v2010.10.26 - Convex Optimization
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v2010.10.26 - Convex Optimization

v2010.10.26 - Convex Optimization v2010.10.26 - Convex Optimization

convexoptimization.com
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416 CHAPTER 5. EUCLIDEAN DISTANCE MATRIXColumns 1 through 6X = 0 -0.1983 -0.4584 0.1657 0.9399 0.74160 0.6863 0.2936 0.6239 -0.2936 0.39270 -0.4835 0.8146 -0.6448 0.0611 -0.42240 0.5059 0.2004 -0.4093 -0.1632 0.3427Columns 7 through 12-0.4815 -0.9399 -0.7416 0.1983 0.4584 -0.28320 0.2936 -0.3927 -0.6863 -0.2936 -0.6863-0.8756 -0.0611 0.4224 0.4835 -0.8146 -0.3922-0.0372 0.1632 -0.3427 -0.5059 -0.2004 -0.5431Columns 13 through 180.2832 -0.2926 -0.6473 0.0943 0.3640 -0.36400.6863 0.9176 -0.6239 -0.2313 -0.0624 0.06240.3922 0.1698 -0.2309 -0.6533 -0.1613 0.16130.5431 -0.2088 0.3721 0.7147 -0.9152 0.9152Columns 19 through 25-0.0943 0.6473 -0.1657 0.2926 -0.5759 0.5759 0.48150.2313 0.6239 -0.6239 -0.9176 0.2313 -0.2313 00.6533 0.2309 0.6448 -0.1698 -0.2224 0.2224 0.8756-0.7147 -0.3721 0.4093 0.2088 -0.7520 0.7520 0.0372This particular optimal solution was found by solving a problem sequence inincreasing number of spheres.Numerical problems begin to arise with matrices of this size due tointerior-point methods of solution. By eliminating some equality constraintsfor this particular problem, matrix size can be reduced: From5.8.3 we have[ 0−VNDV T N = 11 T T− [0 I ] DI] 12(933)(which does not hold more generally) where identity matrix I ∈ S N−1 hascoordinates-expression in dimensions 3 and 4.The first miracle happens in dimension 6. There are better packings than D 6(Conjecture: k(6)=72). It’s a real miracle how dense the packing is in eight dimensions(E 8 =Korkine & Zolotarev packing that was discovered in 1880s) and especially indimension 24, that is the so-called Leech lattice.Actually, people in coding theory have conjectures on the kissing numbers for dimensionsup to 32 (or even greater?). However, sometimes they found better lower bounds. I knowthat Ericson & Zinoviev a few years ago discovered (by hand, no computer) in dimensions13 and 14 better kissing arrangements than were known before. −Oleg Musin

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

416 CHAPTER 5. EUCLIDEAN DISTANCE MATRIXColumns 1 through 6X = 0 -0.1983 -0.4584 0.1657 0.9399 0.74160 0.6863 0.2936 0.6239 -0.2936 0.39270 -0.4835 0.8146 -0.6448 0.0611 -0.42240 0.5059 0.2004 -0.4093 -0.1632 0.3427Columns 7 through 12-0.4815 -0.9399 -0.7416 0.1983 0.4584 -0.28320 0.2936 -0.3927 -0.6863 -0.2936 -0.6863-0.8756 -0.0611 0.4224 0.4835 -0.8146 -0.3922-0.0372 0.1632 -0.3427 -0.5059 -0.2004 -0.5431Columns 13 through 180.2832 -0.2926 -0.6473 0.0943 0.3640 -0.36400.6863 0.9176 -0.6239 -0.2313 -0.0624 0.06240.3922 0.1698 -0.2309 -0.6533 -0.1613 0.16130.5431 -0.2088 0.3721 0.7147 -0.9152 0.9152Columns 19 through 25-0.0943 0.6473 -0.1657 0.2926 -0.5759 0.5759 0.48150.2313 0.6239 -0.6239 -0.9176 0.2313 -0.2313 00.6533 0.2309 0.6448 -0.1698 -0.2224 0.2224 0.8756-0.7147 -0.3721 0.4093 0.2088 -0.7520 0.7520 0.0372This particular optimal solution was found by solving a problem sequence inincreasing number of spheres.Numerical problems begin to arise with matrices of this size due tointerior-point methods of solution. By eliminating some equality constraintsfor this particular problem, matrix size can be reduced: From5.8.3 we have[ 0−VNDV T N = 11 T T− [0 I ] DI] 12(933)(which does not hold more generally) where identity matrix I ∈ S N−1 hascoordinates-expression in dimensions 3 and 4.The first miracle happens in dimension 6. There are better packings than D 6(Conjecture: k(6)=72). It’s a real miracle how dense the packing is in eight dimensions(E 8 =Korkine & Zolotarev packing that was discovered in 1880s) and especially indimension 24, that is the so-called Leech lattice.Actually, people in coding theory have conjectures on the kissing numbers for dimensionsup to 32 (or even greater?). However, sometimes they found better lower bounds. I knowthat Ericson & Zinoviev a few years ago discovered (by hand, no computer) in dimensions13 and 14 better kissing arrangements than were known before. −Oleg Musin

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