B. Lasserre. A new Farkas lemma for positive semidefinite matrices. IEEETransactions on Automatic Control, 40(6):1131–1133, June 1995.[233] Jean B. Lasserre and Eduardo S. Zeron. A Laplace transform algorithm for thevolume of a convex polytope. Journal of the Association for Computing Machinery,48(6):1126–1140, November 2001.http://arxiv.org/abs/math/0106168 (title revision).[234] Monique Laurent. A connection between positive semidefinite and Euclideandistance matrix completion problems. Linear Algebra and its Applications, 273:9–22,1998.[235] Monique Laurent. A tour d’horizon on positive semidefinite and Euclideandistance matrix completion problems. In Panos M. Pardalos and Henry Wolkowicz,editors, Topics in Semidefinite and Interior-Point Methods, pages 51–76. AmericanMathematical Society, 1998.[236] Monique Laurent. Matrix completion problems. In Christodoulos A. Floudas andPanos M. Pardalos, editors, Encyclopedia of Optimization, volume III (Interior-M),pages 221–229. Kluwer, 2001.http://homepages.cwi.nl/~monique/files/opt.ps
BIBLIOGRAPHY 805[237] Monique Laurent and Svatopluk Poljak. On a positive semidefinite relaxation of thecut polytope. Linear Algebra and its Applications, 223/224:439–461, 1995.[238] Monique Laurent and Svatopluk Poljak. On the facial structure of the setof correlation matrices. SIAM Journal on Matrix Analysis and Applications,17(3):530–547, July 1996.[239] Monique Laurent and Franz Rendl. Semidefinite programming and integerprogramming. Optimization Online, 2002.http://www.optimization-online.org/DB HTML/2002/12/585.html[240] Monique Laurent and Franz Rendl. Semidefinite programming and integerprogramming. In K. Aardal, George L. Nemhauser, and R. Weismantel, editors,Discrete Optimization, volume 12 of Handbooks in Operations Research andManagement Science, chapter 8, pages 393–514. Elsevier, 2005.[241] Charles L. Lawson and Richard J. Hanson. Solving Least Squares Problems. SIAM,1995.[242] Jung Rye Lee. The law of cosines in a tetrahedron. Journal of the Korea Societyof Mathematical Education Series B: The Pure and Applied Mathematics, 4(1):1–6,1997.[243] Claude Lemaréchal. Note on an extension of “Davidon” methods to nondifferentiablefunctions. Mathematical Programming, 7(1):384–387, December 1974.http://www.convexoptimization.com/TOOLS/Lemarechal.pdf[244] Vladimir L. Levin. Quasi-convex functions and quasi-monotone operators. Journalof Convex Analysis, 2(1/2):167–172, 1995.[245] Scott Nathan Levine. Audio Representations for Data Compression and CompressedDomain Processing. PhD thesis, Stanford University, Department of ElectricalEngineering, 1999.http://www-ccrma.stanford.edu/~scottl/thesis/thesis.pdf[246] Adrian S. Lewis. Eigenvalue-constrained faces. Linear Algebra and its Applications,269:159–181, 1998.[247] Anhua Lin. Projection algorithms in nonlinear programming. PhD thesis, JohnsHopkins University, 2003.[248] Miguel Sousa Lobo, Lieven Vandenberghe, Stephen Boyd, and Hervé Lebret.Applications of second-order cone programming. Linear Algebra and itsApplications, 284:193–228, November 1998. Special Issue on Linear Algebrain Control, Signals and Image Processing.http://www.stanford.edu/~boyd/socp.html[249] Lee Lorch and Donald J. Newman. On the composition of completely monotonicfunctions and completely monotonic sequences and related questions. Journal ofthe London Mathematical Society (second series), 28:31–45, 1983.http://www.convexoptimization.com/TOOLS/Lorch.pdf
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DATTORROCONVEXOPTIMIZATION&EUCLIDEA
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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 Dista
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CONVEX OPTIMIZATION & EUCLIDEAN DIS
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List of Figures1 Overview 211 Orion
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LIST OF FIGURES 1562 Shrouded polyh
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LIST OF FIGURES 17130 Elliptope E 3
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Chapter 1OverviewConvex Optimizatio
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ˇx 4ˇx 3ˇx 2Figure 2: Applicatio
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25Figure 4: This coarsely discretiz
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(biorthogonal expansion) is examine
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29cardinality Boolean solution to a
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31Figure 8: Robotic vehicles in con
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an elaborate exposition offering in
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Chapter 2Convex geometryConvexity h
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2.1. CONVEX SET 372.1.2 linear inde
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2.1. CONVEX SET 392.1.6 empty set v
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2.1. CONVEX SET 41(a)R(b)R 2(c)R 3F
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2.1. CONVEX SET 43where Q∈ R 3×3
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2.1. CONVEX SET 45Now let’s move
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2.1. CONVEX SET 47By additive inver
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2.1. CONVEX SET 49R nR mR(A T )x px
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2.2. VECTORIZED-MATRIX INNER PRODUC
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2.2. VECTORIZED-MATRIX INNER PRODUC
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2.2. VECTORIZED-MATRIX INNER PRODUC
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2.3. HULLS 61Figure 20: Convex hull
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2.3. HULLS 63Aaffine hull (drawn tr
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2.3. HULLS 65subset of the affine h
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2.3. HULLS 672.3.2.0.2 Example. Nuc
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2.3. HULLS 692.3.2.0.3 Exercise. Co
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2.3. HULLS 71Figure 24: A simplicia
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2.4. HALFSPACE, HYPERPLANE 73H + =
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2.4. HALFSPACE, HYPERPLANE 7511−1
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2.4. HALFSPACE, HYPERPLANE 772.4.2.
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2.4. HALFSPACE, HYPERPLANE 792.4.2.
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2.4. HALFSPACE, HYPERPLANE 81tradit
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2.4. HALFSPACE, HYPERPLANE 832.4.2.
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2.5. SUBSPACE REPRESENTATIONS 852.5
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2.5. SUBSPACE REPRESENTATIONS 87(Ex
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2.5. SUBSPACE REPRESENTATIONS 89(a)
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2.5. SUBSPACE REPRESENTATIONS 91are
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2.6. EXTREME, EXPOSED 93A one-dimen
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2.6. EXTREME, EXPOSED 952.6.1.1 Den
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2.7. CONES 97X(a)00(b)XFigure 33: (
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2.7. CONES 99XXFigure 37: Truncated
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2.7. CONES 101Figure 39: Not a cone
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2.7. CONES 103cone that is a halfli
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2.7. CONES 105A pointed closed conv
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2.8. CONE BOUNDARY 107That means th
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2.8. CONE BOUNDARY 1092.8.1.1 extre
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2.8. CONE BOUNDARY 1112.8.2 Exposed
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2.9. POSITIVE SEMIDEFINITE (PSD) CO
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2.9. POSITIVE SEMIDEFINITE (PSD) CO
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2.9. POSITIVE SEMIDEFINITE (PSD) CO
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2.9. POSITIVE SEMIDEFINITE (PSD) CO
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2.9. POSITIVE SEMIDEFINITE (PSD) CO
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2.9. POSITIVE SEMIDEFINITE (PSD) CO
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2.9. POSITIVE SEMIDEFINITE (PSD) CO
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2.9. POSITIVE SEMIDEFINITE (PSD) CO
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2.10. CONIC INDEPENDENCE (C.I.) 141
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2.10. CONIC INDEPENDENCE (C.I.) 143
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2.10. CONIC INDEPENDENCE (C.I.) 145
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2.12. CONVEX POLYHEDRA 147all dimen
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2.12. CONVEX POLYHEDRA 149convex po
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2.12. CONVEX POLYHEDRA 1512.12.2.2
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2.13. DUAL CONE & GENERALIZED INEQU
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2.13. DUAL CONE & GENERALIZED INEQU
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Chapter 3Geometry of convex functio
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3.1. CONVEX FUNCTION 221f 1 (x)f 2
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3.1. CONVEX FUNCTION 223Rf(b)f(X
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3.2. PRACTICAL NORM FUNCTIONS, ABSO
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3.2. PRACTICAL NORM FUNCTIONS, ABSO
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3.2. PRACTICAL NORM FUNCTIONS, ABSO
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3.2. PRACTICAL NORM FUNCTIONS, ABSO
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3.3. INVERTED FUNCTIONS AND ROOTS 2
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3.4. AFFINE FUNCTION 237rather]x >
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3.4. AFFINE FUNCTION 239f(z)Az 2z 1
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3.5. EPIGRAPH, SUBLEVEL SET 241{a T
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3.5. EPIGRAPH, SUBLEVEL SET 243Subl
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3.5. EPIGRAPH, SUBLEVEL SET 245wher
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3.5. EPIGRAPH, SUBLEVEL SET 249that
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3.6. GRADIENT 251respect to its vec
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3.6. GRADIENT 253Invertibility is g
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3.6. GRADIENT 2553.6.1.0.2 Theorem.
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3.6. GRADIENT 257f(Y )[ ∇f(X)−1
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3.6. GRADIENT 259αβα ≥ β ≥
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3.6. GRADIENT 2613.6.4 second-order
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3.7. CONVEX MATRIX-VALUED FUNCTION
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3.7. CONVEX MATRIX-VALUED FUNCTION
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3.7. CONVEX MATRIX-VALUED FUNCTION
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3.8. QUASICONVEX 269exponential alw
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3.9. SALIENT PROPERTIES 2713.8.0.0.
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Chapter 4Semidefinite programmingPr
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4.1. CONIC PROBLEM 275(confer p.162
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4.1. CONIC PROBLEM 277PCsemidefinit
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4.1. CONIC PROBLEM 279is the affine
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4.1. CONIC PROBLEM 281faces of S 3
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4.1. CONIC PROBLEM 2834.1.2.3 Previ
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4.2. FRAMEWORK 285Semidefinite Fark
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4.2. FRAMEWORK 287On the other hand
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4.2. FRAMEWORK 2894.2.2.1 Dual prob
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4.2. FRAMEWORK 291For symmetric pos
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4.2. FRAMEWORK 293has norm ‖x ⋆
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4.2. FRAMEWORK 295minimize 1 TˆxX
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4.2. FRAMEWORK 297asminimize ‖ỹ
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4.3. RANK REDUCTION 2994.3 Rank red
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4.3. RANK REDUCTION 301A rank-reduc
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4.3. RANK REDUCTION 303(t ⋆ i)
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4.3. RANK REDUCTION 3054.3.3.0.1 Ex
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4.3. RANK REDUCTION 3074.3.3.0.2 Ex
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4.4. RANK-CONSTRAINED SEMIDEFINITE
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4.4. RANK-CONSTRAINED SEMIDEFINITE
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4.5. CONSTRAINING CARDINALITY 333mi
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4.5. CONSTRAINING CARDINALITY 3350R
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4.5. CONSTRAINING CARDINALITY 337it
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4.5. CONSTRAINING CARDINALITY 339m/
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4.5. CONSTRAINING CARDINALITY 341we
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4.5. CONSTRAINING CARDINALITY 343fl
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4.5. CONSTRAINING CARDINALITY 3474.
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4.5. CONSTRAINING CARDINALITY 349R
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4.7. CONSTRAINING RANK OF INDEFINIT
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4.7. CONSTRAINING RANK OF INDEFINIT
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4.7. CONSTRAINING RANK OF INDEFINIT
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4.8. CONVEX ITERATION RANK-1 391whi
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4.8. CONVEX ITERATION RANK-1 393the
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Chapter 5Euclidean Distance MatrixT
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5.2. FIRST METRIC PROPERTIES 397to
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5.3. ∃ FIFTH EUCLIDEAN METRIC PRO
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5.3. ∃ FIFTH EUCLIDEAN METRIC PRO
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5.4. EDM DEFINITION 403The collecti
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5.4. EDM DEFINITION 4055.4.2 Gram-f
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5.4. EDM DEFINITION 407We provide a
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5.4. EDM DEFINITION 4095.4.2.3.1 Ex
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5.4. EDM DEFINITION 411is the fact:
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5.4. EDM DEFINITION 413Figure 120:
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5.4. EDM DEFINITION 415is found fro
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5.4. EDM DEFINITION 417one less dim
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5.4. EDM DEFINITION 419equality con
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5.4. EDM DEFINITION 421How much dis
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5.4. EDM DEFINITION 423105ˇx 4ˇx
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5.4. EDM DEFINITION 425now implicit
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5.4. EDM DEFINITION 427by translate
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5.4. EDM DEFINITION 429Crippen & Ha
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5.4. EDM DEFINITION 431where ([√t
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5.4. EDM DEFINITION 433because (A.3
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5.5. INVARIANCE 4355.5.1.0.1 Exampl
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5.5. INVARIANCE 437x 2 x 2x 3 x 1 x
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5.6. INJECTIVITY OF D & UNIQUE RECO
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5.7. EMBEDDING IN AFFINE HULL 4455.
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5.7. EMBEDDING IN AFFINE HULL 447Fo
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5.7. EMBEDDING IN AFFINE HULL 4495.
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5.8. EUCLIDEAN METRIC VERSUS MATRIX
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5.9. BRIDGE: CONVEX POLYHEDRA TO ED
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5.10. EDM-ENTRY COMPOSITION 465of a
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5.10. EDM-ENTRY COMPOSITION 467Then
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5.11. EDM INDEFINITENESS 4695.11.1
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5.11. EDM INDEFINITENESS 471we have
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5.11. EDM INDEFINITENESS 473So beca
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5.11. EDM INDEFINITENESS 475holds o
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5.12. LIST RECONSTRUCTION 477where
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5.12. LIST RECONSTRUCTION 479(a)(c)
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5.13. RECONSTRUCTION EXAMPLES 481Wi
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5.13. RECONSTRUCTION EXAMPLES 483d
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5.13. RECONSTRUCTION EXAMPLES 485Th
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5.14. FIFTH PROPERTY OF EUCLIDEAN M
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Chapter 6Cone of distance matricesF
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6.1. DEFINING EDM CONE 4996.1 Defin
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6.2. POLYHEDRAL BOUNDS 501This cone
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6.4. EDM DEFINITION IN 11 T 503That
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6.4. EDM DEFINITION IN 11 T 505N(1
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Page 507 and 508:
6.4. EDM DEFINITION IN 11 T 507Then
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Page 509 and 510:
6.4. EDM DEFINITION IN 11 T 5096.4.
-
Page 511 and 512:
6.5. CORRESPONDENCE TO PSD CONE S N
-
Page 513 and 514:
6.5. CORRESPONDENCE TO PSD CONE S N
-
Page 515 and 516:
6.5. CORRESPONDENCE TO PSD CONE S N
-
Page 517 and 518:
6.6. VECTORIZATION & PROJECTION INT
-
Page 519 and 520:
6.6. VECTORIZATION & PROJECTION INT
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Page 521 and 522:
6.6. VECTORIZATION & PROJECTION INT
-
Page 523 and 524:
6.7. A GEOMETRY OF COMPLETION 523(b
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Page 525 and 526:
6.7. A GEOMETRY OF COMPLETION 525[3
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Page 527 and 528:
6.7. A GEOMETRY OF COMPLETION 527wh
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Page 529 and 530:
6.8. DUAL EDM CONE 529to the geomet
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Page 531 and 532:
6.8. DUAL EDM CONE 531Proof. First,
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Page 533 and 534:
6.8. DUAL EDM CONE 533EDM 2 = S 2 h
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Page 535 and 536:
6.8. DUAL EDM CONE 535therefore the
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Page 537 and 538:
6.8. DUAL EDM CONE 537Elegance of t
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Page 539 and 540:
6.8. DUAL EDM CONE 5396.8.1.5 Affin
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Page 541 and 542:
6.8. DUAL EDM CONE 5416.8.1.7 Schoe
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Page 543 and 544:
6.8. DUAL EDM CONE 5430dvec rel ∂
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Page 545 and 546:
6.10. POSTSCRIPT 5456.10 Postscript
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Page 547 and 548:
Chapter 7Proximity problemsIn the
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Page 549 and 550:
549project on the subspace, then pr
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Page 551 and 552:
551HS N h0EDM NK = S N h ∩ R N×N
-
Page 553 and 554:
5537.0.3 Problem approachProblems t
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Page 555 and 556:
7.1. FIRST PREVALENT PROBLEM: 555fi
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Page 557 and 558:
7.1. FIRST PREVALENT PROBLEM: 5577.
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Page 559 and 560:
7.1. FIRST PREVALENT PROBLEM: 559di
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Page 561 and 562:
7.1. FIRST PREVALENT PROBLEM: 5617.
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Page 563 and 564:
7.1. FIRST PREVALENT PROBLEM: 563wh
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Page 565 and 566:
7.1. FIRST PREVALENT PROBLEM: 565Th
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Page 567 and 568:
7.2. SECOND PREVALENT PROBLEM: 567O
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Page 569 and 570:
7.2. SECOND PREVALENT PROBLEM: 569S
-
Page 571 and 572:
7.2. SECOND PREVALENT PROBLEM: 571r
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Page 573 and 574:
7.2. SECOND PREVALENT PROBLEM: 573w
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Page 575 and 576:
7.2. SECOND PREVALENT PROBLEM: 5757
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Page 577 and 578:
7.2. SECOND PREVALENT PROBLEM: 577a
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Page 579 and 580:
7.3. THIRD PREVALENT PROBLEM: 579is
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Page 581 and 582:
7.3. THIRD PREVALENT PROBLEM: 581We
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Page 583 and 584:
7.3. THIRD PREVALENT PROBLEM: 583su
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Page 585 and 586:
7.3. THIRD PREVALENT PROBLEM: 585Gi
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Page 587 and 588:
7.3. THIRD PREVALENT PROBLEM: 587Op
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Page 589 and 590:
7.4. CONCLUSION 589filtering, multi
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Page 591 and 592:
Appendix ALinear algebraA.1 Main-di
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Page 593 and 594:
A.1. MAIN-DIAGONAL δ OPERATOR, λ
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Page 595 and 596:
A.1. MAIN-DIAGONAL δ OPERATOR, λ
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Page 597 and 598:
A.2. SEMIDEFINITENESS: DOMAIN OF TE
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Page 599 and 600:
A.3. PROPER STATEMENTS 599(AB) T
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Page 601 and 602:
A.3. PROPER STATEMENTS 601A.3.1Semi
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Page 603 and 604:
A.3. PROPER STATEMENTS 603For A dia
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Page 605 and 606:
A.3. PROPER STATEMENTS 605Diagonali
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Page 607 and 608:
A.3. PROPER STATEMENTS 607For A,B
-
Page 609 and 610:
A.3. PROPER STATEMENTS 609When B is
-
Page 611 and 612:
A.4. SCHUR COMPLEMENT 611A.4 Schur
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Page 613 and 614:
A.4. SCHUR COMPLEMENT 613A.4.0.0.3
-
Page 615 and 616:
A.4. SCHUR COMPLEMENT 615From Corol
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Page 617 and 618:
A.5. EIGENVALUE DECOMPOSITION 617wh
-
Page 619 and 620:
A.5. EIGENVALUE DECOMPOSITION 619A.
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Page 621 and 622:
A.6. SINGULAR VALUE DECOMPOSITION,
-
Page 623 and 624:
A.6. SINGULAR VALUE DECOMPOSITION,
-
Page 625 and 626:
A.7. ZEROS 625A.6.5SVD of symmetric
-
Page 627 and 628:
A.7. ZEROS 627(Transpose.)Likewise,
-
Page 629 and 630:
A.7. ZEROS 629For X,A∈ S M +[34,2
-
Page 631 and 632:
A.7. ZEROS 631A.7.5.0.1 Proposition
-
Page 633 and 634:
Appendix BSimple matricesMathematic
-
Page 635 and 636:
B.1. RANK-ONE MATRIX (DYAD) 635R(v)
-
Page 637 and 638:
B.1. RANK-ONE MATRIX (DYAD) 637B.1.
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Page 639 and 640:
B.2. DOUBLET 639R([u v ])R(Π)= R([
-
Page 641 and 642:
B.3. ELEMENTARY MATRIX 641has N −
-
Page 643 and 644:
B.4. AUXILIARY V -MATRICES 643is an
-
Page 645 and 646:
B.4. AUXILIARY V -MATRICES 64514. [
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Page 647 and 648:
B.5. ORTHOGONAL MATRIX 647Given X
-
Page 649 and 650:
B.5. ORTHOGONAL MATRIX 649Figure 15
-
Page 651 and 652:
B.5. ORTHOGONAL MATRIX 651which is
-
Page 653 and 654:
Appendix CSome analytical optimal r
-
Page 655 and 656:
C.2. TRACE, SINGULAR AND EIGEN VALU
-
Page 657 and 658:
C.2. TRACE, SINGULAR AND EIGEN VALU
-
Page 659 and 660:
C.2. TRACE, SINGULAR AND EIGEN VALU
-
Page 661 and 662:
C.3. ORTHOGONAL PROCRUSTES PROBLEM
-
Page 663 and 664:
C.4. TWO-SIDED ORTHOGONAL PROCRUSTE
-
Page 665 and 666:
C.4. TWO-SIDED ORTHOGONAL PROCRUSTE
-
Page 667 and 668:
C.4. TWO-SIDED ORTHOGONAL PROCRUSTE
-
Page 669 and 670:
Appendix DMatrix calculusFrom too m
-
Page 671 and 672:
D.1. DIRECTIONAL DERIVATIVE, TAYLOR
-
Page 673 and 674:
D.1. DIRECTIONAL DERIVATIVE, TAYLOR
-
Page 675 and 676:
D.1. DIRECTIONAL DERIVATIVE, TAYLOR
-
Page 677 and 678:
D.1. DIRECTIONAL DERIVATIVE, TAYLOR
-
Page 679 and 680:
D.1. DIRECTIONAL DERIVATIVE, TAYLOR
-
Page 681 and 682:
D.1. DIRECTIONAL DERIVATIVE, TAYLOR
-
Page 683 and 684:
D.1. DIRECTIONAL DERIVATIVE, TAYLOR
-
Page 685 and 686:
D.1. DIRECTIONAL DERIVATIVE, TAYLOR
-
Page 687 and 688:
D.1. DIRECTIONAL DERIVATIVE, TAYLOR
-
Page 689 and 690:
D.1. DIRECTIONAL DERIVATIVE, TAYLOR
-
Page 691 and 692:
D.2. TABLES OF GRADIENTS AND DERIVA
-
Page 693 and 694:
D.2. TABLES OF GRADIENTS AND DERIVA
-
Page 695 and 696:
D.2. TABLES OF GRADIENTS AND DERIVA
-
Page 697 and 698:
D.2. TABLES OF GRADIENTS AND DERIVA
-
Page 699 and 700:
Appendix EProjectionFor any A∈ R
-
Page 701 and 702:
701U T = U † for orthonormal (inc
-
Page 703 and 704:
E.1. IDEMPOTENT MATRICES 703where A
-
Page 705 and 706:
E.1. IDEMPOTENT MATRICES 705order,
-
Page 707 and 708:
E.1. IDEMPOTENT MATRICES 707are lin
-
Page 709 and 710:
E.3. SYMMETRIC IDEMPOTENT MATRICES
-
Page 711 and 712:
E.3. SYMMETRIC IDEMPOTENT MATRICES
-
Page 713 and 714:
E.3. SYMMETRIC IDEMPOTENT MATRICES
-
Page 715 and 716:
E.4. ALGEBRA OF PROJECTION ON AFFIN
-
Page 717 and 718:
E.5. PROJECTION EXAMPLES 717a ∗ 2
-
Page 719 and 720:
E.5. PROJECTION EXAMPLES 719where Y
-
Page 721 and 722:
E.5. PROJECTION EXAMPLES 721(B.4.2)
-
Page 723 and 724:
E.6. VECTORIZATION INTERPRETATION,
-
Page 725 and 726:
E.6. VECTORIZATION INTERPRETATION,
-
Page 727 and 728:
E.6. VECTORIZATION INTERPRETATION,
-
Page 729 and 730:
E.7. PROJECTION ON MATRIX SUBSPACES
-
Page 731 and 732:
E.7. PROJECTION ON MATRIX SUBSPACES
-
Page 733 and 734:
E.8. RANGE/ROWSPACE INTERPRETATION
-
Page 735 and 736:
E.9. PROJECTION ON CONVEX SET 735As
-
Page 737 and 738:
E.9. PROJECTION ON CONVEX SET 737Wi
-
Page 739 and 740:
E.9. PROJECTION ON CONVEX SET 739R(
-
Page 741 and 742:
E.9. PROJECTION ON CONVEX SET 741E.
-
Page 743 and 744:
E.9. PROJECTION ON CONVEX SET 743E.
-
Page 745 and 746:
E.9. PROJECTION ON CONVEX SET 745Un
-
Page 747 and 748:
E.9. PROJECTION ON CONVEX SET 747ac
-
Page 749 and 750:
E.10. ALTERNATING PROJECTION 749bC
-
Page 751 and 752:
E.10. ALTERNATING PROJECTION 7510
-
Page 753 and 754:
E.10. ALTERNATING PROJECTION 753E.1
-
Page 755 and 756:
E.10. ALTERNATING PROJECTION 755y 2
-
Page 757 and 758:
E.10. ALTERNATING PROJECTION 757Def
-
Page 759 and 760:
E.10. ALTERNATING PROJECTION 759Dis
-
Page 761 and 762:
E.10. ALTERNATING PROJECTION 761mat
-
Page 763 and 764:
E.10. ALTERNATING PROJECTION 763K
-
Page 765 and 766:
E.10. ALTERNATING PROJECTION 765E.1
-
Page 767 and 768:
E.10. ALTERNATING PROJECTION 767E.1
-
Page 769 and 770:
Appendix FNotation and a few defini
-
Page 771 and 772:
771A ij or A(i, j) , ij th entry of
-
Page 773 and 774:
773⊞orthogonal vector sum of sets
-
Page 775 and 776:
775x +vector x whose negative entri
-
Page 777 and 778:
777X point list ((76) having cardin
-
Page 779 and 780:
779SDPSVDSNRdBEDMS n 1S n hS n⊥hS
-
Page 781 and 782:
781vectorentrycubixquartixfeasible
-
Page 783 and 784:
783Oorder of magnitude information
-
Page 785 and 786:
785cofmatrix of cofactors correspon
-
Page 787 and 788:
Bibliography[1] Edwin A. Abbott. Fl
-
Page 789 and 790:
BIBLIOGRAPHY 789[23] Dror Baron, Mi
-
Page 791 and 792:
BIBLIOGRAPHY 791[49] Leonard M. Blu
-
Page 793 and 794:
BIBLIOGRAPHY 793[74] Yves Chabrilla
-
Page 795 and 796:
BIBLIOGRAPHY 795[102] Etienne de Kl
-
Page 797 and 798:
BIBLIOGRAPHY 797[129] Carl Eckart a
-
Page 799 and 800:
BIBLIOGRAPHY 799[154] James Gleik.
-
Page 801 and 802:
BIBLIOGRAPHY 801[182] Johan Håstad
-
Page 803:
BIBLIOGRAPHY 803[212] Viren Jain an
-
Page 807 and 808:
BIBLIOGRAPHY 807[265] Sunderarajan
-
Page 809 and 810:
BIBLIOGRAPHY 809Notes in Computer S
-
Page 811 and 812:
BIBLIOGRAPHY 811[319] Anthony Man-C
-
Page 813 and 814:
BIBLIOGRAPHY 813[346] Pham Dinh Tao
-
Page 815 and 816:
BIBLIOGRAPHY 815[375] Bernard Widro
-
Page 817 and 818:
Index∅, see empty set0-norm, 229,
-
Page 819 and 820:
INDEX 819bees, 30, 413Bellman, 276b
-
Page 821 and 822:
INDEX 821circular, 130construction,
-
Page 823 and 824:
INDEX 823measure, 295convex, 21, 36
-
Page 825 and 826:
INDEX 825duality, 160, 410gap, 161,
-
Page 827 and 828:
INDEX 827positive semidefinite, 284
-
Page 829 and 830:
INDEX 829point, 39, 62positive semi
-
Page 831 and 832:
INDEX 831isomorphic, 52, 56, 59, 77
-
Page 833 and 834:
INDEX 833nonsymmetric, 703range, 70
-
Page 835 and 836:
INDEX 835Muller, 624multidimensiona
-
Page 837 and 838:
INDEX 837projection on, 739, 745tra
-
Page 839 and 840:
INDEX 839pseudoinverse, 701trace, 5
-
Page 841 and 842:
INDEX 841coordinate system, 156line
-
Page 843 and 844:
INDEX 843nonconvex, 40, 97-101nulls
-
Page 845 and 846:
INDEX 845faces, 106intersection, 10
-
Page 847 and 848:
INDEX 847projection, 517symmetric,
-
Page 849 and 850:
849
-
Page 852:
Convex Optimization & Euclidean Dis
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