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738 BIBLIOGRAPHY [235] Katta G. Murty and Feng-Tien Yu. Linear Complementarity, Linear and Nonlinear Programming. Heldermann Verlag, Internet edition, 1988. ioe.engin.umich.edu/people/fac/books/murty/linear complementarity webbook [236] Oleg R. Musin. An extension of Delsarte’s method. The kissing problem in three and four dimensions. ArXiv.org, December 2005. http://arxiv.org/abs/math.MG/0512649 [237] Navraj Nandra. Synthesizable analog IP. IP Based Design Workshop, 2002. [238] Stephen G. Nash and Ariela Sofer. Linear and Nonlinear Programming. McGraw-Hill, 1996. [239] Yurii Nesterov and Arkadii Nemirovskii. Interior-Point Polynomial Algorithms in Convex Programming. SIAM, 1994. [240] L. Nirenberg. Functional Analysis. New York University, New York, 1961. Lectures given in 1960-1961, notes by Lesley Sibner. [241] Jorge Nocedal and Stephen J. Wright. Numerical Optimization. Springer-Verlag, 1999. [242] Royal Swedish Academy of Sciences. Nobel prize in chemistry, 2002. http://nobelprize.org/chemistry/laureates/2002/public.html [243] C. S. Ogilvy. Excursions in Geometry. Dover, 1990. Citation: Proceedings of the CUPM Geometry Conference, Mathematical Association of America, No.16 (1967), p.21. [244] Ricardo Oliveira, João Costeira, and João Xavier. Optimal point correspondence through the use of rank constraints. In Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition, volume 2, pages 1016–1021, June 2005. [245] Önder Filiz and Aylin Yener. Rank constrained temporal-spatial filters for CDMA systems with base station antenna arrays. In Proceedings of the Johns Hopkins University Conference on Information Sciences and Systems, March 2003. labs.ee.psu.edu/labs/wcan/Publications/filiz-yener ciss03.pdf [246] Alan V. Oppenheim and Ronald W. Schafer. Discrete-Time Signal Processing. Prentice-Hall, 1989. [247] Robert Orsi, Uwe Helmke, and John B. Moore. A Newton-like method for solving rank constrained linear matrix inequalities, April 2006. users.rsise.anu.edu.au/∼robert/publications/OHM06 extended version.pdf [248] Brad Osgood. Notes on the Ahlfors mapping of a multiply connected domain, 2000. www-ee.stanford.edu/∼osgood/Ahlfors-Bergman-Szego.pdf [249] M. L. Overton and R. S. Womersley. On the sum of the largest eigenvalues of a symmetric matrix. SIAM Journal on Matrix Analysis and Applications, 13:41–45, 1992.
BIBLIOGRAPHY 739 [250] Pythagoras Papadimitriou. Parallel Solution of SVD-Related Problems, With Applications. PhD thesis, Department of Mathematics, University of Manchester, October 1993. [251] Panos M. Pardalos and Henry Wolkowicz, editors. Topics in Semidefinite and Interior-Point Methods. American Mathematical Society, 1998. [252] Gábor Pataki. Cone-LP’s and semidefinite programs: Geometry and a simplex-type method. In William H. Cunningham, S. Thomas McCormick, and Maurice Queyranne, editors, Integer Programming and Combinatorial Optimization, Proceedings of the 5th International IPCO Conference, Vancouver, British Columbia, Canada, June 3-5, 1996, volume 1084 of Lecture Notes in Computer Science, pages 162–174. Springer-Verlag, 1996. [253] Gábor Pataki. On the rank of extreme matrices in semidefinite programs and the multiplicity of optimal eigenvalues. Mathematics of Operations Research, 23(2):339–358, 1998. http://citeseer.ist.psu.edu/pataki97rank.html Erratum: p.601 herein. [254] Gábor Pataki. The geometry of semidefinite programming. In Henry Wolkowicz, Romesh Saigal, and Lieven Vandenberghe, editors, Handbook of Semidefinite Programming: Theory, Algorithms, and Applications, chapter 3. Kluwer, 2000. [255] Teemu Pennanen and Jonathan Eckstein. Generalized Jacobians of vector-valued convex functions. Technical Report RRR 6-97, RUTCOR, Rutgers University, May 1997. http://rutcor.rutgers.edu/pub/rrr/reports97/06.ps [256] Roger Penrose. A generalized inverse for matrices. In Proceedings of the Cambridge Philosophical Society, volume 51, pages 406–413, 1955. [257] Chris Perkins. A convergence analysis of Dykstra’s algorithm for polyhedral sets. SIAM Journal on Numerical Analysis, 40(2):792–804, 2002. [258] Florian Pfender and Günter M. Ziegler. Kissing numbers, sphere packings, and some unexpected proofs. Notices of the American Mathematical Society, 51(8):873–883, September 2004. http://www.convexoptimization.com/TOOLS/Pfender.pdf [259] Benjamin Recht. Convex Modeling with Priors. PhD thesis, Massachusetts Institute of Technology, Media Arts and Sciences Department, 2006. www.media.mit.edu/physics/publications/theses/06.06.Recht.pdf [260] Benjamin Recht, Maryam Fazel, and Pablo A. Parrilo. Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization. Optimization Online, June 2007. http://arxiv.org/abs/0706.4138 [261] Alex Reznik. Problem 3.41, from Sergio Verdú. Multiuser Detection. Cambridge University Press, 1998. www.ee.princeton.edu/∼verdu/mud/solutions/3/3.41.areznik.pdf , 2001.
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DATTORRO CONVEX OPTIMIZATION & EUCL
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Convex Optimization & Euclidean Dis
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for Jennie Columba ♦ Antonio ♦
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Prelude The constant demands of my
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Convex Optimization & Euclidean Dis
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CONVEX OPTIMIZATION & EUCLIDEAN DIS
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List of Figures 1 Overview 19 1 Ori
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LIST OF FIGURES 15 3 Geometry of co
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LIST OF FIGURES 17 126 Decomposing
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Chapter 1 Overview Convex Optimizat
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ˇx 4 ˇx 3 ˇx 2 Figure 2: Applica
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23 Figure 4: This coarsely discreti
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ases (biorthogonal expansion). We e
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27 Figure 7: These bees construct a
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that establish its membership to th
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31 appendices Provided so as to be
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Chapter 2 Convex geometry Convexity
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2.1. CONVEX SET 35 2.1.2 linear ind
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2.1. CONVEX SET 37 2.1.6 empty set
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2.1. CONVEX SET 39 2.1.7.1 Line int
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2.1. CONVEX SET 41 (a) R 2 (b) R 3
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2.1. CONVEX SET 43 This theorem in
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2.2. VECTORIZED-MATRIX INNER PRODUC
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2.3. HULLS 55 Figure 16: Convex hul
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2.3. HULLS 57 The affine hull of tw
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2.3. HULLS 59 2.3.2 Convex hull The
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2.3. HULLS 61 In case k = N , the F
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2.3. HULLS 63 2.3.2.0.3 Exercise. C
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2.3. HULLS 65 Figure 20: A simplici
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2.4. HALFSPACE, HYPERPLANE 67 H + a
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2.4. HALFSPACE, HYPERPLANE 69 1 1
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2.4. HALFSPACE, HYPERPLANE 71 Recal
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2.4. HALFSPACE, HYPERPLANE 73 C H
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2.4. HALFSPACE, HYPERPLANE 75 2.4.2
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2.4. HALFSPACE, HYPERPLANE 77 (conf
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2.5. SUBSPACE REPRESENTATIONS 79 Ra
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2.5. SUBSPACE REPRESENTATIONS 81 If
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2.6. EXTREME, EXPOSED 83 2.6 Extrem
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2.6. EXTREME, EXPOSED 85 A B C D Fi
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2.7. CONES 87 2.6.1.3.1 Definition.
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2.7. CONES 89 0 Figure 30: Boundary
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2.7. CONES 91 2.7.2 Convex cone We
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2.7. CONES 93 Then a pointed closed
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2.7. CONES 95 A pointed closed conv
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2.8. CONE BOUNDARY 97 So the ray th
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2.8. CONE BOUNDARY 99 2.8.1.1 extre
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2.8. CONE BOUNDARY 101 2.8.2 Expose
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2.8. CONE BOUNDARY 103 From Theorem
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2.9. POSITIVE SEMIDEFINITE (PSD) CO
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2.10. CONIC INDEPENDENCE (C.I.) 127
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2.12. CONVEX POLYHEDRA 133 It follo
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2.12. CONVEX POLYHEDRA 135 Coeffici
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2.12. CONVEX POLYHEDRA 137 2.12.3 U
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2.12. CONVEX POLYHEDRA 139
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2.13. DUAL CONE & GENERALIZED INEQU
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Chapter 3 Geometry of convex functi
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3.1. CONVEX FUNCTION 197 f 1 (x) f
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3.1. CONVEX FUNCTION 199 3.1.3 norm
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3.1. CONVEX FUNCTION 201 A B 1 Figu
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3.1. CONVEX FUNCTION 203 k/m 1 0.9
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k∑ i=1 3.1. CONVEX FUNCTION 205 S
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3.1. CONVEX FUNCTION 207 rather x >
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3.1. CONVEX FUNCTION 209 rather ] x
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3.1. CONVEX FUNCTION 211 3.1.6.0.2
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3.1. CONVEX FUNCTION 213 q(x) f(x)
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3.1. CONVEX FUNCTION 215 3.1.7.0.2
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3.1. CONVEX FUNCTION 217 We learned
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3.1. CONVEX FUNCTION 219 Since opti
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3.1. CONVEX FUNCTION 221 2 1.5 1 0.
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3.1. CONVEX FUNCTION 223 Setting th
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3.1. CONVEX FUNCTION 225 Similarly,
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3.1. CONVEX FUNCTION 227 For vector
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3.1. CONVEX FUNCTION 229 This means
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3.1. CONVEX FUNCTION 231 f(Y ) −
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3.2. MATRIX-VALUED CONVEX FUNCTION
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3.3. QUASICONVEX 239 exponential al
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3.3. QUASICONVEX 241 Unlike convex
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3.4. SALIENT PROPERTIES 243 6. (af
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Chapter 4 Semidefinite programming
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4.1. CONIC PROBLEM 247 where K is a
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4.1. CONIC PROBLEM 249 4.1.1.2 Redu
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4.1. CONIC PROBLEM 251 In any SDP f
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4.1. CONIC PROBLEM 253 Proposition
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4.2. FRAMEWORK 255 sets are closed
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4.2. FRAMEWORK 257 4.2.1.1.3 Exampl
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4.2. FRAMEWORK 259 4.2.2 Duals The
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4.2. FRAMEWORK 261 When equality is
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4.2. FRAMEWORK 263 The pseudoinvers
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4.2. FRAMEWORK 265 For the data giv
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4.2. FRAMEWORK 267 minimizes an aff
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4.3. RANK REDUCTION 269 whose rank
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4.3. RANK REDUCTION 271 and where m
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4.3. RANK REDUCTION 273 4.3.3 Optim
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4.3. RANK REDUCTION 275 Initialize:
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4.4. RANK-CONSTRAINED SEMIDEFINITE
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4.5. CONSTRAINING CARDINALITY 295 m
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4.5. CONSTRAINING CARDINALITY 297 m
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4.5. CONSTRAINING CARDINALITY 299 a
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4.5. CONSTRAINING CARDINALITY 301 f
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4.5. CONSTRAINING CARDINALITY 303 n
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4.5. CONSTRAINING CARDINALITY 305 W
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4.5. CONSTRAINING CARDINALITY 307 t
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4.6. CARDINALITY AND RANK CONSTRAIN
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4.7. CONVEX ITERATION RANK-1 341 fi
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4.7. CONVEX ITERATION RANK-1 343 Gi
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Chapter 5 Euclidean Distance Matrix
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5.2. FIRST METRIC PROPERTIES 347 co
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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 353 The collect
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5.4. EDM DEFINITION 355 5.4.2 Gram-
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5.4. EDM DEFINITION 357 D ∈ EDM N
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5.4. EDM DEFINITION 359 5.4.2.2.1 E
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5.4. EDM DEFINITION 361 ten affine
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5.4. EDM DEFINITION 363 spheres: Th
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5.4. EDM DEFINITION 365 By eliminat
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5.4. EDM DEFINITION 367 where Φ ij
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5.4. EDM DEFINITION 369 5.4.2.2.6 D
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5.4. EDM DEFINITION 371 10 5 ˇx 4
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5.4. EDM DEFINITION 373 corrected b
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5.4. EDM DEFINITION 375 by translat
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5.4. EDM DEFINITION 377 Crippen & H
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5.4. EDM DEFINITION 379 where ([√
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5.4. EDM DEFINITION 381 because (A.
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5.5. INVARIANCE 383 5.5.1.0.1 Examp
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5.5. INVARIANCE 385 x 2 x 2 x 3 x 1
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5.6. INJECTIVITY OF D & UNIQUE RECO
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5.7. EMBEDDING IN AFFINE HULL 393 5
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5.7. EMBEDDING IN AFFINE HULL 395 F
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5.7. EMBEDDING IN AFFINE HULL 397 5
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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 413 of
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5.10. EDM-ENTRY COMPOSITION 415 The
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5.11. EDM INDEFINITENESS 417 5.11.1
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5.11. EDM INDEFINITENESS 419 we hav
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5.11. EDM INDEFINITENESS 421 So bec
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5.11. EDM INDEFINITENESS 423 where
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5.12. LIST RECONSTRUCTION 425 where
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5.12. LIST RECONSTRUCTION 427 (a) (
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5.13. RECONSTRUCTION EXAMPLES 429 D
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5.13. RECONSTRUCTION EXAMPLES 431 T
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5.13. RECONSTRUCTION EXAMPLES 433 w
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5.14. FIFTH PROPERTY OF EUCLIDEAN M
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Chapter 6 Cone of distance matrices
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6.1. DEFINING EDM CONE 447 6.1 Defi
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6.2. POLYHEDRAL BOUNDS 449 This con
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6.3. √ EDM CONE IS NOT CONVEX 451
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6.4. A GEOMETRY OF COMPLETION 453 (
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6.4. A GEOMETRY OF COMPLETION 455 (
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6.4. A GEOMETRY OF COMPLETION 457 F
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6.5. EDM DEFINITION IN 11 T 459 by
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6.5. EDM DEFINITION IN 11 T 461 6.5
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6.5. EDM DEFINITION IN 11 T 463 1 0
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6.5. EDM DEFINITION IN 11 T 465 6.5
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6.6. CORRESPONDENCE TO PSD CONE S N
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6.7. VECTORIZATION & PROJECTION INT
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6.7. VECTORIZATION & PROJECTION INT
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6.8. DUAL EDM CONE 477 When the Fin
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6.8. DUAL EDM CONE 479 Proof. First
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6.8. DUAL EDM CONE 481 EDM 2 = S 2
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6.8. DUAL EDM CONE 483 whose veraci
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6.8. DUAL EDM CONE 485 6.8.1.3.1 Ex
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6.8. DUAL EDM CONE 487 has dual aff
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6.8. DUAL EDM CONE 489 6.8.1.7 Scho
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6.9. THEOREM OF THE ALTERNATIVE 491
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6.10. POSTSCRIPT 493 When D is an E
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Chapter 7 Proximity problems In sum
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497 project on the subspace, then p
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499 H S N h 0 EDM N K = S N h ∩ R
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501 7.0.3 Problem approach Problems
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7.1. FIRST PREVALENT PROBLEM: 503 f
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7.1. FIRST PREVALENT PROBLEM: 505 7
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7.1. FIRST PREVALENT PROBLEM: 507 d
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7.1. FIRST PREVALENT PROBLEM: 509 7
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7.1. FIRST PREVALENT PROBLEM: 511 w
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7.1. FIRST PREVALENT PROBLEM: 513 T
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7.2. SECOND PREVALENT PROBLEM: 515
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7.3. THIRD PREVALENT PROBLEM: 525 g
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7.3. THIRD PREVALENT PROBLEM: 527 w
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7.3. THIRD PREVALENT PROBLEM: 529 7
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7.3. THIRD PREVALENT PROBLEM: 531 7
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7.3. THIRD PREVALENT PROBLEM: 533 O
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7.4. CONCLUSION 535 The rank constr
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Appendix A Linear algebra A.1 Main-
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A.1. MAIN-DIAGONAL δ OPERATOR, λ
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A.1. MAIN-DIAGONAL δ OPERATOR, λ
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A.2. SEMIDEFINITENESS: DOMAIN OF TE
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A.3. PROPER STATEMENTS 545 (AB) T
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A.3. PROPER STATEMENTS 547 A.3.1 Se
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A.3. PROPER STATEMENTS 549 For A di
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A.3. PROPER STATEMENTS 551 Diagonal
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A.3. PROPER STATEMENTS 553 For A,B
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A.3. PROPER STATEMENTS 555 A.3.1.0.
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A.4. SCHUR COMPLEMENT 557 A.4 Schur
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A.4. SCHUR COMPLEMENT 559 A.4.0.0.2
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A.5. EIGEN DECOMPOSITION 561 When B
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A.5. EIGEN DECOMPOSITION 563 dim N(
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A.6. SINGULAR VALUE DECOMPOSITION,
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A.6. SINGULAR VALUE DECOMPOSITION,
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A.7. ZEROS 569 Given symmetric matr
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A.7. ZEROS 571 (TRANSPOSE.) Likewis
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A.7. ZEROS 573 For X,A∈ S M + [31
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A.7. ZEROS 575 A.7.5.0.1 Propositio
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Appendix B Simple matrices Mathemat
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B.1. RANK-ONE MATRIX (DYAD) 579 R(v
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B.1. RANK-ONE MATRIX (DYAD) 581 ran
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B.2. DOUBLET 583 R([u v ]) R(Π)= R
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B.3. ELEMENTARY MATRIX 585 If λ
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B.4. AUXILIARY V -MATRICES 587 the
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B.4. AUXILIARY V -MATRICES 589 18.
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B.5. ORTHOGONAL MATRIX 591 B.5 Orth
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B.5. ORTHOGONAL MATRIX 593 Figure 1
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Appendix C Some analytical optimal
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C.2. TRACE, SINGULAR AND EIGEN VALU
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C.3. ORTHOGONAL PROCRUSTES PROBLEM
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C.4. TWO-SIDED ORTHOGONAL PROCRUSTE
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C.4. TWO-SIDED ORTHOGONAL PROCRUSTE
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Appendix D Matrix calculus From too
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D.1. DIRECTIONAL DERIVATIVE, TAYLOR
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D.2. TABLES OF GRADIENTS AND DERIVA
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Appendix E Projection For any A∈
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641 U T = U † for orthonormal (in
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E.1. IDEMPOTENT MATRICES 643 where
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E.1. IDEMPOTENT MATRICES 645 order,
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E.1. IDEMPOTENT MATRICES 647 When t
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E.3. SYMMETRIC IDEMPOTENT MATRICES
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E.5. PROJECTION EXAMPLES 655 E.4.0.
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E.5. PROJECTION EXAMPLES 657 a ∗
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E.5. PROJECTION EXAMPLES 659 E.5.0.
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E.6. VECTORIZATION INTERPRETATION,
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E.7. ON VECTORIZED MATRICES OF HIGH
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E.7. ON VECTORIZED MATRICES OF HIGH
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E.8. RANGE/ROWSPACE INTERPRETATION
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E.9. PROJECTION ON CONVEX SET 675 A
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E.9. PROJECTION ON CONVEX SET 677 W
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E.9. PROJECTION ON CONVEX SET 679 P
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E.9. PROJECTION ON CONVEX SET 681 E
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E.9. PROJECTION ON CONVEX SET 683 T
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E.9. PROJECTION ON CONVEX SET 685
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Page 705 and 706: Appendix F Notation and a few defin
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Page 717 and 718: 717 O O sort-index matrix order of
Page 719 and 720: (x,y) angle between vectors x and y
Page 721 and 722: Bibliography [1] Suliman Al-Homidan
Page 723 and 724: BIBLIOGRAPHY 723 [24] Alexander I.
Page 725 and 726: BIBLIOGRAPHY 725 [52] Stephen Boyd,
Page 727 and 728: BIBLIOGRAPHY 727 [78] Frank Critchl
Page 729 and 730: BIBLIOGRAPHY 729 [105] Richard L. D
Page 731 and 732: BIBLIOGRAPHY 731 [132] Michel X. Go
Page 733 and 734: BIBLIOGRAPHY 733 [162] T. Herrmann,
Page 735 and 736: BIBLIOGRAPHY 735 [191] Mark Kahrs a
Page 737: BIBLIOGRAPHY 737 [220] K. V. Mardia
Page 741 and 742: BIBLIOGRAPHY 741 [277] Anthony Man-
Page 743 and 744: BIBLIOGRAPHY 743 [306] Michael W. T
Page 745 and 746: [333] Margaret H. Wright. The inter
Page 747 and 748: Index 0-norm, 203, 261, 294, 296, 2
Page 749 and 750: INDEX 749 product, 43, 92, 147, 254
Page 751 and 752: INDEX 751 coordinates, 140, 170, 17
Page 753 and 754: INDEX 753 affine dimension, 485 fea
Page 755 and 756: INDEX 755 affine, 209 nonlinear, 19
Page 757 and 758: INDEX 757 of point, 37 ray, 90 rela
Page 759 and 760: INDEX 759 normal, 47, 548, 563 norm
Page 761 and 762: INDEX 761 strictly, 515, 520 functi
Page 763 and 764: INDEX 763 vector, 45, 241, 248, 325
Page 765 and 766: INDEX 765 convex envelope, see conv
Page 767 and 768: INDEX 767 cone, 418, 420, 507 dual,
Page 769 and 770: INDEX 769 trilateration, 21, 42, 36
Page 772: Convex Optimization & Euclidean Dis
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