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
- Page 1 and 2: DATTORRO CONVEX OPTIMIZATION & EUCL
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LIST OF FIGURES 17<br />
126 Decomposing a member of polar EDM cone . . . . . . . . . . 486<br />
127 Ordinary dual EDM cone projected on S 3 h . . . . . . . . . . . 492<br />
7 Proximity problems 495<br />
128 Pseudo-Venn diagram . . . . . . . . . . . . . . . . . . . . . . 498<br />
129 Elbow placed in path of projection . . . . . . . . . . . . . . . 499<br />
130 <strong>Convex</strong> envelope . . . . . . . . . . . . . . . . . . . . . . . . . 519<br />
A Linear algebra 537<br />
131 Geometrical interpretation of full SVD . . . . . . . . . . . . . 568<br />
B Simple matrices 577<br />
132 Four fundamental subspaces of any dyad . . . . . . . . . . . . 579<br />
133 Four fundamental subspaces of a doublet . . . . . . . . . . . . 583<br />
134 Four fundamental subspaces of elementary matrix . . . . . . . 584<br />
135 Gimbal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593<br />
D Matrix calculus 609<br />
136 <strong>Convex</strong> quadratic bowl in R 2 × R . . . . . . . . . . . . . . . . 620<br />
E Projection 639<br />
137 Action of the pseudoinverse . . . . . . . . . . . . . . . . . . . 640<br />
138 Nonorthogonal projection of x∈ R 3 on R(U)= R 2 . . . . . . . 646<br />
139 Biorthogonal expansion of point x∈aff K . . . . . . . . . . . . 657<br />
140 Dual interpretation of projection on convex set . . . . . . . . . 676<br />
141 Projection product on convex set in subspace . . . . . . . . . 685<br />
142 von Neumann-style projection of point b . . . . . . . . . . . . 688<br />
143 Alternating projection on two halfspaces . . . . . . . . . . . . 689<br />
144 Distance, optimization, feasibility . . . . . . . . . . . . . . . . 691<br />
145 Alternating projection on nonnegative orthant and hyperplane 694<br />
146 Geometric convergence of iterates in norm . . . . . . . . . . . 694<br />
147 Distance between PSD cone and iterate in A . . . . . . . . . . 699<br />
148 Dykstra’s alternating projection algorithm . . . . . . . . . . . 700<br />
149 Polyhedral normal cones . . . . . . . . . . . . . . . . . . . . . 702<br />
150 Normal cone to elliptope . . . . . . . . . . . . . . . . . . . . . 703
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