v2007.11.26 - Convex Optimization

266 CHAPTER 4. SEMIDEFINITE PROGRAMMINGMARKET St.Figure 69: Uncertainty ellipsoid in R 2 for each of 15 sensors • located withinthree city blocks in downtown San Francisco. Data by Polaris Wireless. [243]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 (66)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; [183,3.6]⎡⎤‖x 1 ‖ 2 x T 1x 2 x T 1x 3 · · · x T 1x NxG = ∆ T 2x 1 ‖x 2 ‖ 2 x T 2x 3 · · · x T 2x NX T X =x T 3x 1 x T 3x 2 ‖x 3 ‖ 2 ... x T 3x N∈ S N + (743)⎢⎥⎣ . ....... . ⎦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 thereal symmetric 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