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Proceedings of GO 2005, pp. 241 – 246.Pruning a box from Baumann point in an Interval GlobalOptimization Algorithm ∗Boglárka Tóth 1,† and L.G. Casado 21 Dpt. Statistics and Operations Research, University of Murcia, Spain, boglarka@um.es2 Dpt. of Architecture and Electronics, University of Almeria, Spain, leo@ace.ual.esAbstractThis work is in the context of reliable global optimization algorithms, which use branch and boundtechniques and interval arithmetic to obtain inclusion functions. The search region have to haveedges parallel to the axes (box) in order to allow the use of interval arithmetic to obtain boundsof the objective function in that region. We study a method to determine the largest region insidethe current box where the global minimum cannot exist, based on the gradient information. If thecurrent box cannot be rejected completely, the removed region have to satisfy that the generatedsubproblems (that can contain the global minimum) have to have a box shape, in order to apply thebranch and bound algorithm, but we are also interested in generate the smaller number of them toreduce the overall computational cost. The method is presented for two and n dimensional problemsand numerical results show that is worth to incorporate it to interval global optimization algorithms.1. IntroductionThe context of our study is unconstrained global optimization. Thus, the general problem canbe defined as min x∈S f(x), where f is a continuously differentiable function defined over then-dimensional interval S ⊂ R n , where the minimum have to be found.In our context, real numbers are denoted by x, y, . . . and compact intervals by X = [x L , x U ], Y =[y L , y U ], . . ., where x L = min{x ∈ X} and x U = max{x ∈ X} are the lower and upper boundsof X, respectively. The set of compact intervals is denoted by I := {[a, b] | a, b ∈ R, a ≤ b}. Thenotation x = (x 1 , . . . , x n ) T , x i ∈ R and X = (X 1 , . . . , X n ) T , X i ∈ I (i = 1, . . . , n) is used forreal and interval vectors, respectively. The set of n-dimensional interval vectors (also calledboxes) is denoted by I n .Let f : Y ⊆ R n → R be a continuous function, and I(Y ) = {X | X ∈ I, X ⊆ Y }. Thefunction F : I(Y ) ⊆ I n → I is an inclusion function of f, if for every X ∈ I(Y ) and x ∈ X,f(x) ∈ F (X), i.e. f(X) = {f(x) |x ∈ X} ⊆ F (X). This inclusion can be obtained, forinstance, by Interval Arithmetic [2].Let X be such a box that x, c ∈ X. If G(X), i.e. an inclusion function of the gradient vectorg(X) is known, then the centered form is defined as F c (X) = F (c)+G(X)(X −c). Many timesc is the midpoint of X, but it can be anywhere in X.∗ This work has been partially supported by the Ministry of Education and Science of Spain through grant CICYT-TIC2002-00228.† On leave from the Research Group on Artificial Intelligence of the Hungarian Academy of Sciences and the University ofSzeged, H-6720 Szeged, Aradi vértanúk tere 1., Hungary.