v2009.01.01 - Convex Optimization

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24 CHAPTER 1. OVERVIEW Figure 5: Swiss roll from Weinberger & Saul [325]. The problem of manifold learning, illustrated for N =800 data points sampled from a “Swiss roll” 1Ç. A discretized manifold is revealed by connecting each data point and its k=6 nearest neighbors 2Ç. An unsupervised learning algorithm unfolds the Swiss roll while preserving the local geometry of nearby data points 3Ç. Finally, the data points are projected onto the two dimensional subspace that maximizes their variance, yielding a faithful embedding of the original manifold 4Ç.
ases (biorthogonal expansion). We explain conversion between halfspaceand vertex-description of a convex cone, we motivate the dual cone and provide formulae for finding it, and we show how first-order optimality conditions or alternative systems of linear inequality or linear matrix inequality [51] can be explained by generalized inequalities with respect to convex cones and their duals. The conic analogue to linear independence, called conic independence, is introduced as a new tool in the study of cones; the logical next step in the progression: linear, affine, conic. Any convex optimization problem can be visualized geometrically. Desire to visualize in high dimension is deeply embedded in the mathematical psyche. Chapter 2 provides tools to make visualization easier, and we teach how to visualize in high dimension. The concepts of face, extreme point, and extreme direction of a convex Euclidean body are explained here; crucial to understanding convex optimization. The convex cone of positive semidefinite matrices, in particular, is studied in depth: We interpret, for example, inverse image of the positive semidefinite cone under affine transformation. (Example 2.9.1.0.2) Subsets of the positive semidefinite cone, discriminated by rank exceeding some lower bound, are convex. In other words, high-rank subsets of the positive semidefinite cone boundary united with its interior are convex. (Theorem 2.9.2.6.3) There is a closed form for projection on those convex subsets. The positive semidefinite cone is a circular cone in low dimension, while Geršgorin discs specify inscription of a polyhedral cone into that positive semidefinite cone. (Figure 42) Chapter 3, Geometry of convex functions, observes analogies between convex sets and functions: We explain, for example, how the real affine function relates to convex functions as the hyperplane relates to convex sets. Partly a cookbook for the most useful of convex functions and optimization problems, methods are drawn from matrix calculus for determining convexity and discerning geometry. Semidefinite programming is reviewed in chapter 4 with particular attention to optimality conditions for prototypical primal and dual programs, their interplay, and the perturbation method of rank reduction of optimal solutions (extant but not well known). Positive definite Farkas’ lemma 25
Page 1 and 2: DATTORRO CONVEX OPTIMIZATION & EUCL
Page 3 and 4: Convex Optimization & Euclidean Dis
Page 5 and 6: for Jennie Columba ♦ Antonio ♦
Page 7 and 8: Prelude The constant demands of my
Page 9 and 10: Convex Optimization & Euclidean Dis
Page 11 and 12: CONVEX OPTIMIZATION & EUCLIDEAN DIS
Page 13 and 14: List of Figures 1 Overview 19 1 Ori
Page 15 and 16: LIST OF FIGURES 15 3 Geometry of co
Page 17 and 18: LIST OF FIGURES 17 126 Decomposing
Page 19 and 20: Chapter 1 Overview Convex Optimizat
Page 21 and 22: ˇx 4 ˇx 3 ˇx 2 Figure 2: Applica
Page 23: 23 Figure 4: This coarsely discreti
Page 27 and 28: 27 Figure 7: These bees construct a
Page 29 and 30: that establish its membership to th
Page 31 and 32: 31 appendices Provided so as to be
Page 33 and 34: Chapter 2 Convex geometry Convexity
Page 35 and 36: 2.1. CONVEX SET 35 2.1.2 linear ind
Page 37 and 38: 2.1. CONVEX SET 37 2.1.6 empty set
Page 39 and 40: 2.1. CONVEX SET 39 2.1.7.1 Line int
Page 41 and 42: 2.1. CONVEX SET 41 (a) R 2 (b) R 3
Page 43 and 44: 2.1. CONVEX SET 43 This theorem in
Page 45 and 46: 2.2. VECTORIZED-MATRIX INNER PRODUC
Page 55 and 56: 2.3. HULLS 55 Figure 16: Convex hul
Page 57 and 58: 2.3. HULLS 57 The affine hull of tw
Page 59 and 60: 2.3. HULLS 59 2.3.2 Convex hull The
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Page 63 and 64: 2.3. HULLS 63 2.3.2.0.3 Exercise. C
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Page 67 and 68: 2.4. HALFSPACE, HYPERPLANE 67 H + a
Page 69 and 70: 2.4. HALFSPACE, HYPERPLANE 69 1 1
Page 71 and 72: 2.4. HALFSPACE, HYPERPLANE 71 Recal
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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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E.10. ALTERNATING PROJECTION 687 E.
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E.10. ALTERNATING PROJECTION 689 b
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E.10. ALTERNATING PROJECTION 691 a
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E.10. ALTERNATING PROJECTION 693 (a
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E.10. ALTERNATING PROJECTION 695 wh
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E.10. ALTERNATING PROJECTION 699 10
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E.10. ALTERNATING PROJECTION 703 E
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Appendix F Notation and a few defin
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707 a.i. c.i. l.i. w.r.t affinely i
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709 is or ← → t → 0 + as in
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711 ∑ π(γ) Ξ Π ∏ ψ(Z) D D
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713 R m×n Euclidean vector space o
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715 H − H + ∂H ∂H ∂H −
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717 O O sort-index matrix order of
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(x,y) angle between vectors x and y
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Bibliography [1] Suliman Al-Homidan
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BIBLIOGRAPHY 723 [24] Alexander I.
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BIBLIOGRAPHY 725 [52] Stephen Boyd,
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BIBLIOGRAPHY 727 [78] Frank Critchl
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BIBLIOGRAPHY 729 [105] Richard L. D
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BIBLIOGRAPHY 731 [132] Michel X. Go
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BIBLIOGRAPHY 733 [162] T. Herrmann,
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BIBLIOGRAPHY 735 [191] Mark Kahrs a
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BIBLIOGRAPHY 737 [220] K. V. Mardia
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BIBLIOGRAPHY 739 [250] Pythagoras P
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BIBLIOGRAPHY 741 [277] Anthony Man-
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BIBLIOGRAPHY 743 [306] Michael W. T
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[333] Margaret H. Wright. The inter
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Index 0-norm, 203, 261, 294, 296, 2
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INDEX 749 product, 43, 92, 147, 254
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INDEX 751 coordinates, 140, 170, 17
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INDEX 753 affine dimension, 485 fea
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INDEX 755 affine, 209 nonlinear, 19
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INDEX 757 of point, 37 ray, 90 rela
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INDEX 759 normal, 47, 548, 563 norm
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INDEX 761 strictly, 515, 520 functi
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INDEX 763 vector, 45, 241, 248, 325
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INDEX 765 convex envelope, see conv
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INDEX 767 cone, 418, 420, 507 dual,
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INDEX 769 trilateration, 21, 42, 36
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Convex Optimization & Euclidean Dis
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