v2007.09.13 - Convex Optimization

4.4. RANK-CONSTRAINED SEMIDEFINITE PROGRAM 269d ij = d ij (1 + √ 3 rand(1)η) 2d ij = d ij (1 − √ 3 rand(1)η) 2 (643)where η = 0.1 is a constant noise factor, rand(1) is the Matlab functionproviding one sample of uniformly distributed noise in the interval [0, 1] ,and d ij is actual distance-square from i th to j th sensor. Because of theseparate function calls rand(1) , each range of distance-square [d ij , d ij ]is not necessarily centered on actual distance-square d ij . The factor √ 3provides unit variance on the stochastic range.Figure 68 through Figure 71 each illustrate one realization of numericalsolution to the standardized lattice problems posed by Figure 63 throughFigure 66 respectively. Exact localization is impossible because ofmeasurement noise. Certainly, by inspection of their published graphicaldata, our new results are competitive with those of Carter & Jin. Obviouslyour solutions do not suffer from those compaction-type errors (clustering oflocalized sensors) exhibited by Biswas’ graphical results for the same noisefactor η ; which is all we intended to demonstrate.localization example conclusionSolution to this sensor-network localization problem became apparent byunderstanding geometry of optimization. Trace of a matrix, to a student oflinear algebra, is perhaps a sum of eigenvalues. But to us, trace representsthe normal I to some hyperplane in Euclidean vector space.The legacy of Carter & Jin [51] is a sobering demonstration of the needfor more efficient methods for solution of semidefinite programs, whilethat of So & Ye [237] is the bonding of distance geometry to semidefiniteprogramming. Elegance of our semidefinite problem statement (640) for asensor-network localization problem in any dimension should provide someimpetus to focus more research on computational intensity. Higher speedand greater accuracy from a simplex-like solver is what is required. We numerically tested the foregoing technique for constraining rank ona wide range of problems including localization of randomized positions,stress (7.2.2.7.1), ball packing (5.4.2.2.3), and cardinality problems. Wehave had some success introducing the direction vector inner-product (641)as a regularization term (Pareto optimization) whose purpose is to constrainrank, affine dimension, or cardinality: