Adaptive cluster covering and evolutionary approach: comparison, differences and similarities Dimitri P. Solomatine UNESCO-IHE Institute for Water Education P.O. Box 3015 Delft The Netherlands sol@ihe.nl Abstract- In case the objective function to be minimized is not known analytically and no assumption can be made about the single extremum, global optimization (GO) methods must be used. Paper gives a brief overview of GO methods, with the special attention to principles of clustering, covering and evolution. Nine algorithms, including a simple GA, are compared in terms of effectiveness (accuracy), efficiency (number of the needed function evaluations) and reliability on several problems. Particular features of Adaptive cluster covering algorithm (ACCO) leading to its high efficiency are analyzed and compared with those of an evolutionary approach. The possibilities of (partially) attributing ACCO and other GO algorithms to the group of EA are considered. 1 Introduction During the last decade the wide spread of evolutionary and genetic algorithms (EGA) lead to successful solution of many global optimization problems that previously were not even posed. The success of EGAs is fully deserved and can be explained by their methodological appeal, relative simplicity, robustness and the existence of a well-organized community. The recent developments in hybrid EAs, in particular, memetic algorithms, lead to considerable improvements in effectiveness of this class of algorithms. The author’s interest in global optimization started from very practical problems related to hydrological modeling – mainly problems of calibration (parameters identification). This problem is typically multi-extremum and the function to be minimized is not known analytically, so the derivative-based methods cannot be used. A number of non-derivative methods for GO are known; in engineering community EAs and GAs seem to be the most popular. A considerable number of publications related, for example, to hydraulic applications started to appear already in the beginning and mid-1990s (Wang 1991; Babovic et al. 1994; Cieniawski et al., 1995; Solomatine 1995a, Savic & Walters 1997; Franchini & Galeati 1997), and by now EGAs (at least their standard implementations) have become a widely- spread technology. One can see, however, that quite often the use of EGAs is not supported by the proper analysis and not always justified. The author’s motivation for this paper was the experience with GO methods in solving practical problems related to hydrology and water management. The paper addresses the following topics: • classification and briefly description of GO algorithms and the position of EGAs; • comparison of several GO algorithms, including GAs, on a suite of problems with respect to effectiveness, efficiency and reliability; • making an attempt to explain the reasons behind the performance of some of the algorithms; • identifying differences and similarities between some GOs and EAs. 2 Approaches to solving optimization problems A global minimization problem with box constraints is considered: find an optimizer x * such that generates a minimum of the objective function f (x) where x∈X and f(x) is defined in the finite interval (box) region of the n- dimensional Euclidean space: X = {x∈R n : a≤x≤b} (componentwise). This constrained optimization problem can be transformed to an unconstrained optimization problem by introducing the penalty function with a high value outside the specified constraints. In cases when the exact value of an optimizer cannot be found, we speak about its estimate and, correspondingly, about its minimum estimate. Approaches to solving this problem depend on the properties of f(x): 1. f(x) is a single-extremum function expressed analytically. If its derivatives can be computed, then gradient-based methods may be used: conjugate gradient methods; quasi-Newton or variable metric methods, like DFP and BFGS methods (Jacobs 1977, Press et al. 1991). Many engineering applications use minimization techniques for single-extremum functions, but often without investigating whether the functions are indeed