v2007.11.26 - Convex Optimization

332 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX5.4.2.2.6 Example. Tandem trilateration in wireless sensor network.Given three known absolute point-positions in R 2 (three anchors ˇx 3 , ˇx 4 , ˇx 5 )and two unknown sensors x 1 , x 2 ∈ R 2 , the sensors’ absolute positions aredeterminable from their noiseless distances-square (as indicated in Figure 86)assuming the anchors exhibit no rotational or reflective symmetry in theiraffine hull (5.5.2). This example differs from Example 5.4.2.2.4 in so faras trilateration of each sensor is now in terms of one unknown position, theother sensor. We express this localization as a convex optimization problem(a semidefinite program,4.1) in terms of list X ∆ = [x 1 x 2 ˇx 3 ˇx 4 ˇx 5 ]∈ R 2×5and Gram matrix G∈ S 5 (743) via relaxation (778):minimize trGG∈S 5 , X∈R2×5 subject to tr(GΦ i1 ) = ďi1 , i = 2, 4, 5tr(GΦ i2 ) = ďi2 , i = 3, 5tr ( )Ge i e T i = ‖ˇx i ‖ 2 , i = 3, 4, 5tr(G(e i e T j + e j e T i )/2) = ˇx T i ˇx j , 3 ≤ i < j = 4, 5X(:, 3:5) = [ ˇx 3 ˇx 4 ˇx 5 ][ ] I XX T≽ 0G(785)whereΦ ij = (e i − e j )(e i − e j ) T ∈ S N + (732)This problem realization is fragile because of the unknown distances betweensensors and anchors. Yet there is no more information we may include beyondthe 11 independent equality constraints on the Gram matrix (nonredundantconstraints not antithetical) to reduce the feasible set 5.14 . (By virtue of theirdimensioning, the sensors are already constrained to R 2 the affine hull of theanchors.)Exhibited in Figure 87 are two mistakes in solution X ⋆ (:,1:2) dueto a rank-3 optimal Gram matrix G ⋆ . The trace objective is a heuristicminimizing convex envelope of quasiconcave function 5.15 rankG. (2.9.2.6.2,7.2.2.1) A rank-2 optimal Gram matrix can be found and the errors5.14 the presumably nonempty convex set of all points G and X satisfying the constraints.5.15 Projection on that nonconvex subset of all N ×N-dimensional positive semidefinitematrices, in an affine subset, whose rank does not exceed 2 is a problem considered difficultto solve. [266,4]