v2010.10.26 - Convex Optimization

424 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX5.4.2.3.8 Example. Tandem trilateration in wireless sensor network.Given three known absolute point-positions in R 2 (three anchors ˇx 3 , ˇx 4 , ˇx 5 ),two unknown sensors x 1 , x 2 ∈ R 2 have absolute position determinable fromtheir noiseless distances-square (as indicated in Figure 123) assuming theanchors exhibit no rotational or reflective symmetry in their affine hull(5.5.2). This example differs from Example 5.4.2.3.5 in so far as trilaterationof each sensor is now in terms of one unknown position: the other sensor. Weexpress this localization as a convex optimization problem (a semidefiniteprogram,4.1) in terms of list X [x 1 x 2 ˇx 3 ˇx 4 ˇx 5 ]∈ R 2×5 and Grammatrix G∈ S 5 (900) via relaxation (937):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(944)whereΦ ij = (e i − e j )(e i − e j ) T ∈ S N + (889)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. 5.17Exhibited in Figure 124 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 5.18 rankG. (2.9.2.9.2,7.2.2.1) A rank-2 optimal Gram matrix can be found and the errorscorrected by choosing a different normal for the linear objective function,5.17 By virtue of their dimensioning, the sensors are already constrained to R 2 the affinehull of the anchors.5.18 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. [353,4]