v2010.10.26 - Convex Optimization
v2010.10.26 - Convex Optimization v2010.10.26 - Convex Optimization
322 CHAPTER 4. SEMIDEFINITE PROGRAMMING1718 21 19 2013 14221516910 23 11 125 6 24 7 8122534Figure 89: 5-lattice in R 2 . Nodes 21 through 25 are anchors.0 • ? ? • • ? ? • ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?• 0 ? ? • • ? ? ? • ? ? ? ? ? ? ? ? ? ? ? ? ? • •? ? 0 • ? • • • ? ? • • ? ? ? ? ? ? ? ? ? ? • • •? ? • 0 ? ? • • ? ? ? • ? ? ? ? ? ? ? ? ? ? ? • ?• • ? ? 0 • ? ? • • ? ? • • ? ? • ? ? ? ? ? • ? •• • • ? • 0 • ? • • • ? ? • ? ? ? ? ? ? ? ? • • •? ? • • ? • 0 • ? ? • • ? ? • • ? ? ? ? ? ? • • •? ? • • ? ? • 0 ? ? • • ? ? • • ? ? ? ? ? ? ? • ?• ? ? ? • • ? ? 0 • ? ? • • ? ? • • ? ? ? ? ? ? ?? • ? ? • • ? ? • 0 • ? • • ? ? ? • ? ? • • • • •? ? • ? ? • • • ? • 0 • ? • • • ? ? • ? ? • • • •? ? • • ? ? • • ? ? • 0 ? ? • • ? ? • • ? • • • ?? ? ? ? • ? ? ? • • ? ? 0 • ? ? • • ? ? • • ? ? ?? ? ? ? • • ? ? • • • ? • 0 • ? • • • ? • • • • ?? ? ? ? ? ? • • ? ? • • ? • 0 • ? ? • • • • • • ?? ? ? ? ? ? • • ? ? • • ? ? • 0 ? ? • • ? • ? ? ?? ? ? ? • ? ? ? • ? ? ? • • ? ? 0 • ? ? • ? ? ? ?? ? ? ? ? ? ? ? • • ? ? • • ? ? • 0 • ? • • • ? ?? ? ? ? ? ? ? ? ? ? • • ? • • • ? • 0 • • • • ? ?? ? ? ? ? ? ? ? ? ? ? • ? ? • • ? ? • 0 • • ? ? ?? ? ? ? ? ? ? ? ? • ? ? • • • ? • • • • 0 ◦ ◦ ◦ ◦? ? ? ? ? ? ? ? ? • • • • • • • ? • • • ◦ 0 ◦ ◦ ◦? ? • ? • • • ? ? • • • ? • • ? ? • • ? ◦ ◦ 0 ◦ ◦? • • • ? • • • ? • • • ? • • ? ? ? ? ? ◦ ◦ ◦ 0 ◦? • • ? • • • ? ? • • ? ? ? ? ? ? ? ? ? ◦ ◦ ◦ ◦ 0(749)
4.4. RANK-CONSTRAINED SEMIDEFINITE PROGRAM 323MARKET St.Figure 90: Location uncertainty ellipsoid in R 2 for each of 15 sensors • withinthree city blocks in downtown San Francisco. Data by Polaris Wireless. [322]problem statementAscribe points in a list {x l ∈ R n , l=1... N} to the columns of a matrix X ;X = [x 1 · · · x N ] ∈ R n×N (76)where N is regarded as cardinality of list X . Positive semidefinite matrixX T X , formed from inner product of the list, is a Gram matrix; [250,3.6]⎡⎤‖x 1 ‖ 2 x T 1x 2 x T 1x 3 · · · x T 1x Nx T 2x 1 ‖x 2 ‖ 2 x T 2x 3 · · · x T 2x NG = X T X =x T 3x 1 x T 3x 2 ‖x 3 ‖ 2 ... x T 3x N∈ S N + (900)⎢⎥⎣ . ....... . ⎦xN Tx 1 xN Tx 2 xN Tx 3 · · · ‖x N ‖ 2where S N + is the convex cone of N ×N positive semidefinite matrices in thesymmetric matrix subspace S N .Existence of noise precludes measured distance from the input data. Weinstead assign measured distance to a range estimate specified by individualupper and lower bounds: d ij is an upper bound on distance-square from i th
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4.4. RANK-CONSTRAINED SEMIDEFINITE PROGRAM 323MARKET St.Figure 90: Location uncertainty ellipsoid in R 2 for each of 15 sensors • withinthree city blocks in downtown San Francisco. Data by Polaris Wireless. [322]problem statementAscribe points in a list {x l ∈ R n , l=1... N} to the columns of a matrix X ;X = [x 1 · · · x N ] ∈ R n×N (76)where N is regarded as cardinality of list X . Positive semidefinite matrixX T X , formed from inner product of the list, is a Gram matrix; [250,3.6]⎡⎤‖x 1 ‖ 2 x T 1x 2 x T 1x 3 · · · x T 1x Nx T 2x 1 ‖x 2 ‖ 2 x T 2x 3 · · · x T 2x NG = X T X =x T 3x 1 x T 3x 2 ‖x 3 ‖ 2 ... x T 3x N∈ S N + (900)⎢⎥⎣ . ....... . ⎦xN Tx 1 xN Tx 2 xN Tx 3 · · · ‖x N ‖ 2where S N + is the convex cone of N ×N positive semidefinite matrices in thesymmetric matrix subspace S N .Existence of noise precludes measured distance from the input data. Weinstead assign measured distance to a range estimate specified by individualupper and lower bounds: d ij is an upper bound on distance-square from i th