326 IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. 11, NO. 3, JUNE 2007 Clustering-Based Adaptive Crossover and Mutation Probabilities for Genetic Algorithms Jun Zhang, Member, IEEE, Henry Shu-Hung Chung, Senior Member, IEEE, and Wai-Lun Lo, Member, IEEE Abstract—Research into adjusting the probabilities of crossover and mutation in genetic algorithms (GAs) is one of the most sig- nificant and promising areas in evolutionary computation. and greatly determine whether the algorithm will find a near-op- timum solution or whether it will find a solution efficiently. Instead of using fixed values of and , this paper presents the use of fuzzy logic to adaptively adjust the values of and in GA. By applying the -means algorithm, distribution of the population in the search space is clustered in each generation. A fuzzy system is used to adjust the values of and . It is based on considering the relative size of the cluster containing the best chromosome and the one containing the worst chromosome. The proposed method has been applied to optimize a buck regulator that requires sat- isfying several static and dynamic operational requirements. The optimized circuit component values, the regulator’s performance, and the convergence rate in the training are favorably compared with the GA using fixed values of and . The effectiveness of the fuzzy-controlled crossover and mutation probabilities is also demonstrated by optimizing eight multidimensional mathematical functions. Index Terms—Evolutionary computation, fuzzy logics, genetic algorithms (GA), power electronics. I. INTRODUCTION T HE CONVENTIONAL approach to circuit optimization is to develop a formal model that can resemble actual circuit responses closely, but is solvable by means of available mathe- matical methods, such as linear and nonlinear programming. In the area of power electronics, state-space averaging and the vari- ants [1]–[3] have been the dominant modeling techniques since 1970. By recognizing that power electronic circuits (PECs) typ- ically have output filter cutoff frequency that is much lower than the switching frequency, linear time-invariant models, such as the control-to-output or input-to-output transfer functions, can be formulated to approximate the time-variant and piece- wise-linear properties of the circuits. Although this approach has been proven to be very successful in many applications, it has the drawbacks of oversimplifying the circuit behaviors and of having limitations on particular operating mode and control Manuscript received June 15, 2005; revised January 11, 2006, April 4, 2006, and May 4, 2006. J. Zhang is with the Department of Computer Science, Sun Yat-sen Univer- sity, Guangzhou, China (e-mail: junzhang@ieee.org). H. S.-H. Chung is with the Department of Electronic Engineering, City Uni- versity of Hong Kong, Kowloon Tong, Hong Kong (e-mail: eeshc@cityu.edu. hk). W.-L. Lo is with the Department of Computer Science, Chu Hai College of Higher Education, Tsuen Wan, Hong Kong. Digital Object Identifier 10.1109/TEVC.2006.880727 schemes. As a circuit has been converted into a mathematical model and its state variables have been averaged, no detailed in- formation about the exact waveforms and the response profiles can be obtained. Circuit designers would sometimes find it diffi- cult to predict precisely the circuit responses under large-signal variations [3]. As power electronics technology continues to develop, a large number of combinatorial issues, including circuit complexity, static and dynamic responses, thermal problems, electromag- netic compatibility, control schemes, costing, etc., are associ- ated. A plethora of such multimodal functions exist in a PEC. In particular, there is a growing need for automated synthesis that starts with high-level statements of the desired behaviors and optimizes the circuit component values for meeting required specifications. Optimization strategies that are based on satisfying con- strained equations might be subject to becoming trapped into local minima, leading to suboptimal parameter values, and thus, having a limitation on operating in large, multimodal, and noisy spaces. Since 1950, other strategies that employ Darwin’s evolution theory have been proposed [4]–[6]. The most significant advantage of using this evolutionary search lies in the gain of flexibility and adaptability to the task at hand and the global search characteristics. Among various evolutionary computation methods (ECM), genetic algorithms (GA), which have been applied to many optimization problems [7], [8], employ a random, yet directed, search for locating the global optimal solution. They are superior to gradient descent techniques, as the search is not biased towards the local optimal solution. They differ from random sampling algorithms, as they can direct the search towards relatively prospective regions in the search space [9]. However, the usage of GA was progressed slowly in real applications. Apart from the shortcomings of early approaches, it was also largely due to the lack of powerful computer platforms at that time [10], [11]. Due to the recent advancements in computer technology, much research effort has been emphasized on developing new GA-based optimization methods. There are many new design schemes for analog circuits, like voltage reference circuit [12], transconductance amplifier [13], and analog circuit synthesis [14], [15]. Recently, GA have been applied to PEC optimiza- tion [16]–[18]. The circuit behaviors [16], [17] and controller functions [18] are described by well-defined mathematical functions with unknown optimal component values. The parameters of the search space in GA are encoded in the form of a chromosome-like structure. A group of these chro- mosomes constitutes a population. An index of merit (fitness value) is assigned to each individual chromosome, according 1089-778X/$20.00 © 2006 IEEE matlab1.com matlab1.com MATLAB code is ready for download matlab1.com