v2010.10.26 - Convex Optimization

5.4. EDM DEFINITION 417one less dimension than EDM D . By defining an EDM principal submatrix[ ] 0TˆD [0 I ] D ∈ S N−1Ih(934)we get a convex problem equivalent to (929)minimize − tr(W ˆD)ˆD∈S N−1subject to ˆDij ≥ 1 , 1 ≤ i < j = 2... N −111 T − ˆD 1 2 ≽ 0δ( ˆD) = 0(935)Any feasible solution 11 T − ˆD 1 2 belongs to an elliptope (5.9.1.0.1). This next example shows how finding the common point of intersectionfor three circles in a plane, a nonlinear problem, has convex expression.5.4.2.3.5 Example. Trilateration in wireless sensor network.Given three known absolute point positions in R 2 (three anchors ˇx 2 , ˇx 3 , ˇx 4 )and only one unknown point (one sensor x 1 ), the sensor’s absolute position isdetermined from its noiseless measured distance-square ďi1 to each of threeanchors (Figure 2, Figure 121a). This trilateration can be expressed as aconvex optimization problem in terms of list X [x 1 ˇx 2 ˇx 3 ˇx 4 ]∈ R 2×4 andGram matrix G∈ S 4 (900):minimize trGG∈S 4 , X∈R2×4 subject to tr(GΦ i1 ) = ďi1 , i = 2, 3, 4tr ( )Ge i e T i = ‖ˇx i ‖ 2 , i = 2, 3, 4tr(G(e i e T j + e j e T i )/2) = ˇx T i ˇx j , 2 ≤ i < j = 3, 4X(:, 2:4) = [ ˇx 2 ˇx 3 ˇx 4 ][ ] I XX T≽ 0G(936)whereΦ ij = (e i − e j )(e i − e j ) T ∈ S N + (889)and where the constraint on distance-square ďi1 is equivalently written asa constraint on the Gram matrix via (902). There are 9 independent affine