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140 Kenneth Holmströmsurrogate model, e.g. Kriging models, is needed in such iteration steps. Checking for an interiorpoint makes it possible to detect the problem and do something else before computing thecostly f(x). The new ARBF algorithm is very efficient when the RBF interpolation has globalminima that most often are interior points. It has better local convergence properties thanthe original RBF algorithm. The experimental version of ARBFMIP takes about 2-4 secondsper iteration for up to 200 function values, and 3-7 seconds per iteration if doing 400 functionvalues. The final version should be fast enough for practical use. The use of radial basis interpolationmethods for costly (expensive) mixed-integer nonlinear optimization problems ispromising.References[1] M. Björkman and K. Holmström. Global Optimization of Costly Nonconvex Functions Using Radial BasisFunctions. Optimization and Engineering, 1(4):373–397, 2000.[2] L. C. W. Dixon and G. P. Szegö. The global optimisation problem: An introduction. In L. Dixon and G Szego,editors, Toward Global Optimization, pages 1–15, New York, 1978. North-Holland Publishing Company.[3] Hans-Martin Gutmann. On the semi-norm of radial basis function interpolants. Journal of ApproximationTheory, 2001.[4] Hans-Martin Gutmann. A radial basis function method for global optimization. Journal of Global Optimization,19:201–227, 2001.[5] K. Holmström. The TOMLAB Optimization Environment in Matlab. Advanced Modeling and Optimization,1(1):47–69, 1999.[6] Kenneth Holmström and Marcus M. Edvall. Chapter 19: The TOMLAB optimization environment. In LudwigshafenGermany Josef Kallrath, BASF AB, editor, Modeling Languages in Mathematical Optimization, AP-PLIED OPTIMIZATION 88, ISBN 1-4020-7547-2, Boston/Dordrecht/London, January 2004. Kluwer AcademicPublishers.[7] Waltraud Huyer and Arnold Neumaier. Global optimization by multilevel coordinate search. Journal ofGlobal Optimization, 14:331–355, 1999.[8] D. R. Jones, C. D. Perttunen, and B. E. Stuckman. Lipschitzian optimization without the Lipschitz constant.Journal of Optimization Theory and Applications, 79(1):157–181, October 1993.[9] Donald R. Jones. DIRECT. Encyclopedia of Optimization, 2001.[10] Donald R. Jones. A taxonomy of global optimization methods based on response surfaces. Journal of GlobalOptimization, 21:345–383, 2002.[11] Donald R. Jones, Matthias Schonlau, and William J. Welch. Efficient global optimization of expensive Black-Box functions. Journal of Global Optimization, 13:455–492, 1998.[12] M. J. D. Powell. The theory of radial basis function approximation in 1990. In W.A. Light, editor, Advancesin Numerical Analysis, Volume 2: Wavelets, Subdivision Algorithms and Radial Basis Functions, pages 105–210.Oxford University Press, 1992.[13] M. J. D. Powell. Recent research at Cambridge on radial basis functions. In M. D. Buhmann, M. Felten,D. Mache, and M. W. Müller, editors, New Developments in Approximation Theory, pages 215–232. Birkhäuser,Basel, 1999.[14] Rommel G. Regis and Christine A. Shoemaker. Constrained Global Optimization of Expensive Black BoxFunctions Using Radial Basis Functions. Journal of Global Optimization, 31(1):153–171, 2005.