v2010.10.26 - Convex Optimization

330 CHAPTER 4. SEMIDEFINITE PROGRAMMINGdirection matrix WDenote by Z ⋆ an optimal composite matrix from semidefinite program (752).Then for Z ⋆ ∈ S N+n whose eigenvalues λ(Z ⋆ )∈ R N+n are arranged innonincreasing order, (Ky Fan)N+n∑λ(Z ⋆ ) ii=n+1= minimizeW ∈ S N+n 〈Z ⋆ , W 〉subject to 0 ≼ W ≼ ItrW = N(1700a)which has an optimal solution that is known in closed form (p.658,4.4.1.1).This eigenvalue sum is zero when Z ⋆ has rank n or less.Foreknowledge of optimal Z ⋆ , to make possible this search for W ,implies iteration; id est, semidefinite program (752) is solved for Z ⋆initializing W = I or W = 0. Once found, Z ⋆ becomes constant insemidefinite program (1700a) where a new normal direction W is found asits optimal solution. Then this cycle (752) (1700a) iterates until convergence.When rankZ ⋆ = n , solution via this convex iteration solves sensor-networklocalization problem (745) and its equivalent (750).numerical solutionIn all examples to follow, number of anchorsm = √ N (754)equals square root of cardinality N of list X . Indices set I identifying allmeasurable distances • is ascertained from connectivity matrix (746), (747),(748), or (749). We solve iteration (752) (1700a) in dimension n = 2 for eachrespective example illustrated in Figure 86 through Figure 89.In presence of negligible noise, true position is reliably localized for everystandardized example; noteworthy in so far as each example represents anincomplete graph. This implies that the set of all optimal solutions havinglowest rank must be small.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 (752); id est, for each and every (i,j)∈ Id ij = d ij (1 + √ 3η χ l) 2d ij = d ij (1 − √ 3η χ l+1) 2 (755)