View - Universidad de Almería

The Small Octagon with Longest Perimeter 25are the length of two sides of the octagon, and [v i , v j ], [v i , v j+1 ] and [v i , v j+2 ] are edges of thethrackleation graph; i.e. ‖v i − v j ‖ = ‖v i − v j+1 ‖ = ‖v i − v j+2 ‖ = 1.•v jc jc j+1β....α.....α i ...•v j+1.v i • ..• v j+2Figure 2.Equal angles.Using the same demonstration, one proves in [2] the two following properties:Proposition 1. Consider a small octagon based on a thrackleation with maximal perimeter. If c j is atype I edge, as represented in Figure 2 (v i is an extreme point of two edges), and v i its associated vertex,then ∠v j v i v j+1 ∈ [0.317, 0.465]. Moreover, if c j+1 is also a type I edge, consecutive to c j and with thesame associated vertex v i then ∠v j v i v j+2 ∈ [0.688, 0.881].The next result concerns thrackleations containing a pending diameter.Proposition 2. Consider a small octagon based on a thrackleation with maximal perimeter, with twoconsecutive type I edges c j = [v j , v j+1 ] and c j+1 = [v j+1 , v j+2 ] sharing the same associated vertex v i .Then ∠v j v i v j+1 = ∠v j+1 v i v j+2 = ∠v jv i v j+22; i.e. α = β = α i2 .These two properties allow to reduce the remaining studied problems.4. Exact algorithmIn that stage, one needs the use of an exact global optimization algorithm to discard somecases and to determine the global solution.The global optimization algorithm used here is a Fortran 90/95 implementation of a branchand-boundmethod where bounds are computed with interval analysis techniques. It is calledIBBA (for Interval Branch and Bound Algorithm). All computations were performed on acluster of thirty bi-processors PC’s ranging from 1GHz to 2.4GHz at the University of Pau.Interval analysis was introduced by Moore [12] in order to control the propagation of numericalerrors due to floating point computations. Thus, Moore proposes to enclose all realvalues by an interval where the bounds are the two closest floating point numbers. Thenexpanding the classical operations - addition, subtraction, multiplication and division- intointervals, defines interval arithmetic. A straightforward generalization allows computation ofreliable bounds (excluding the problem of numerical errors) of a function over a hypercube(or box) defined by an interval vector. Moreover, classical tools of analysis such as Taylor expansionscan be used together with interval arithmetic to compute more precise bounds [12].Other bounding techniques due to Messine et al. [11] combines linear underestimations (orhyperplanes) obtained at all vertices of the box [11].The principle of IBBA is to bisect the initial domain where the solution is sought for intosmaller and smaller boxes, and then to eliminate the boxes where the global optimum cannotoccur:by proving, using interval bounds, that no point in a box can produce a better solutionthan the current best one;