v2009.01.01 - Convex Optimization
v2009.01.01 - Convex Optimization v2009.01.01 - Convex Optimization
16 LIST OF FIGURES 5 Euclidean Distance Matrix 345 92 Convex hull of three points . . . . . . . . . . . . . . . . . . . . 346 93 Complete dimensionless EDM graph . . . . . . . . . . . . . . 349 94 Fifth Euclidean metric property . . . . . . . . . . . . . . . . . 351 95 Arbitrary hexagon in R 3 . . . . . . . . . . . . . . . . . . . . . 360 96 Kissing number . . . . . . . . . . . . . . . . . . . . . . . . . . 362 97 Trilateration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366 98 This EDM graph provides unique isometric reconstruction . . 370 99 Two sensors • and three anchors ◦ . . . . . . . . . . . . . . . 370 100 Two discrete linear trajectories of sensors . . . . . . . . . . . . 371 101 Coverage in cellular telephone network . . . . . . . . . . . . . 374 102 Contours of equal signal power . . . . . . . . . . . . . . . . . . 374 103 Depiction of molecular conformation . . . . . . . . . . . . . . 376 104 Square diamond . . . . . . . . . . . . . . . . . . . . . . . . . . 385 105 Orthogonal complements in S N abstractly oriented . . . . . . . 388 106 Elliptope E 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 107 Elliptope E 2 interior to S 2 + . . . . . . . . . . . . . . . . . . . . 408 108 Smallest eigenvalue of −VN TDV N . . . . . . . . . . . . . . . . 414 109 Some entrywise EDM compositions . . . . . . . . . . . . . . . 414 110 Map of United States of America . . . . . . . . . . . . . . . . 427 111 Largest ten eigenvalues of −VN TOV N . . . . . . . . . . . . . . 430 112 Relative-angle inequality tetrahedron . . . . . . . . . . . . . . 437 113 Nonsimplicial pyramid in R 3 . . . . . . . . . . . . . . . . . . . 441 6 Cone of distance matrices 445 114 Relative boundary of cone of Euclidean distance matrices . . . 448 115 Intersection of EDM cone with hyperplane . . . . . . . . . . . 450 116 Neighborhood graph . . . . . . . . . . . . . . . . . . . . . . . 453 117 Trefoil knot untied . . . . . . . . . . . . . . . . . . . . . . . . 455 118 Trefoil ribbon . . . . . . . . . . . . . . . . . . . . . . . . . . . 457 119 Example of V X selection to make an EDM . . . . . . . . . . . 460 120 Vector V X spirals . . . . . . . . . . . . . . . . . . . . . . . . . 463 121 Three views of translated negated elliptope . . . . . . . . . . . 470 122 Halfline T on PSD cone boundary . . . . . . . . . . . . . . . . 474 123 Vectorization and projection interpretation example . . . . . . 475 124 Orthogonal complement of geometric center subspace . . . . . 480 125 EDM cone construction by flipping PSD cone . . . . . . . . . 481
LIST OF FIGURES 17 126 Decomposing a member of polar EDM cone . . . . . . . . . . 486 127 Ordinary dual EDM cone projected on S 3 h . . . . . . . . . . . 492 7 Proximity problems 495 128 Pseudo-Venn diagram . . . . . . . . . . . . . . . . . . . . . . 498 129 Elbow placed in path of projection . . . . . . . . . . . . . . . 499 130 Convex envelope . . . . . . . . . . . . . . . . . . . . . . . . . 519 A Linear algebra 537 131 Geometrical interpretation of full SVD . . . . . . . . . . . . . 568 B Simple matrices 577 132 Four fundamental subspaces of any dyad . . . . . . . . . . . . 579 133 Four fundamental subspaces of a doublet . . . . . . . . . . . . 583 134 Four fundamental subspaces of elementary matrix . . . . . . . 584 135 Gimbal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593 D Matrix calculus 609 136 Convex quadratic bowl in R 2 × R . . . . . . . . . . . . . . . . 620 E Projection 639 137 Action of the pseudoinverse . . . . . . . . . . . . . . . . . . . 640 138 Nonorthogonal projection of x∈ R 3 on R(U)= R 2 . . . . . . . 646 139 Biorthogonal expansion of point x∈aff K . . . . . . . . . . . . 657 140 Dual interpretation of projection on convex set . . . . . . . . . 676 141 Projection product on convex set in subspace . . . . . . . . . 685 142 von Neumann-style projection of point b . . . . . . . . . . . . 688 143 Alternating projection on two halfspaces . . . . . . . . . . . . 689 144 Distance, optimization, feasibility . . . . . . . . . . . . . . . . 691 145 Alternating projection on nonnegative orthant and hyperplane 694 146 Geometric convergence of iterates in norm . . . . . . . . . . . 694 147 Distance between PSD cone and iterate in A . . . . . . . . . . 699 148 Dykstra’s alternating projection algorithm . . . . . . . . . . . 700 149 Polyhedral normal cones . . . . . . . . . . . . . . . . . . . . . 702 150 Normal cone to elliptope . . . . . . . . . . . . . . . . . . . . . 703
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16 LIST OF FIGURES<br />
5 Euclidean Distance Matrix 345<br />
92 <strong>Convex</strong> hull of three points . . . . . . . . . . . . . . . . . . . . 346<br />
93 Complete dimensionless EDM graph . . . . . . . . . . . . . . 349<br />
94 Fifth Euclidean metric property . . . . . . . . . . . . . . . . . 351<br />
95 Arbitrary hexagon in R 3 . . . . . . . . . . . . . . . . . . . . . 360<br />
96 Kissing number . . . . . . . . . . . . . . . . . . . . . . . . . . 362<br />
97 Trilateration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366<br />
98 This EDM graph provides unique isometric reconstruction . . 370<br />
99 Two sensors • and three anchors ◦ . . . . . . . . . . . . . . . 370<br />
100 Two discrete linear trajectories of sensors . . . . . . . . . . . . 371<br />
101 Coverage in cellular telephone network . . . . . . . . . . . . . 374<br />
102 Contours of equal signal power . . . . . . . . . . . . . . . . . . 374<br />
103 Depiction of molecular conformation . . . . . . . . . . . . . . 376<br />
104 Square diamond . . . . . . . . . . . . . . . . . . . . . . . . . . 385<br />
105 Orthogonal complements in S N abstractly oriented . . . . . . . 388<br />
106 Elliptope E 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407<br />
107 Elliptope E 2 interior to S 2 + . . . . . . . . . . . . . . . . . . . . 408<br />
108 Smallest eigenvalue of −VN TDV N . . . . . . . . . . . . . . . . 414<br />
109 Some entrywise EDM compositions . . . . . . . . . . . . . . . 414<br />
110 Map of United States of America . . . . . . . . . . . . . . . . 427<br />
111 Largest ten eigenvalues of −VN TOV N . . . . . . . . . . . . . . 430<br />
112 Relative-angle inequality tetrahedron . . . . . . . . . . . . . . 437<br />
113 Nonsimplicial pyramid in R 3 . . . . . . . . . . . . . . . . . . . 441<br />
6 Cone of distance matrices 445<br />
114 Relative boundary of cone of Euclidean distance matrices . . . 448<br />
115 Intersection of EDM cone with hyperplane . . . . . . . . . . . 450<br />
116 Neighborhood graph . . . . . . . . . . . . . . . . . . . . . . . 453<br />
117 Trefoil knot untied . . . . . . . . . . . . . . . . . . . . . . . . 455<br />
118 Trefoil ribbon . . . . . . . . . . . . . . . . . . . . . . . . . . . 457<br />
119 Example of V X selection to make an EDM . . . . . . . . . . . 460<br />
120 Vector V X spirals . . . . . . . . . . . . . . . . . . . . . . . . . 463<br />
121 Three views of translated negated elliptope . . . . . . . . . . . 470<br />
122 Halfline T on PSD cone boundary . . . . . . . . . . . . . . . . 474<br />
123 Vectorization and projection interpretation example . . . . . . 475<br />
124 Orthogonal complement of geometric center subspace . . . . . 480<br />
125 EDM cone construction by flipping PSD cone . . . . . . . . . 481
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