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174 Frédéric Messine and Ahmed Touhamiforms which correspond to each occurrence of each variable, the computations are performedusing operations between general quadratic form and then, the result is converted into aninterval. This resulting interval encloses rigorously the range of the function over the consideredbox (interval vector). This technique defines a new automatic tool for the computationsof bounds for a function over a box. One denotes by GQF an inclusion function constructedsuch a way (using general quadratic forms).Example 1. The interest of this general quadratic form can be illustrated on this simple examplef(x 1 , x 2 ) = x 2 1 x 2 − x 1 x 2 2 over [−1, 3]2 . We have:x 2 1x 2 − x 1 x 2 2 = (1 − 2ɛ 1 ) 2 (1 − 2ɛ 2 ) − (1 − 2ɛ 1 )(1 − 2ɛ 2 ) 2= (1 − 4ɛ 1 + 4ɛ 2 1)(1 − 2ɛ 2 ) − (1 − 4ɛ 2 + 4ɛ 2 2)(1 − 2ɛ 1 )= ( 1 − 4ɛ 1 + 4ɛ 2 1 − 2ɛ 2 + 8ɛ 1 ɛ 2 − 8ɛ 2 1 ɛ (2)+ −1 + 4ɛ2 − 4ɛ 2 2 + 2ɛ 1 − 8ɛ 1 ɛ 2 + 8ɛ 1 ɛ 2 )2= −2ɛ 1 + 2ɛ 2 + 4ɛ 2 1 − 4ɛ 2 2 − 8ɛ 2 1ɛ 2 + 8ɛ 1 ɛ 2 2AF(X) = [−2ɛ 1 + 2ɛ 2 + 40ɛ 3 ] = [−44, 44], with ɛ n+1 = ɛ 3 ∈ [−1, 1].GQF(X) = [−2ɛ 1 + 2ɛ 2 + 4ɛ 2 1 − 4ɛ2 2 + 16ɛ 5] = [−24, 24], with ɛ n+3 = ɛ 5 ∈ [−1, 1].NE(X) = X 2 1 X 2 − X 1 X 2 2 = [−1, 3]2 [−1, 3] − [−1, 3] 2 [−1, 3] = [−9, 27] − [−9, 27] =[−36, 36].In that case, NE is better than AF but GQF is the most efficient.In [10], some theoretical properties about the efficiency of the bounds of such a techniqueare reported. Furthermore, the extension into reliable forms are defined by replacing all thefloating point coefficients by intervals and by replacing all the operations by rounded intervaloperations. Thus, the bounds computed by introducing interval components inside thegeneral quadratic form become reliable; no numerical error can occur performing these computations.The bounds are also guaranteed.3. Application to Rigorous Global OptimizationThe main global optimization algorithm is based on a classical Branch-and-Bound techniquedue to Ichida-Fujii [12]. To show the efficiency of the use of general quadratic forms, thecomputations of the bounds are replaced inside this algorithm. One can speak here about rigorousglobal optimization because the bounds are numerically guaranteed; all the computedbounds are reliable using the four techniques presented above NE, T 1 , AF and GQF. In orderto show the efficiency of such an inclusion function based on the general quadratic formsome multivariate polynomial problems are taken into account, see Table 1. All these problemsare solved by a basic interval Branch-and-Bound algorithm due to Ichida-Fujii, see [12].This algorithm is modified in order to determine the global minimum f ∗ with a maximal accuracy(by solving the problem with an accuracy divided by 10), see [10]. In fact, the algorithmstops when the new precision (divided by 10) of the global solution cannot be improved aftera consequent computational effort (here 100000 iterations).In Table 1, we summarize for each polynomial problem, the initial domain of research andthe global minimum f ∗ . Generally, these functions came from the literature [8, 9, 12]; f 2 is thewell known Golstein Price function and f 5 a function due to Ratschek.All these numerical tests have been performed on a HP-UX, 9000/800, 4 GB memory, quadriprocessor64-bit processor, computer from the Laboratoire d’Electrotechnique etd’Electronique Industrielle du CNRS/UMR 5828 ENSEEIHT-INPT Toulouse. The codeshave been developed in Fortran 90. The algorithm uses iteratively the following inclusionfunctions: natural extension into interval NE, a technique based on Taylor expansions at the