v2007.11.26 - Convex Optimization

5.4. EDM DEFINITION 325By eliminating some equality constraints for this particular problem,matrix variable dimension can be reduced. From5.8.3 we have[ ] 0−VNDV T N = 11 T T 1− [0 I ] D(774)I 2(which does not generally hold) where identity matrix I ∈ S N−1 has one lessdimension than EDM D . By defining an EDM principal submatrix[ ]ˆD = ∆ 0T[0 I ] D ∈ S N−1Ih(775)we get a convex problem equivalent to (770)minimize − tr(W ˆD)ˆD∈S N−1subject to ˆDij ≥ 1 , 1 ≤ i < j = 2... N −111 T − ˆD 1 2 ≽ 0δ( ˆD) = 0(776)Any feasible matrix 11 T − ˆD 1 2belongs 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.5.4.2.2.4 Example. So & Ye 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 ∈ R 2 ), the sensor’s absoluteposition is determined from its noiseless measured distance-square ď i1to each of three anchors (Figure 2, Figure 84(a)). This trilaterationcan be expressed as a convex optimization problem in terms of listX = ∆ [x 1 ˇx 2 ˇx 3 ˇx 4 ]∈ R 2×4 and Gram matrix G∈ S 4 (743):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(777)