Funneling algorithms multiscale optimization on rugged terrains

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Global optimization problems for which it is either intractable or undesirable to find all stationary points and that contain so-called "rough" or rugged objective function landscapes are studied. Such problems often show considerable differences between the small-scale (or local) geometry and the large-scale (or more global) geometry. A new class of functions called generalized exponential funnel functions is proposed for modeling large-scale geometry. This class of functions is capable of modeling folds, cups, cones, funnels, and other geometric objects that recur in many physical applications. The basic mathematics of exponential funnels is described, along with their numerical and algorithmic implications. It is shown that funnels provide a nonconvex model of large-scale geometry that, when approximated correctly, exhibits a unique and easily calculated minimum that can be estimated from a small amount of local objective-function and derivative information. Ways of extracting "average" gradient and Hessian matrix information are presented, and novel interpolating formulas for building funnel approximations of large-scale geometries are given. A funneling algorithm, intended to guide or "funnel" iterates to regions where the most promising global optimizers are expected to lie, is described. A multiscale global optimization algorithm based on the combined use of terrain methods and funneling algorithms is proposed. The terrain methodology is used to gather small-scale information while a funneling algorithm is used to guide the overall optimization calculations and to make "large" moves within the feasible region. Communication between scales is also addressed. Typical empirical force field functionality, many stationary points, and changes in convexity are used to clearly illustrate by example, but not to prove theoretically, that the combined terrain/funneling methodology is capable of finding a global minimum without calculating all stationary points and can lead to significant reductions in overall computational work.

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Industrial and Engineering Chemistry Research