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26 Charles Audet, Pierre Hansen, and Frédéric Messineby proving (with interval arithmetic), that at least one constraint cannot be satisfied byany point in such a box.To accelerate convergence, constraint propagation techniques are used in some steps ofIBBA, see [9, 10] for details. The principle is to use, a priori, the implicit relations between thevariables (induced by the constraints) to reduce the size of a box.5. Optimal solutionThe optimal solution appeared by solving case corresponding to thrackleation 10, see Figure3, which is a relaxation of case 29: Add edge [v 0 , v 4 ] to case 10 then you obtain thrackleation 29given in Figure 1. This just changes the constraint ‖v 0 − v 4 ‖ ≤ 1 into the inequality constraint‖v 0 − v 4 ‖ = 1.v 3v 4v ′ 1••α 3 .. ..v• 0•..v 1..α•1 .. α 2..•v 2• v 3′•Figure 3. Thrackleation 10The non-convex program is:max 4 sin α 1α 4 + 4 sin α 24 + 4 sin α 34 + ||v 1 − v 4 || + ||v 0 − v 3 ||s.t. ||v 0 − v 4 || ≤ 10.688 ≤ α i ≤ 0.881 i = 1, 2, 3,where v 0 = (cos α 1 , sin α 1 ), v 1 = (0, 0), v 2 = (1, 0), v 3 = (1 − cos α 2 , sin α 2 ) and v 4 = (1 −cos α 2 + cos(α 2 + α 3 ), sin α 2 − sin(α 2 + α 3 )).Solving this program by using Algorithm IBBA, one obtains in 3 hours the optimal solutionwhich has a perimeter about p ∗ = 3.121147, with an error less than 10 −6 . In this solution, theconstraint ||v 0 − v 4 || ≤ 1 is binding and thus the optimal configuration corresponds to case 29.In [2], one found, using MAPLE T M , an analytical solution for p ∗ .Furthermore, by adding the following constraints derived from Proposition 2:∂ (‖v 2 − v ′ 1 ‖ + ‖v′ 1 − v 0‖ + ‖v 0 − v 3 ‖)∂α 1= 0and,∂ (‖v 2 − v ′ 3 ‖ + ‖v′ 3 − v 4‖ + ‖v 4 − v 1 ‖)∂α 3= 0where v 1 ′ = ( cos( α 12), sin( α 12) ) and v 3 ′ = ( x 3 + cos(α 2 + α 32), y 3 − sin(α 2 + α 32) ) , IBBA showedin only 0.12 seconds that there are no feasible solution. Therefore, case 10 is eliminated.