v2010.10.26 - Convex Optimization

324 CHAPTER 4. SEMIDEFINITE PROGRAMMINGto j th sensor, while d ij is a lower bound. These bounds become the inputdata. Each measurement range is presumed different from the others becauseof measurement uncertainty; e.g., Figure 90.Our mathematical treatment of anchors and sensors is notdichotomized. 4.29 A sensor position that is known a priori to highaccuracy (with absolute certainty) ˇx i is called an anchor. Then thesensor-network localization problem (745) can be expressed equivalently:Given a number of anchors m , and I a set of indices (corresponding to allmeasurable distances • ), for 0 < n < Nfind XG∈S N , X∈R n×Nsubject to d ij ≤ 〈G , (e i − e j )(e i − e j ) T 〉 ≤ d ij ∀(i,j)∈ I〈G , e i e T i 〉 = ‖ˇx i ‖ 2 , i = N − m + 1... N〈G , (e i e T j + e j e T i )/2〉 = ˇx T i ˇx j , i < j , ∀i,j∈{N − m + 1... N}X(:, N − m + 1:N) = [ ˇx N−m+1 · · · ˇx N ][ ] I XZ =X T≽ 0GrankZ= nwhere e i is the i th member of the standard basis for R N . Distance-squared ij = ‖x i − x j ‖ 2 2 = 〈x i − x j , x i − x j 〉 (887)is related to Gram matrix entries G[g ij ] by vector inner-product(750)d ij = g ii + g jj − 2g ij= 〈G , (e i − e j )(e i − e j ) T 〉 = tr(G T (e i − e j )(e i − e j ) T )(902)hence the scalar inequalities. Each linear equality constraint in G∈ S Nrepresents a hyperplane in isometrically isomorphic Euclidean vector spaceR N(N+1)/2 , while each linear inequality pair represents a convex Euclideanbody known as slab (an intersection of two parallel but opposing halfspaces,Figure 11). By Schur complement (A.4), any solution (G, X) providescomparison with respect to the positive semidefinite coneG ≽ X T X (937)4.29 Wireless location problem thus stated identically; difference being: fewer sensors.