v2007.09.13 - Convex Optimization

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24 CHAPTER 1. OVERVIEWFigure 5: Swiss roll from Weinberger & Saul [278]. The problem of manifoldlearning, 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=6nearest neighbors 2Ç. An unsupervised learning algorithm unfolds the Swissroll while preserving the local geometry of nearby data points 3Ç. Finally, thedata points are projected onto the two dimensional subspace that maximizestheir variance, yielding a faithful embedding of the original manifold 4Ç.
ases (biorthogonal expansion). We explain conversion between halfspaceandvertex-description of a convex cone, we motivate the dual cone andprovide formulae for finding it, and we show how first-order optimalityconditions or alternative systems of linear inequality or linear matrixinequality [44] can be explained by generalized inequalities with respect toconvex 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 themathematical 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 areexplained here; crucial to understanding convex optimization. The convexcone of positive semidefinite matrices, in particular, is studied in depth:We interpret, for example, inverse image of the positive semidefinitecone under affine transformation. (Example 2.9.1.0.2)Subsets of the positive semidefinite cone, discriminated by rankexceeding some lower bound, are convex. In other words, high-ranksubsets of the positive semidefinite cone boundary united with itsinterior are convex. (Theorem 2.9.2.6.3) There is a closed form forprojection 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 thatpositive semidefinite cone. (Figure 36)Chapter 3, Geometry of convex functions, observes analogiesbetween convex sets and functions: We explain, for example, how the realaffine function relates to convex functions as the hyperplane relates to convexsets. Multidimensional convex functions are the focus; methods drawing frommatrix calculus for determining convexity and discerning their geometry.Semidefinite programming is reviewed in chapter 4 with particularattention to optimality conditions of prototypical primal and dual conicprograms, their interplay, and the perturbation method of rank reductionof optimal solutions (extant but not well-known). Positive definite Farkas’lemma is derived, and we also show how to determine if a feasible set belongs25
Page 1 and 2: DATTORROCONVEXOPTIMIZATION&EUCLIDEA
Page 3 and 4: Convex Optimization&Euclidean Dista
Page 5 and 6: for Jennie Columba♦Antonio♦♦&
Page 7 and 8: PreludeThe constant demands of my d
Page 9 and 10: Convex Optimization&Euclidean Dista
Page 11 and 12: CONVEX OPTIMIZATION & EUCLIDEAN DIS
Page 13 and 14: List of Figures1 Overview 191 Orion
Page 15 and 16: LIST OF FIGURES 1559 Quadratic func
Page 17 and 18: LIST OF FIGURES 17E Projection 5791
Page 19 and 20: Chapter 1OverviewConvex Optimizatio
Page 21 and 22: ˇx 4ˇx 3ˇx 2Figure 2: Applicatio
Page 23: 23Figure 4: This coarsely discretiz
Page 27 and 28: 27Figure 7: These bees construct a
Page 29 and 30: its membership to the EDM cone. The
Page 31 and 32: 31appendicesProvided so as to be mo
Page 33 and 34: Chapter 2Convex geometryConvexity h
Page 35 and 36: 2.1. CONVEX SET 35Figure 9: A slab
Page 37 and 38: 2.1. CONVEX SET 372.1.6 empty set v
Page 39 and 40: 2.1. CONVEX SET 392.1.7.1 Line inte
Page 41 and 42: 2.1. CONVEX SET 41(a)R 2(b)R 3(c)(d
Page 43 and 44: 2.1. CONVEX SET 43This theorem in c
Page 45 and 46: 2.2. VECTORIZED-MATRIX INNER PRODUC
Page 53 and 54: 2.3. HULLS 53Figure 12: Convex hull
Page 55 and 56: 2.3. HULLS 55Aaffine hull (drawn tr
Page 57 and 58: 2.3. HULLS 57The union of relative
Page 59 and 60: 2.4. HALFSPACE, HYPERPLANE 59of dim
Page 61 and 62: 2.4. HALFSPACE, HYPERPLANE 61H +ay
Page 63 and 64: 2.4. HALFSPACE, HYPERPLANE 63Inters
Page 65 and 66: 2.4. HALFSPACE, HYPERPLANE 65Conver
Page 67 and 68: 2.4. HALFSPACE, HYPERPLANE 67A 1A 2
Page 69 and 70: 2.4. HALFSPACE, HYPERPLANE 69tradit
Page 71 and 72: 2.4. HALFSPACE, HYPERPLANE 71There
Page 73 and 74: 2.5. SUBSPACE REPRESENTATIONS 732.5
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2.5. SUBSPACE REPRESENTATIONS 752.5
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2.6. EXTREME, EXPOSED 77In other wo
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2.6. EXTREME, EXPOSED 792.6.1 Expos
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2.7. CONES 812.6.1.3.1 Definition.
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2.7. CONES 830Figure 24: Boundary o
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2.7. CONES 852.7.2 Convex coneWe ca
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2.7. CONES 87Thus the simplest and
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2.7. CONES 89nomenclature generaliz
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2.8. CONE BOUNDARY 91Proper cone {0
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2.8. CONE BOUNDARY 93the same extre
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96 CHAPTER 2. CONVEX GEOMETRYBCADFi
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98 CHAPTER 2. CONVEX GEOMETRYThe po
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100 CHAPTER 2. CONVEX GEOMETRY2.9.0
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102 CHAPTER 2. CONVEX GEOMETRYwhere
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104 CHAPTER 2. CONVEX GEOMETRY√2
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106 CHAPTER 2. CONVEX GEOMETRYwhich
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108 CHAPTER 2. CONVEX GEOMETRY2.9.2
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110 CHAPTER 2. CONVEX GEOMETRYA con
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2.9. POSITIVE SEMIDEFINITE (PSD) CO
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2.10. CONIC INDEPENDENCE (C.I.) 121
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2.12. CONVEX POLYHEDRA 127It follow
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2.12. CONVEX POLYHEDRA 129Coefficie
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2.12. CONVEX POLYHEDRA 1312.12.3 Un
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2.12. CONVEX POLYHEDRA 133
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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 185f 1 (x)f 2
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3.1. CONVEX FUNCTION 1873.1.3 norm
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3.1. CONVEX FUNCTION 189where the n
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3.1. CONVEX FUNCTION 191where k ∈
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3.1. CONVEX FUNCTION 193f(z)Az 2z 1
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3.1. CONVEX FUNCTION 195{a T z 1 +
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3.1. CONVEX FUNCTION 197When an epi
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3.1. CONVEX FUNCTION 199orthonormal
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3.1. CONVEX FUNCTION 201[30,1.1] Ex
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3.1. CONVEX FUNCTION 20321.510.5Y 2
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3.1. CONVEX FUNCTION 2053.1.8.0.1 E
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3.1. CONVEX FUNCTION 207This equiva
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3.1. CONVEX FUNCTION 2093.1.8.1 mon
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3.1. CONVEX FUNCTION 211[ Yt]∈ ep
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3.1. CONVEX FUNCTION 213→Y −Xwh
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3.2. MATRIX-VALUED CONVEX FUNCTION
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3.3. QUASICONVEX 221A quasiconcave
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3.4. SALIENT PROPERTIES 2236.A nonn
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Chapter 4Semidefinite programmingPr
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4.1. CONIC PROBLEM 227where K is a
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4.1. CONIC PROBLEM 229C0PΓ 1Γ 2S+
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4.1. CONIC PROBLEM 231faces of S 3
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4.1. CONIC PROBLEM 2334.1.1.3 Previ
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4.2. FRAMEWORK 235Equivalently, pri
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4.2. FRAMEWORK 237is positive semid
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4.2. FRAMEWORK 239Optimal value of
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4.2. FRAMEWORK 2414.2.3.0.2 Example
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4.2. FRAMEWORK 243where δ is the m
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4.2. FRAMEWORK 2454.2.3.0.3 Example
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4.3. RANK REDUCTION 2474.3 Rank red
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4.3. RANK REDUCTION 249A rank-reduc
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4.3. RANK REDUCTION 251(t ⋆ i)
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4.3. RANK REDUCTION 2534.3.3.0.1 Ex
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4.3. RANK REDUCTION 2554.3.3.0.2 Ex
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4.4. RANK-CONSTRAINED SEMIDEFINITE
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Chapter 5Euclidean Distance MatrixT
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5.2. FIRST METRIC PROPERTIES 291cor
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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 297The collecti
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5.4. EDM DEFINITION 2995.4.2 Gram-f
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5.4. EDM DEFINITION 301D ∈ EDM N
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5.4. EDM DEFINITION 3035.4.2.2.1 Ex
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5.4. EDM DEFINITION 305ten affine e
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5.4. EDM DEFINITION 307spheres:Then
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5.4. EDM DEFINITION 309By eliminati
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5.4. EDM DEFINITION 311whereΦ ij =
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5.4. EDM DEFINITION 3135.4.2.2.5 De
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5.4. EDM DEFINITION 315105ˇx 4ˇx
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5.4. EDM DEFINITION 317corrected by
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5.4. EDM DEFINITION 319aptly be app
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5.4. EDM DEFINITION 321As before, a
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5.4. EDM DEFINITION 323where ([√t
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5.4. EDM DEFINITION 325because (A.3
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5.5. INVARIANCE 3275.5.1.0.1 Exampl
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5.6. INJECTIVITY OF D & UNIQUE RECO
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5.7. EMBEDDING IN AFFINE HULL 3355.
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5.7. EMBEDDING IN AFFINE HULL 337Fo
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5.7. EMBEDDING IN AFFINE HULL 3395.
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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 357(ii)
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5.11. EDM INDEFINITENESS 3595.11.1
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5.11. EDM INDEFINITENESS 361(confer
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5.11. EDM INDEFINITENESS 363we have
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5.11. EDM INDEFINITENESS 365For pre
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5.12. LIST RECONSTRUCTION 367where
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5.12. LIST RECONSTRUCTION 369(a)(c)
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5.13. RECONSTRUCTION EXAMPLES 371D
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5.13. RECONSTRUCTION EXAMPLES 373Th
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5.13. RECONSTRUCTION EXAMPLES 375wh
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5.14. FIFTH PROPERTY OF EUCLIDEAN M
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Chapter 6EDM coneFor N > 3, the con
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6.1. DEFINING EDM CONE 3896.1 Defin
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6.2. POLYHEDRAL BOUNDS 391This cone
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6.3.√EDM CONE IS NOT CONVEX 393N
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6.4. A GEOMETRY OF COMPLETION 3956.
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6.4. A GEOMETRY OF COMPLETION 397(a
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6.4. A GEOMETRY OF COMPLETION 399Fi
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6.5. EDM DEFINITION IN 11 T 401and
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6.5. EDM DEFINITION IN 11 T 403then
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6.5. EDM DEFINITION IN 11 T 4056.5.
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6.5. EDM DEFINITION IN 11 T 407D =
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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 419When the Fins
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6.8. DUAL EDM CONE 421Proof. First,
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6.8. DUAL EDM CONE 423EDM 2 = S 2 h
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6.8. DUAL EDM CONE 425whose veracit
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6.8. DUAL EDM CONE 4276.8.1.3.1 Exe
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6.8. DUAL EDM CONE 429has dual affi
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6.8. DUAL EDM CONE 4316.8.1.7 Schoe
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6.9. THEOREM OF THE ALTERNATIVE 433
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6.10. POSTSCRIPT 435When D is an ED
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Chapter 7Proximity problemsIn summa
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In contrast, order of projection on
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441HS N h0EDM NK = S N h ∩ R N×N
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4437.0.3 Problem approachProblems t
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7.1. FIRST PREVALENT PROBLEM: 445fi
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7.1. FIRST PREVALENT PROBLEM: 4477.
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7.1. FIRST PREVALENT PROBLEM: 449di
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7.1. FIRST PREVALENT PROBLEM: 4517.
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7.1. FIRST PREVALENT PROBLEM: 453wh
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7.1. FIRST PREVALENT PROBLEM: 455Th
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7.2. SECOND PREVALENT PROBLEM: 457O
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7.2. SECOND PREVALENT PROBLEM: 459S
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7.2. SECOND PREVALENT PROBLEM: 461r
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7.2. SECOND PREVALENT PROBLEM: 463c
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7.2. SECOND PREVALENT PROBLEM: 4657
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7.3. THIRD PREVALENT PROBLEM: 467fo
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7.3. THIRD PREVALENT PROBLEM: 469a
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7.3. THIRD PREVALENT PROBLEM: 4717.
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7.3. THIRD PREVALENT PROBLEM: 4737.
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7.3. THIRD PREVALENT PROBLEM: 475Ou
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478 CHAPTER 7. PROXIMITY PROBLEMSth
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480 APPENDIX A. LINEAR ALGEBRAA.1.1
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484 APPENDIX A. LINEAR ALGEBRAonly
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486 APPENDIX A. LINEAR ALGEBRA(AB)
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490 APPENDIX A. LINEAR ALGEBRAFor A
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492 APPENDIX A. LINEAR ALGEBRADiago
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494 APPENDIX A. LINEAR ALGEBRAFor A
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498 APPENDIX A. LINEAR ALGEBRAA.4 S
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502 APPENDIX A. LINEAR ALGEBRAA.5 e
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504 APPENDIX A. LINEAR ALGEBRAs i w
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508 APPENDIX A. LINEAR ALGEBRAΣq 2
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510 APPENDIX A. LINEAR ALGEBRAA.7 Z
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512 APPENDIX A. LINEAR ALGEBRAThere
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516 APPENDIX A. LINEAR ALGEBRA
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518 APPENDIX B. SIMPLE MATRICESB.1
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520 APPENDIX B. SIMPLE MATRICESProo
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522 APPENDIX B. SIMPLE MATRICESB.1.
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524 APPENDIX B. SIMPLE MATRICESN(u
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526 APPENDIX B. SIMPLE MATRICESDue
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530 APPENDIX B. SIMPLE MATRICEShas
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532 APPENDIX B. SIMPLE MATRICESFigu
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536 APPENDIX C. SOME ANALYTICAL OPT
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550 APPENDIX D. MATRIX CALCULUSThe
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552 APPENDIX D. MATRIX CALCULUSGrad
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554 APPENDIX D. MATRIX CALCULUSBeca
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556 APPENDIX D. MATRIX CALCULUSwhic
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558 APPENDIX D. MATRIX CALCULUS⎡
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560 APPENDIX D. MATRIX CALCULUS→Y
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562 APPENDIX D. MATRIX CALCULUSD.1.
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564 APPENDIX D. MATRIX CALCULUSwhic
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566 APPENDIX D. MATRIX CALCULUSIn t
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570 APPENDIX D. MATRIX CALCULUSD.2
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572 APPENDIX D. MATRIX CALCULUSalge
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574 APPENDIX D. MATRIX CALCULUStrac
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578 APPENDIX D. MATRIX CALCULUS
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580 APPENDIX E. PROJECTIONThe follo
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582 APPENDIX E. PROJECTIONFor matri
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584 APPENDIX E. PROJECTION(⇐) To
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586 APPENDIX E. PROJECTIONNonorthog
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588 APPENDIX E. PROJECTIONE.2.0.0.1
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590 APPENDIX E. PROJECTIONE.3.2Orth
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592 APPENDIX E. PROJECTIONE.3.5Unif
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594 APPENDIX E. PROJECTIONE.4 Algeb
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596 APPENDIX E. PROJECTIONa ∗ 2K
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598 APPENDIX E. PROJECTIONwhere Y =
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600 APPENDIX E. PROJECTION(B.4.2).
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602 APPENDIX E. PROJECTIONis a nono
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604 APPENDIX E. PROJECTIONE.6.4.1Or
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606 APPENDIX E. PROJECTIONq i q T i
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608 APPENDIX E. PROJECTIONThe test
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610 APPENDIX E. PROJECTIONPerpendic
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612 APPENDIX E. PROJECTIONE.8 Range
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614 APPENDIX E. PROJECTIONAs for su
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616 APPENDIX E. PROJECTIONWith refe
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618 APPENDIX E. PROJECTIONProjectio
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620 APPENDIX E. PROJECTIONE.9.2.2.2
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622 APPENDIX E. PROJECTIONThe foreg
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624 APPENDIX E. PROJECTION❇❇❇
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626 APPENDIX E. PROJECTIONE.10 Alte
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628 APPENDIX E. PROJECTIONbH 1H 2P
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630 APPENDIX E. PROJECTIONa(a){y |
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632 APPENDIX E. PROJECTION(a feasib
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634 APPENDIX E. PROJECTIONwhile, th
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636 APPENDIX E. PROJECTIONE.10.2.1.
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638 APPENDIX E. PROJECTION10 0dist(
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640 APPENDIX E. PROJECTIONE.10.3.1D
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642 APPENDIX E. PROJECTIONE 3K ⊥
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644 APPENDIX E. PROJECTION
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646 APPENDIX F. MATLAB PROGRAMSif n
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648 APPENDIX F. MATLAB PROGRAMSend%
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650 APPENDIX F. MATLAB PROGRAMSF.1.
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652 APPENDIX F. MATLAB PROGRAMScoun
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654 APPENDIX F. MATLAB PROGRAMSF.3
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658 APPENDIX F. MATLAB PROGRAMS% so
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660 APPENDIX F. MATLAB PROGRAMS% tr
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664 APPENDIX F. MATLAB PROGRAMSbrea
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666 APPENDIX F. MATLAB PROGRAMSwhil
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668 APPENDIX F. MATLAB PROGRAMSF.7
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670 APPENDIX F. MATLAB PROGRAMS
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672 APPENDIX G. NOTATION AND A FEW
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688 BIBLIOGRAPHY[7] Abdo Y. Alfakih
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690 BIBLIOGRAPHY[27] Aharon Ben-Tal
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692 BIBLIOGRAPHY[48] Lev M. Brègma
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694 BIBLIOGRAPHY[67] Joel Dawson, S
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696 BIBLIOGRAPHY[85] Alan Edelman,
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698 BIBLIOGRAPHY[102] Philip E. Gil
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700 BIBLIOGRAPHYWeiss, editors, Pol
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702 BIBLIOGRAPHY[146] Jean-Baptiste
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704 BIBLIOGRAPHY[168] Jean B. Lasse
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706 BIBLIOGRAPHY[189] Rudolf Mathar
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708 BIBLIOGRAPHY[211] M. L. Overton
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710 BIBLIOGRAPHY[229] C. K. Rushfor
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712 BIBLIOGRAPHY[252] Jos F. Sturm
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714 BIBLIOGRAPHY[274] È. B. Vinber
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[294] Yinyu Ye. Semidefinite progra
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718 INDEXobtuse, 62positive semidef
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720 INDEXelliptope, 642orthant, 177
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722 INDEXdistancegeometry, 20, 317m
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724 INDEX-dimensional, 37, 89rank,
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726 INDEXGram form, 331is, 674isedm
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728 INDEXdiscretized, 152, 431in su
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730 INDEXboundary, 115dimension, 10
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732 INDEXlinear operator, 587, 591,
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734 INDEXlargest entries, 188monoto
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736 INDEXsimilarity, 606translation
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