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10 Bernardetta Addis and Sven Leyffer1098765432100 1 2 3 4 5 6 7 8 9 10Figure 1.Example of a funnel functionWe note that the piecewise constant nature of L(x) implies that the minima of (1) and (3) neednot agree. In fact, any global minimum of (1) is also a global minimum of (3), but not viceversa. Because L(x) is implicitly defined, however, we can simply recordx min := LS(x) :={ argmin f(y) starting from xysubject to y ∈ S.(4)It follows that x min is also a local minimum of f(x), and we can recover a global minimum off(x) by solving (3) in this way.Multistart is an elementary example of a two-phase method aimed at minimizing L(x); inpractice, it reduces to a purely random (uniform) sampling applied to L(x). It is in principlepossible to apply any known global optimization method to solve the transformed problem(3), but many difficulties arise. First, function evaluation becomes much more expensive: wehave to perform a local search on the original problem in order to observe the function L(x) ata single point. Second, the analytical form of L(x) is not available, and it is a discontinuous,piecewise constant function.In many applications, such as molecular conformation problems [3], it is widely believedthat the local optimum points are not randomly displaced but that the objective function f(x)displays a so-called funnel structure. A univariate example of such a function is given in Figure1, where the function to be minimized is represented by the solid line and the underlyingfunnel structure is given by the dotted line. In general, we say f(x) has funnel structure if itis a perturbation of an underlying function with a low number of local minima. Motivatedby examples of this kind, some authors [5,6,8] have proposed filtering approaches: if one canfilter the high frequencies that perturb the funnel structure, then one can recover the underlyingfunnel structure and use a standard global optimization method on the filtered function(which is much easier to globally optimize) in order to reach the global optimum.In contrast, we believe that it is better to filter the piecewise linear function L(x) becauseit is less oscillatory than f(x); Figure 2 shows L(x) for the simple funnel function previouslypresented. This follows the approach of [2], and much of the analysis in [2] also applies here.In this paper we make two important contributions to global optimization. First, we removethe need for the arbitrary parameters in [2] by interpreting these parameters as aradius. Weembed the algorithm from [2], called ALSO, in aframework and show that our new algorithmis more robust than other methods. Second, we introduce the concept of global quality. Thisconcept is motivated by the fact that theframework is essentially a local optimization schemeand therefore requires modifications to be effective as a global method.