v2010.10.26 - Convex Optimization

4.2. FRAMEWORK 295minimize 1 TˆxX∈ S n , ˆx∈R nsubject to A(ˆx + 1) 1[ = b 2] X ˆxG =ˆx T 1δ(X) = 1≽ 0(692)whose optimal solution x ⋆ (688) is identical to that of minimal cardinalityBoolean problem (680) if and only if rankG ⋆ =1.Hope 4.16 of acquiring a rank-1 solution is not ill-founded because 2 nelliptope vertices have rank 1, and we are minimizing an affine function ona subset of the elliptope (Figure 130) containing rank-1 vertices; id est, byassumption that the feasible set of minimal cardinality Boolean problem(680) is nonempty, a desired solution resides on the elliptope relativeboundary at a rank-1 vertex. 4.17For that data given in (682), our semidefinite program solversdpsol [384][385] (accurate in solution to approximately 1E-8) 4.18 finds optimal solutionto (692)⎡round(G ⋆ ) =⎢⎣1 1 1 −1 1 1 −11 1 1 −1 1 1 −11 1 1 −1 1 1 −1−1 −1 −1 1 −1 −1 11 1 1 −1 1 1 −11 1 1 −1 1 1 −1−1 −1 −1 1 −1 −1 1⎤⎥⎦(693)near a rank-1 vertex of the elliptope in S n+1 (Theorem 5.9.1.0.2); its sorted4.16 A more deterministic approach to constraining rank and cardinality is in4.6.0.0.11.4.17 Confinement to the elliptope can be regarded as a kind of normalization akin tomatrix A column normalization suggested in [120] and explored in Example 4.2.3.1.2.4.18 A typically ignored limitation of interior-point solution methods is their relativeaccuracy of only about 1E-8 on a machine using 64-bit (double precision) floating-pointarithmetic; id est, optimal solution x ⋆ cannot be more accurate than square root ofmachine epsilon (ǫ=2.2204E-16). Nonzero primal−dual objective difference is not a goodmeasure of solution accuracy.