v2007.09.13 - Convex Optimization

5.4. EDM DEFINITION 311whereΦ ij = (e i − e j )(e i − e j ) T ∈ S N + (703)and where the constraint on distance-square ďi1 is equivalently written asa constraint on the Gram matrix via (716). There are 9 independent affineequality constraints on that Gram matrix while the sensor is constrainedonly by dimensioning to lie in R 2 . Although trG the objective ofminimization 5.12 insures a solution on the boundary of positive semidefinitecone S 4 + , we claim that the set of feasible Gram matrices forms a line(2.5.1.1) in isomorphic R 10 tangent (2.1.7.2) to the positive semidefinitecone boundary. (confer4.2.1.3)By Schur complement (A.4,2.9.1.0.1) any feasible G and X provideG ≽ X T X (749)which is a convex relaxation of the desired (nonconvex) equality constraint[ ] [ ]I X IX T =G X T [ I X ] (750)expected positive semidefinite rank-2 under noiseless conditions. But by(1288), the relaxation admits(3 ≥) rankG ≥ rankX (751)(a third dimension corresponding to an intersection of three spheres(not circles) were there noise). If rank[ ]I Xrank⋆X ⋆T G ⋆ = 2 (752)of an optimal solution equals 2, then G ⋆ = X ⋆T X ⋆ by Theorem A.4.0.0.3.As posed, this localization problem does not require affinely independent(Figure 18, three noncollinear) anchors. Assuming the anchors exhibitno rotational or reflective symmetry in their affine hull (5.5.2) andassuming the sensor x 1 lies in that affine hull, then sensor position solutionx ⋆ 1= X ⋆ (:, 1) is unique under noiseless measurement. [237] 5.12 Trace (trG = 〈I , G〉) minimization is a heuristic for rank minimization. (7.2.2.1) Itmay be interpreted as squashing G which is bounded below by X T X as in (749).