v2007.09.13 - Convex Optimization

266 CHAPTER 4. SEMIDEFINITE PROGRAMMINGExistence 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 thto 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 67.Our mathematical treatment of anchors and sensors is notdichotomized. 4.25 A known sensor position to high accuracy ˇx i is ananchor. Then the sensor-network localization problem (634) can beexpressed equivalently: Given a number of anchors m , and I a setof indices (corresponding to all existing distance measurements •),for 0 < n < Nminimize trZG∈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-square(639)d ij = ‖x i − x j ‖ 2 2∆= 〈x i − x j , x i − x j 〉 (701)is related to Gram matrix entries G ∆ =[g ij ] by a vector inner-productd 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 )(716)hence the scalar inequalities. The objective function trZ is a heuristic whosesole purpose is to represent the convex envelope of rankZ . (7.2.2.1.1) BySchur complement (A.4) any feasible G and X provide a comparison withrespect to the positive semidefinite cone4.25 Wireless location problem thus stated identically; difference being: fewer sensors.