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134 Kenneth Holmströmcomputed by RBF and the algorithm handled cheap box constraints. Global non-costly subproblems were solved with TOMLAB [5] implementations of DIRECT methods [8, 9] combinedwith good local solvers.This paper presents an important extension to the method in [1] — the Adaptive-RBF(ARBF) method implemented in TOMLAB [6]. The standard RBF algorithm very often resultin points sampled on the boundary, which leads to poor performance and non-convergence.Therefor we propose a one-dimensional search for a suitable target value f ∗ n to improve convergence.This leads to a sequence of global optimization problems to be solved in each iteration.In addition the standard RBF algorithm has bad local convergence properties and wesuggest new ways to improve local convergence.The algorithm is described in detail and we analyze its efficiency on the Shekel test problems,that are part of the standard test problem set of Dixon-Szegö [2]. The results show thatthis improved implementation of the RBF algorithm is very efficient on the standard test problemscompared to the standard RBF algorithm.2. The RBF methodThe idea of the RBF algorithm is to use radial basis function interpolation and a measure of‘bumpiness’ of a radial function, σ say. A target value f ∗ n is chosen that is an estimate of theglobal minimum of f. For each x /∈ {x 1 , . . . , x n } there exists a radial basis function s n (x) thatsatisfies the interpolation conditionss n (x i ) = f(x i ), i = 1, . . . , n,s n (x) = f ∗ n . (2)The next point x n+1 is then calculated as the value of x in the feasible region that minimizesσ(s n ). It turns out that the function x ↦→ σ(s n ) is much cheaper to compute than the originalfunction.The smoothest radial basis interpolation is obtained by minimizing the semi-norm [3]s n = arg mins< s, s > . (3)Here, the radial basis function interpolant s n has the formn∑s n (x) = λ i φ (‖x − x i ‖ 2) + b T x + a, (4)i=1with λ 1 , . . . , λ n ∈ R, b ∈ R d , a ∈ R and φ is either cubic with φ(r) = r 3 or the thin plate splineφ(r) = r 2 log r, see Table 1. The unknown parameters λ i , b and a are obtained as the solutionof the system of linear equations( )ΦPP T 0) (λc(F=0), (5)where Φ is the n × n matrix with Φ ij = φ ( )‖x i − x j ‖ 2 and⎛ ⎞⎛x T ⎞ ⎛ ⎞ b1 1λ 1⎛ ⎞1x T 2 1P =⎜ . .⎟⎝ . . ⎠ , λ = λ 2b 2 f(x 1 )⎜ .⎟⎝ . ⎠ , c = .f(x 2 )⎜ ., F =⎜ .⎟⎟ ⎝⎝x T bn 1λ d⎠. ⎠ . (6)n f(xan )( ) Φ PIf the rank of P is d + 1, then the matrixP T is nonsingular and the linear system (5)0has a unique solution [12].