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Proceedings of GO 2005, pp. 9 – 14.A Trust-Region Algorithmfor Global OptimizationBernardetta Addis 1 and Sven Leyffer 21 Università di Firenze, Firenze, Italy b.addis@ing.unifi.it2 Argonne National Laboratory, Argonne, IL, USA leyffer@mcs.anl.govAbstractKeywords:We consider the global minimization of a box-constrained function with a so-called funnel structure.We develop a two-phase procedure that uses sampling, local optimization, and Gaussian smoothingto construct a smooth model of the underlying funnel. The procedure is embedded in a trust-regionframework that avoids the pitfalls of a fixed sampling radius. We present a numerical comparisonto popular methods and show that the new algorithm is robust and uses fewer local minimizationssteps.Global optimization, smoothing, trust region.1. IntroductionWe consider the global optimization problem{minimizexf(x)subject to x ∈ S ⊂ IR n ,(1)where f is sufficiently smooth and S ⊂ IR n is a compact set with simple structure, such as abounded box.Problems of type (1) arise in diverse fields, in particular, well-known conformational problemssuch as protein folding and atomic/cluster problems. In these applications we are interestedin finding the lowest free energy conformation in three-dimensional space. A box canbe defined that eventually will contain all “interesting” molecular conformations.If the problem allows the use of a sufficiently efficient local optimization algorithm, a twophaseprocedure is a good candidate for global optimization [7]. Such a procedure involvessampling coupled with local searches started from the sampled points. We define the localminimization operator as{minimize f(y) starting from xL(x) := y(2)subject to y ∈ S.We note that this operator is implicitly defined and depends on the local minimizer used.In general, L(x) is a piecewise constant function whose pieces correspond to the basins ofattraction of the local minima of f(x).Clearly, the global optimization problem (1) has the same optimal objective value as thefollowing problem:{minimize L(x)x(3)subject to x ∈ S.