v2010.10.26 - Convex Optimization

412 CHAPTER 5. EUCLIDEAN DISTANCE MATRIXx 3x 4x 5x 6Figure 119: Arbitrary hexagon in R 3 whose vertices are labelled clockwise.x 1x 2zero distance between unknowns is revealed. An optimal solution to (924)gravitates toward gradient discontinuities (D.2.1), as in Figure 72, whereasoptimal solution to (926) is less compact in the unknowns. 5.11 5.4.2.3.3 Example. Hexagon.Barvinok [25,2.6] poses a problem in geometric realizability of an arbitraryhexagon (Figure 119) having:1. prescribed (one-dimensional) face-lengths l2. prescribed angles ϕ between the three pairs of opposing faces3. a constraint on the sum of norm-square of each and every vertex x ;ten affine equality constraints in all on a Gram matrix G∈ S 6 (912). Let’srealize this as a convex feasibility problem (with constraints written in thesame order) also assuming 0 geometric center (911):findD∈S 6 h−V DV 1 2 ∈ S6subject to tr ( )D(e i e T j + e j e T i ) 1 2 = l2ij , j−1 = (i = 1... 6) mod 6tr ( − 1V DV (A 2 i + A T i ) 2) 1 = cos ϕi , i = 1, 2, 3tr(− 1 2 V DV ) = 1−V DV ≽ 0 (927)5.11 Optimal solution to (924) has mechanical interpretation in terms of interconnectingsprings with constant force when distance is nonzero; otherwise, 0 force. Problem (926) isinterpreted instead using linear springs.