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Proceedings of GO 2005, pp. 171 – 176.An Interval Branch-and-Bound Algorithm based onthe General Quadratic FormFrédéric Messine ∗ and Ahmed TouhamiENSEEIHT-IRIT, 2 rue C Camichel, 31071 Toulouse Cedex, France {Frederic.Messine, Ahmed.Touhami}@n7.frAbstractIn this paper, an extension of affine arithmetic introduced by Stolfi et al. [1–5] is proposed. It isbased on a general quadratic form which leads to most efficient computations of errors due to nonaffineoperations. This work recalls the main results published in [10] and emphasizes the interest ofsuch a technique for solving unconstrained global optimization problems of multivariate polynomialfunctions.1. IntroductionWe focus in this paper on unconstrained bounded minimization problems of polynomial functions,written as follows:min f(x), (1)n x∈X⊆IRwhere X is a hypercube of IR n and f is a polynomial function.The Branch-and-Bound algorithms can be based on interval arithmetic for the computationsof the bounds of a function over a box. The outwardly rounded interval arithmetic was thenintroduced by Moore to deal with numerical errors, [11]. Thus, its utilization inside Branchand-Boundalgorithms makes these methods rigorous [7]; we speak about rigorous globaloptimization methods when no numerical error can render wrong the computations of thebounds. The principle of a Branch-and-Bound algorithm is to bisect the initial domain wherethe function is sought for into smaller and smaller boxes, and then to eliminate the boxeswhere the global optimum cannot occur; i.e. by proving that a current solution is lower (resp.upper for a maximization problem) than a lower (resp. upper) bound of the function over thisbox. Therefore in such a box, there does not exist a point such that its value is lower (resp.upper) than the current solution already found.An inclusion function is an interval function such that the interval result encloses the rangeof the associated real function over the studied box; i.e. if F (X) denotes the inclusion functionof a function f over a box X, one has by definition that [min x∈X f(x), max x∈X f(x)] ⊆ F (X).The extension of an expression of a function into interval (by replacing all the occurrences ofthe variables by the corresponding intervals, and all the operations by interval operations)defines an inclusion function which is called the natural extension inclusion function; it isdenoted NE(X). Nevertheless, the direct use of NE is generally inefficient and then, thecomputed bounds are not sufficiently accurate to solve both rapidly and with a high numericalprecision certain optimization problems of type (1). Therefore, for the past several years, alot of new techniques which permit to improve the computations of the bounds have beenstudied. Generally, these techniques are based on the combinations of first or second order∗ Work of the first author was also supported by the ENSEEIHT-LEEI-EM 3 , 2 rue C Camichel, 31071 Toulouse Cedex