A simple but effective evolutionary algorithm is proposed in this paper for solving complicated optimization problems. The new algorithm presents two hybridization operations incorporated with the conventional genetic algorithm. It takes only 4.1% ~ 4.7% number of function evaluations required by the conventional genetic algorithm to obtain global optima for the benchmark functions tested. Application example is also provided to demonstrate its effectiveness.Singapore-MIT Alliance (SMA

Liu, Guirong

Xu, Y.G.

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  1   Abstract -- A simple but effective evolutionary algorithm is proposed in this paper for solving complicated optimization problems. The new algorithm presents two hybridization operations incorporated with the conventional genetic algorithm. It takes only 4.1% ~ 4.7% number of function evaluations required by the conventional genetic algorithm to obtain global optima for the benchmark functions tested. Application example is also provided to demonstrate its effectiveness. Keywords -- Evolutionary algorithm, optimization.  I. INTRODUCTION volutionary algorithms have been known as the effective technique for solving complicated optimization problems such as those with multi-modal, non-differentiable and non-continuous objective functions [1, 2]. Of the present evolutionary algorithms, hybrid genetic algorithms (GAs) have received the increasing attention and investigation in recent years [3]. This is because the hybrid GAs combine the globe explorative power of conventional GAs with the local exploitation behaviors of deterministic optimization methods, they usually outperform the conventional GAs or deterministic optimization methods to be individually used in engineering practice. In this study, a new hybrid GA (called nhGA) is proposed. It presents two hybridization operations incorporated with the conventional GA. The first one is to use a simple interpolation method to move the best individual produced by the conventional genetic operations to an even better neighboring point in each of generations. The second one is to use a hill-climbing search to move a randomly selected individual to its local optimum. This is may be done only when the first hybrid operation fails to improve the best individual consecutively in several generations. Compared with the other hybrid GAs, the nhGA is not only excellent in the convergence performance, but also very simple and easy to be implemented in engineering practice.  II. EVOLUTIONARY ALGORITHM - nhGA   Basically, the newly proposed evolutionary algorithm nhGA is the further development for the hybrid GA called hGA [4].  G. R. Liu is a Fellow with Singapore-MIT Alliance (HPCES), Associate Professor with Department of Mechanical Engineering, The National University of Singapore. E-mail: mpeliugr@nus.edu.sg. Y. G. Xu is a Research Fellow with Singapore-MIT (HPCES). E-mail: smaxuyg@nus.edu.sg. As the hGA has been discussed in detail in Ref. [4], which may be used as a reference to explain the mechanism of nhGA, it is decided herein to only give a brief description for the implementation process of nhGA as follows:  (1) j=0, start up the evolutionary process.  (a)  Select the operation parameters including population size N, crossover possibility pc, mutation possibility pm, random seed id, control parameter α and β [4], etc. (b) Initialize N individuals, P(j)=(pj1, pj2,…,pjN), using a random method. Every individual pji (i=1,…,N) is a candidate solution. (c) Evaluate the fitness values of P(j). (2) Check the termination condition. If “yes”, the evolutionary process ends. Otherwise, j=j+1 and proceed to next step. (3) Carry out the conventional genetic operations in order to generate the offspring, i.e. the next generation of solutions, C(j)=(cj1, cj2,…,cjN). These operations to be used include niching [5], selection [1], crossover [1], elitism [5], etc.  (4) Implement the first hybridization operation.  (a) Construct the move direction −d of best individual.                  )( ccd bj −=−                    (1)         =≠=−−−b1jbjsjb1jbjb1jcc                    cc        c          cc        (2) Where bjc 1−  is the best individual in C(j-1) at the j-1-th generation, bjc  and sjc  are the best and second best individuals in C(j) at the j-th generation, respectively. (b) Generate two new individuals c1, c2, and evaluate their fitness values;                      −+= dcc bj α1                                (3)                      −−+= dcc bj β12                             (4) where α and β are control parameters. They are recommended to be within 0.1 ~ 05 and 0.3 ~ 0.7, respectively [4]. (c) Select a better individual cm, A Simple But Effective Evolutionary Algorithm for Complicated Optimization Problems Y. G. Xu,  G. R. Liu  E   2                              f(cm) = max{f(c1), f(c2)}    cm∈{c1, c2}       (5) f(.) is the fitness function. (d) Replace the individual bjc  in C(j) with the individual cm. This results in a upgraded offspring Cu(j)=(cj1, cj2,…,cm,…,cjN-1). (e) Check if there occurs population convergence in Cu(j). If “yes”, implement restarting strategy [4] to generate the new C(j). (5) Check if the best individual keeps unimproved consecutively in the M generations (M=3~5). If “yes’, implement the second hybrid operation.  (a) Randomly select a individual cji in Cu(j). (b) Take cji as an initial point to start the hill-climbing search. (c) Replace individual cji with the local optimum cjL obtained by the hill-climbing search. (6) Go back to step (2).   It is clear from the above description that the newly proposed nhGA, compared with the previous hGA, does not incur any deterioration of population diversity when incorporated with the hybridization operations.  III. PERFORMANCE TEST A. Benchmark functions    Three benchmark functions are used to test the proposed nhGA. Each of benchmark functions has lots of local optima and one or more global optima. Figure 1 shows the search space of function F1. F1:  ∏=−−+=2164.0/)0667.0(2log480212)5.01.5sin(),(ixiiexxxf π                  π=3.14159, 0.10 << ix , i=1,2 F2: ∑=−−−−−−−=1012310/10/10/321 }][{) x,,( 21iiiixix xeeeexxf  155 <<− ix , i=1,2,3 F3: 2151 )sin(10{) x,...,( xxf ππ=                        ∑=+ −++−+4125212 )1(5/]})sin(101()1[(iii xxx π  π=3.14159,  1010 <<− ix , i=1,…,5                Fig. 1.  Search space of benchmark function F1. TABE I MEAN NUMBERS OF FUNCTION EVALUATIONS TO CONVERGENCE  No. Global Optimum Func. Value n  mn  mnn /  (%) F1 (0.0669, 0.0669) 1.0 141 3365 4.2 F2 (1, 10, 1) 0.0 237 5745 4.1 F3 (1, 1, 1, 1, 1, 1) 0.0 6637 139915 4.7          0 200 400 600 8000.00.20.40.60.81.0Number of generationsmGAnhGA (a)   point cjL point cji (b)  Fig. 2.  (a) Convergence process in view of generations. (b) Hill-climbing process in hybridization operation.      For each of benchmark functions, the nhGA runs 10 times with the different random seed id. The 10 random seeds are -X1 X2   3 1×102, -5×102, -1×104, -1.5×104, -2×104, -3×104, -3.5×104, -4×104, -4.5×104, -5×104, respectively. The other operation parameters are N=5, pc=0.5, pm=0.02, α=0.2, β = 0.5 and M=3. Tournament selection, one child, niching, elitism are chosen to use. Table 1 shows the mean numbers of function evaluations, n  and mn , that are taken to reach the global optima using the nhGA and conventional mGA [5], respectively. It can be found that the nhGA demonstrates a much faster convergence than the conventional mGA.   Figure 2 shows the convergence processes of benchmark function F1 using the nhGA against the mGA, from which comparison of the convergence processes between nhGA and mGA can be seen more clearly.   B. Application example   Figure 3 schematically shows a valve-less micropump. The pressure-loss coefficients, ζp and ζn, in the flow channels can be optimally solved from the following objective function [6]:   2112nn )) ,(() ,(min ∑=−=nimipip QQE ζζζζ      i=1,…, K   (6) ζpmax≤ζp≤ζpmin,  ζnmax≤ζn≤ζnmin   Qi(ζp, ζn) is the mean flux calculated from a complicated model [5] using the trial ζp and ζn, imQ  is the measured mean flux at the i-th trial. K is the number of trials.                    Fig. 3.  Cross-sectional view of a micropump.   TABLE II SOLUTIONS FOR 3 SIMULATED CASES  n ζp ζn e(ζp) (%) e(ζn) (%) Case I 790 1.389 0.918 -4.9 -3.4 Case II 767 1.307 0.894 2.1 2.8 Case III 525 1.112 0.443 5.9 5.5    The nhGA is used for solving this problem. Table 2 shows the corresponding solutions for 3 simulated cases. In Table 2, n is the number of function evaluations taken by the nhGA, ζp and ζn are the solved pressure-loss coefficients, e(ζp) and e(ζn) are the errors with respect to their true values, respectively. It can be seen that nhGA converges to the satisfactory results very fast. The maximal error of solved ζp and ζn are only -4.9%, 2.8% and 5.9% for 3 simulated cases, respectively. IV. CONCLUSION   In this study, a simple but effective evolutionary algorithm nhGA is proposed. Numerical examples have demonstrated its effectiveness and efficiency. This provides a new choice for solving the complicated optimization problems in engineering practice.  REFERENCES [1] T. Back, U. Hammel and H. P. Schwefel, “Evolutionary computation: comments on the history and current state,” IEEE Trans. Evol. Comput., vol. 1, pp. 3-17, 1997. [2] D. E. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning, Addison-Wesley publishing company, USA, 1989. [3] M. Gen and R. W. Chen, Genetic Algorithms and Engineering Design, John Wiley & Sons, New York, 1997. [4] Y. G. Xu, G. R. Liu and Z. P. Wu, “A novel hybrid genetic algorithm using local optimizer based on heuristic pattern move,” Appl. Artif. Intell. – An Int. J. vol. 15, pp. 601-631, 2001.  [5] D. L. Carroll, “Genetic algorithms and optimizing chemical oxygen-iodine lasers,” Developments in Theor.  and Appl. Mech. Uni. of Alabama, vol. 10, pp. 411-424, 1996. [6] Y. G. Xu, G. R. Liu, L. S. Pan and N. Y. Ng, “Parameter Identification of Dynamic Flow-Pressure Characteristics in Valve-less Micropumps,” J. of Micromech. Microeng., Submitted for publication, July 2001.     Y. G. Xu is currently a research fellow with Singapore-MIT Alliance. He received the Ph. D from Huazhong University of Science and Technology, China, in 1996. His research interests cover (a) simulation and optimization: large-scale structures, smart structures and MEMS, (b) structural damage detection and health monitoring: modal analysis methods, elasticwave propagation methods and PZT actuator-sensor impedance methods, (c) inverse problem algorithms: adaptive neural networks techniques and hybrid genetic algorithms, and (d) Vibration & noise control, probability fracture mechanics and reliability analysis, wavelet analysis.  Chamber  Inlet Excitation force Diffuser Diffuser Outlet 

A Simple But Effective Evolutionary Algorithm for Complicated Optimization Problems

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A Simple But Effective Evolutionary Algorithm for Complicated Optimization Problems

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