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172 Frédéric Messine and Ahmed TouhamiTaylor expansions and on interval computations of an enclosure of the gradient or the Hessianmatrix [6, 7, 9]; one uses here an centered inclusion function based on a Taylor expansion atthe first order, it is denoted T 1 (X) in the following.Affine arithmetic was proposed and developed recently by Stolfi et al. [1–5], although asimilar tool, the generalized interval arithmetic, have been developed in 1975 by Hansen [6].Its principle is to keep some affine informations during the computations of bounds. Likegeneralized interval arithmetic [6], affine arithmetic was developed to take into account theproblems of the dependencies of the variables generated by interval computations. This toolpermits to limit some negative effects due to interval arithmetic: The links between differentoccurrences of a same variable. The interest of the bounds computed by using this affine arithmeticwas shown in [8]. Furthermore in [8], two new affine forms and a first quadratic formwere introduced and have proved their real efficiency solving optimization problems suchas defined in (1). The purpose in this paper is to consider the complete quadratic form suchas defined in [10] and to show its efficiency for solving unconstrained global optimizationproblems for multivariate polynomial functions. Affine arithmetic such as defined in [4] isreliable. In this article, one uses a different way to render reliable affine arithmetic by convertingall the floating point coefficients into interval ones and by introducing rounded intervalcomputations. This reliable inclusion function is denoted by AF(X) in the following.The general quadratic form aims to conserve some affine informations and also some quadraticinformations (about the error due to non-affine operations) during the computations. Therefore,these added informations permit to improve the quality of the so-computed bounds (inspite of an expansion of the complexity of such an algorithm). The gain of this technique isvalidated on some global optimization polynomial problems.A complete work on the general quadratic form, presenting some properties, are detailed in[10]. Here, one recalls the principle of this technique to compute bounds and we emphasizesthe interest of this tool in global optimization.2. Reliable General Quadratic FormA general quadratic form is represented by ̂x:⎧⎪⎨⎪⎩̂x = ɛ T Aɛ + b T ɛ + c + e + ɛ n+1 + e − ɛ n+2 + eɛ n+3 ,n∑n∑= a ij ɛ i ɛ j + b i ɛ i + c + e + ɛ n+1 + e − ɛ n+2 + eɛ n+3 ,i,j=1i=1(2)where A ∈ IR n×n , b ∈ IR n , c ∈ IR, (e + , e − , e) ∈ (IR + ) 3 and ɛ i ∈ [−1, 1], ∀ i ∈ {1, . . . , n},ɛ n+1 ∈ [0, 1], ɛ n+2 ∈ [−1, 0] and ɛ n+3 ∈ [−1, 1]. The symbolic variables ɛ n+1 , ɛ n+2 and ɛ n+3represent the noise for the errors generated by performing non-affine computations.The conversions between interval and a general quadratic form are performed as follows:Interval −→ General Quadratic Form:X = [x L , x U ]−→ ̂x = ɛ T Aɛ + b T ɛ + c + e + ɛ n+1 + e − ɛ n+2 + eɛ n+3