v2007.09.17 - Convex Optimization

268 CHAPTER 4. SEMIDEFINITE PROGRAMMINGdirection matrix WDenote by Z ⋆ an optimal composite matrix from semidefiniteprogram (640). Then for Z ⋆ ∈ S N+n whose eigenvalues λ(Z ⋆ )∈ R N+n arearranged in nonincreasing order, (Fan)N+n∑λ(Z ⋆ ) ii=n+1= minimizeW ∈ S N+n 〈Z ⋆ , W 〉subject to 0 ≼ W ≼ ItrW = N(1480a)whose optimal solution is known in closed form. This eigenvalue sum is zerowhen Z ⋆ has rank n or less.Foreknowledge of optimal Z ⋆ , to make possible this search for W , impliesrecursion; id est, semidefinite program (640) is solved for Z ⋆ initializingW = I or W = 0. Once found, Z ⋆ becomes constant in semidefinite program(1480a) where a new normal direction W is found as its optimal solution.Then the cycle (640) (1480a) iterates until convergence. When rankZ ⋆ = n ,solution via this convex iteration is optimal for sensor-network localizationproblem (634) and its equivalent (639).numerical solutionIn all examples to follow, number of anchorsm = √ N (642)equals square root of cardinality N of list X . Indices set I identifying allexisting distance measurements • is ascertained from connectivity matrix(635), (636), (637), or (638). We solve iteration (640) (1480a) in dimensionn = 2 for each respective example illustrated in Figure 63 through Figure 66.In presence of negligible noise, actual position is reliably localized forevery standardized example; noteworthy in so far as each example representsan incomplete graph. This means the set of all solutions having lowest rankis a single point, to within a rigid transformation.To make the examples interesting and consistent with previous work, werandomize each range of distance-square that bounds 〈G, (e i −e j )(e i −e j ) T 〉in (640); id est, for each and every (i,j)∈ I