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Proceedings of GO 2005, pp. 133 – 140.An Adaptive Radial Basis Algorithm (ARBF) forMixed-Integer Expensive Constrained Global OptimizationKenneth HolmströmDepartment of Mathematics and Physics, Mälardalen University, P.O. Box 883, SE-721 23 Västerås, Sweden,kenneth.holmstrom@mdh.seAbstractKeywords:A mixed-integer constrained extension of the radial basis function (RBF) interpolation algorithmfor computationally costly global non-convex optimization is presented. Implementation in TOM-LAB (http://tomlab.biz) solver rbfSolve is discussed. The algorithm relies on mixed-integer nonlinear(MINLP) sub solvers in TOMLAB, e.g. OQNLP, MINLPBB or the constrained DIRECT solvers(glcDirect or glcSolve). Depending on the initial experimental design, the basic RBF algorithm sometimesfails and make no progress. A new method how to detect when there is a problem is presented.We discuss the causes and present a new faster and more robust Adaptive RBF (ARBF) algorithm.Test results for unconstrained problems are discussed.Expensive, global, mixed-integer, nonconvex, optimization, software, black box.1. IntroductionGlobal optimization with emphasis on costly objective functions and mixed-integer variablesis considered, i.e., the problem of finding the global minimum to (1) when each function valuef(x) takes considerable CPU time, e.g. more than 30 minutes to compute [7, 10, 11, 13].The Mixed-Integer Expensive (Costly) Global Black-Box Nonconvex Problemminxf(x)s/t−∞ < x L ≤ x ≤ x U < ∞b L ≤ Ax ≤ b Uc L ≤ c(x) ≤ c U , x j ∈ N ∀j ∈I,where f(x) ∈ R, x L , x, x U ∈ R d , A ∈ R m 1×d , b L , b U ∈ R m 1and c L , c(x), c U ∈ R m 2. Thevariables x I are restricted to be integers, where I is an index subset of {1,. . . ,d}. Let Ω ∈ R d bethe feasible set defined by the constraints in (1).Such problems often arise in industrial and financial applications, where a function valuecould be the result of a complex computer program, or an advanced simulation, e.g., CFD,tuning of trading strategies, or design optimization. In such cases, derivatives are most oftenhard to obtain (the algorithms discussed make no use of such information) and f(x) is oftennoisy or nonsmooth. One of the methods for this problem type utilizes radial basis functions(RBF) and was presented by Gutmann and Powell in [4, 13]. The idea of the RBF algorithm isto use radial basis function interpolation to define a utility function. The next point, wherethe original objective function should be evaluated, is determined by optimizing on this utilityfunction. The combination of our need for efficient global optimization software and theinteresting ideas of Gutmann led to the development of an improved RBF algorithm [1] implementedin MATLAB. This method was based on interpolating the function values so far(1)