The Modified IWO Algorithm for Optimization of Numerical Functions Daniel Kostrzewa and Henryk Josiński Silesian University of Technology, Akademicka 16 PL-44-100 Gliwice, Poland {Daniel.Kostrzewa,Henryk.Josinski}@polsl.pl Abstract. The Invasive Weed Optimization algorithm (IWO) is an optimization method inspired by dynamic growth of weeds colony. The authors of the present paper have modified the IWO algorithm introducing a hybrid strategy of the search space exploration. The goal of the project was to evaluate the modified version by testing its usefulness for numerical s minimization. The optimized multidimensional s: Griewank, Rastrigin, and Rosenbrock are frequently used as benchmarks which allows to compare the experimental results with outcomes reported in the literature. Both the results produced by the original version of the IWO algorithm and the Adaptive Particle Swarm Optimization (APSO) method served as the reference points. Keywords: Invasive Weed Optimization algorithm, Griewank, Rastrigin fuction, Rosenbrock. 1 Introduction The Invasive Weed Optimization (IWO) algorithm is an optimization method, in which the exploration strategy of the search space (similarly to the evolutionary algorithm) is based on the transformation of a complete solution into another one. Its idea was inspired by observation of dynamic spreading of weeds and their quick adaptation to environmental conditions [1]. The authors of the method from University of Tehran emphasized its usefulness for continuous optimization tasks. Their research was focused inter alia on minimization of the multimodal s and tuning of a second order compensator [1], antenna configurations [2], electricity market dynamics [3], and a recommender system [4]. The authors of the present paper have modified the IWO algorithm introducing a hybrid strategy of the search space exploration (described in detail in section 2) and broadened the scope of the IWO algorithm application dealing skillfully with an important discrete optimization task from the databases area the join ordering problem for both centralized and distributed data [5]-[6]. The symmetric TSP was also successfully solved by the modified IWO equipped with the inver-over operator [7]. The goal of the present paper is an evaluation of the modified IWO based on the effects of the optimization of the Griewank, Rastrigin and Rosenbrock L. Rutkowski et al. (Eds.): SIDE 2012 and EC 2012, LNCS 7269, pp. 267 274, 2012. c Springer-Verlag Berlin Heidelberg 2012
268 D. Kostrzewa and H. Josiński s. The results (minima values) produced by the modified IWO were compared with the outcomes generated by the original IWO as well as with the results of the Adaptive Particle Swarm Optimization (APSO) [8], [9]. The organization of this paper is as follows section 2 contains a brief description of the IWO algorithm taking into serious consideration the proposed hybrid method of the search space penetration. Optimized s are presented in section 3. Section 4 deals with procedure of the experimental research along with its results. The conclusions are formulated in section 5. 2 Description of the Modified IWO Algorithm The modified version of the IWO algorithm provides the opportunity to experiment with different search space exploration strategies. The pseudocode mentioned below describes the algorithm by means of terminological convention consistent with the,,natural inspiration of its idea: Create the first population composed of n randomly generated individuals. For each individual { Compute the value of the fitness as the reciprocal of the minimized. } While the stop criterion is not satisfied { For each individual from the population { Compute the number of seeds depending on the value of the fitness. For each seed { Determine a place of fall of the seed choosing with the fixed probability one of the following methods: dispersing, spreading or rolling down. Create a new individual according to the randomly chosen method. Compute the value of the fitness for the new individual. } } Create a new population composed of n best adapted individuals taking into account members of the former population as well as new individuals. } The number of seeds S ind produced by a single individual depends on the value of its fitness f ind the greater the degree of individual s adaptation, the greater its reproduction ability according to the following formula: S ind = S min + (f ind f min ) S max S min, (1) f max f min
Modified IWO Algorithm 269 where S max, S min denote maximum and minimum admissible number of seeds generated, respectively, by the best population member (fitness f max )andby the worst one (fitness f min ). The character of the operation described as,,determine a place of fall of the seed differs depending on the method chosen randomly for its realization. Probability values of selection assigned to the particular methods: dispersing, spreading and rolling down form parameters of the algorithm. In case of dispersing the aforementioned operation computes the distance between the place where the seed falls on the ground and the parent individual (Fig. 1a). The distance is described by normal distribution with a mean equal to 0 and a standard deviation truncated to nonnegative values. The standard deviation is decreased in each algorithm iteration (i.e. for each population) and computed for the iteration iter, iter [1,iter max ] according to the following formula: ( ) m itermax iter σ iter = (σ init σ fin)+σ fin. (2) iter max The total number of iterations (iter max ), equivalent to the total number of populations, can be either used in form of the algorithm parameter with the purpose of determination of the stop criterion or can be estimated based on the stop criterion defined as the execution time limit. The symbols σ init, σ fin represent, respectively, initial and final values of the standard deviation, whereas m is a nonlinear modulation factor. According to the dispersing method construction of a new individual represented by a vector of a length equal to n, where element i, i [1,n] contains argument x i of the optimized n-dimensional, is based on the random generation of the values for all arguments x i. Those values determine the direction of the seed s,,fly. Because the seed has to fall on the ground at the determined distance from the parent individual, the values of arguments are scaled so that this condition is fulfilled. The spreading is a random disseminating seeds over the whole of the search space. Therefore, this operation is equivalent to the random construction of new individuals (Fig. 1b). The rolling down is basedonthe examinationofthe neighbourhoodofthe parent individual. In case of continuous optimization the term,,neighbours stands for individuals located at the same randomly generated distance from the considered one. The best adapted individual is chosen from among the determined number of neighbours, whereupon its neighbourhood is analyzed in search of the next best adapted individual. This procedure is repeated k times (k is a parameter of the method) giving the opportunity to select the best adapted individual found in the k-th iteration as a new one (Fig. 1c). 3 Characterization of the Optimized Functions According to [10] there are following classes of s used as benchmarks for numerical optimization problems:
270 D. Kostrzewa and H. Josiński Fig. 1. Idea of: a) dispersing b) spreading c) rolling down (k =3) 1. Unimodal, convex, multidimensional. 2. Multimodal, two-dimensional with a small number of local extremes. 3. Multimodal, two-dimensional with a huge number of local extremes. 4. Multimodal, multidimensional with a huge number of local extremes. Griewank and Rastrigin s belong to the 4. class. The classical Rosenbrock is a two-dimensional unimodal, whereas the n-dimensional (n =4 30) Rosenbrock has 2 minima [11]. The global minimum for all s is equal to 0. The formula defining the n-dimensional Griewank (Fig. 2a) is as follows: f (x) = 1 4000 n n x 2 i i=1 i=1 ( ) xi cos +1. (3) i The n-dimensional Rastrigin (Fig. 2b) is described by the following formula: n [ f (x) =10n + x 2 i 10 cos (2πx i ) ]. (4) i=1 The following formula defines the n-dimensional (n >1) Rosenbrock (Fig. 2c): n 1 [ f (x) = 100 ( x i+1 x 2 ) 2 i +(1 xi ) 2]. (5) i=1 4 Experimental Research The goal of the experiments was to compare the results (minima values) found by the modified IWO with the outcomes generated by other methods. As reference
Modified IWO Algorithm 271 Fig. 2. a) The Griewank b) the Rastrigin c) the Rosenbrock points served the results achieved from experiments with the original version of the IWO algorithm and those reported in [9] as minima found by the APSO method. For purpose of comparison the initial scope of the search space for particular s as well as other optimization parameters correspond with values proposed in the literature. The initial scopes given in Figures 5, 6, 7 are asymmetric according to the suggestion expressed in [9]. Values of the modified IWO parameters describing the hybrid strategy of the search space exploration were collected in Table 1. They were found during the research as the most appropriate values for the considered problem. The workstation used for experiments is described by the following parameters: Intel Core2 Quad Q6600 2.4GHz processor, RAM 2GB 800MHz. The number of Table 1. Modified IWO parameters describing the search space exploration strategy Description Griewank (n = 10) Griewank (n =20, 30) Rastrigin Rosenbrock Number k of neighbourhoods examined 1 1 1 1 during the rolling down Probability of the dispersing 0.7 0.3 0.8 0.1 Probability of the spreading 0.2 0.2 0 0.1 Probability of the rolling down 0.1 0.5 0.2 0.8
272 D. Kostrzewa and H. Josiński trial runs for each in the presence of a single parameters configuration of the optimization method was equal to 500. Minima of the 30-dimensional Rastrigin and Griewank s found by the original and modified versions of the IWO algorithm are presented in Figures 3, 4, respectively. The X values denote the optimization time. Fig. 3. Comparison between the original and modified IWO based on the Rastrigin Fig. 4. Comparison between the original and modified IWO based on the Griewank Minima of the n-dimensional Rastrigin, Rosenbrock, and Griewank s (n =10, 20, 30) found by the modified IWO algorithm and the APSO method are presented in Figures 5, 6, 7, respectively. The n value is strictly related to the number of algorithm iterations (respectively: 1000, 1500, 2000) used as a stop criterion. The X values denote the number of individuals.
Modified IWO Algorithm 273 Fig. 5. Comparison between the APSO and IWO algorithms based on the Rastrigin Fig. 6. Comparison between the APSO and IWO algorithms based on the Rosenbrock Fig. 7. Comparison between the APSO and IWO algorithms based on the Griewank
274 D. Kostrzewa and H. Josiński 5 Conclusion The experiments revealed the usefulness of the modified IWO for solving continuous optimization tasks. The method can compete with other algorithms, although it should be compared with some methods mentioned in the literature as successful ones (e.g. Artificial Bee Colony). The hybrid strategy of the search space exploration turned out to be more efficient than the dissemination used in the original IWO. However, the influence of the strategy components (dispersing, spreading, rolling down) on the solution found by the modified IWO requires further research. In the area of discrete optimization the modified IWO takes part at present in the World TSP Challenge (www.tsp.gatech.edu/world/index.html) and in the Mona Lisa TSP Challenge (www.tsp.gatech.edu/data/ml/monalisa.html). References 1. Mehrabian, R., Lucas, C.: A novel numerical optimization algorithm inspired from weed colonization. Ecological Informatics 1(4), 355 366 (2006) 2. Mallahzadeh, A.R., Oraizi, H., Davoodi-Rad, Z.: Application of the Invasive Weed Optimization Technique for Antenna Configurations. Progress in Electromagnetics Research, 137 150 (2008) 3. Sahraei-Ardakani, M., Roshanaei, M., Rahimi-Kian, A., Lucas, C.: A Study of Electricity Market Dynamics Using Invasive Weed Colonization Optimization. In: IEEE Symposium on Computational Intelligence and Games, pp. 276 282 (2008) 4. Sepehri Rad, H., Lucas, C.: A Recommender System based on Invasive Weed Optimization Algorithm. In: IEEE Congress on Evolutionary Computation, pp. 4297 4304 (2007) 5. Kostrzewa, D., Josiński, H.: Verification of the Search Space Exploration Strategy Based on the Solutions of the Join Ordering Problem. In: Man-Machine Interactions, AISC, pp. 447 455 (2011) 6. Kostrzewa, D., Josiński, H.: The Comparison of an Adapted Evolutionary Algorithm with the Invasive Weed Optimization Algorithm Based on the Problem of Predetermining the Progress of Distributed Data Merging Process. In: Man- Machine Interactions. AISC, pp. 505 514 (2009) 7. Kostrzewa, D., Josiński, H.: Application of the IWO Algorithm for the Travelling Salesman Problem. In: Proceedings of the 3rd KKNTPD, pp. 238 247 (2010) (in Polish) 8. Hossen, S., Rabbi, F., Rahman, M.: Adaptive Particle Swarm Optimization (APSO) for multimodal optimization. International Journal of Engineering and Technology 1(3), 98 103 (2009) 9. Xie, X.-F., Zhang, W.-J., Yang, Z.-L.: Adaptive Particle Swarm Optimization on Individual Level. In: International Conference on Signal Processing, pp. 1215 1218 (2002) 10. Molga, M., Smutnicki, C.: Test s for optimization needs (2005), http://www.zsd.ict.pwr.wroc.pl/files/docs/s.pdf 11. Shang, Y.-W., Huang-Qiu, Y.: A Note on the Extended Rosenbrock Function. Evolutionary Computation 14(1), 119 126 (2006)