v2007.09.13 - Convex Optimization
v2007.09.13 - Convex Optimization v2007.09.13 - Convex Optimization
16 LIST OF FIGURES92 Relative-angle inequality tetrahedron . . . . . . . . . . . . . . 37993 Nonsimplicial pyramid in R 3 . . . . . . . . . . . . . . . . . . . 3836 EDM cone 38794 Relative boundary of cone of Euclidean distance matrices . . . 39095 Intersection of EDM cone with hyperplane . . . . . . . . . . . 39296 Neighborhood graph . . . . . . . . . . . . . . . . . . . . . . . 39497 Trefoil knot untied . . . . . . . . . . . . . . . . . . . . . . . . 39798 Trefoil ribbon . . . . . . . . . . . . . . . . . . . . . . . . . . . 39999 Example of V X selection to make an EDM . . . . . . . . . . . 402100 Vector V X spirals . . . . . . . . . . . . . . . . . . . . . . . . . 404101 Three views of translated negated elliptope . . . . . . . . . . . 412102 Halfline T on PSD cone boundary . . . . . . . . . . . . . . . . 416103 Vectorization and projection interpretation example . . . . . . 417104 Orthogonal complement of geometric center subspace . . . . . 422105 EDM cone construction by flipping PSD cone . . . . . . . . . 423106 Decomposing a member of polar EDM cone . . . . . . . . . . 428107 Ordinary dual EDM cone projected on S 3 h . . . . . . . . . . . 4347 Proximity problems 437108 Pseudo-Venn diagram . . . . . . . . . . . . . . . . . . . . . . 440109 Elbow placed in path of projection . . . . . . . . . . . . . . . 441110 Convex envelope . . . . . . . . . . . . . . . . . . . . . . . . . 461A Linear algebra 479111 Geometrical interpretation of full SVD . . . . . . . . . . . . . 508B Simple matrices 517112 Four fundamental subspaces of any dyad . . . . . . . . . . . . 519113 Four fundamental subspaces of a doublet . . . . . . . . . . . . 523114 Four fundamental subspaces of elementary matrix . . . . . . . 524115 Gimbal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532D Matrix calculus 549116 Convex quadratic bowl in R 2 × R . . . . . . . . . . . . . . . . 561
LIST OF FIGURES 17E Projection 579117 Nonorthogonal projection of x∈ R 3 on R(U)= R 2 . . . . . . . 585118 Biorthogonal expansion of point x∈aff K . . . . . . . . . . . . 596119 Dual interpretation of projection on convex set . . . . . . . . . 615120 Projection product on convex set in subspace . . . . . . . . . 624121 von Neumann-style projection of point b . . . . . . . . . . . . 627122 Alternating projection on two halfspaces . . . . . . . . . . . . 628123 Distance, optimization, feasibility . . . . . . . . . . . . . . . . 630124 Alternating projection on nonnegative orthant and hyperplane 633125 Geometric convergence of iterates in norm . . . . . . . . . . . 633126 Distance between PSD cone and iterate in A . . . . . . . . . . 638127 Dykstra’s alternating projection algorithm . . . . . . . . . . . 639128 Polyhedral normal cones . . . . . . . . . . . . . . . . . . . . . 641129 Normal cone to elliptope . . . . . . . . . . . . . . . . . . . . . 642
- 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: LIST OF FIGURES 1559 Quadratic func
- Page 19 and 20: Chapter 1OverviewConvex Optimizatio
- Page 21 and 22: ˇx 4ˇx 3ˇx 2Figure 2: Applicatio
- Page 23 and 24: 23Figure 4: This coarsely discretiz
- Page 25 and 26: ases (biorthogonal expansion). We e
- 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
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- 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
LIST OF FIGURES 17E Projection 579117 Nonorthogonal projection of x∈ R 3 on R(U)= R 2 . . . . . . . 585118 Biorthogonal expansion of point x∈aff K . . . . . . . . . . . . 596119 Dual interpretation of projection on convex set . . . . . . . . . 615120 Projection product on convex set in subspace . . . . . . . . . 624121 von Neumann-style projection of point b . . . . . . . . . . . . 627122 Alternating projection on two halfspaces . . . . . . . . . . . . 628123 Distance, optimization, feasibility . . . . . . . . . . . . . . . . 630124 Alternating projection on nonnegative orthant and hyperplane 633125 Geometric convergence of iterates in norm . . . . . . . . . . . 633126 Distance between PSD cone and iterate in A . . . . . . . . . . 638127 Dykstra’s alternating projection algorithm . . . . . . . . . . . 639128 Polyhedral normal cones . . . . . . . . . . . . . . . . . . . . . 641129 Normal cone to elliptope . . . . . . . . . . . . . . . . . . . . . 642
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