Innovative Strategy of SOMA Control Parameter Setting

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1 Innovative Strategy of SOMA Control Parameter Setting PAVEL VAŘACHA Tomas Bata University in Zlin Faculty of Applied Informatics nam. T.G. Masaryka 5555, 76 1 Zlin CZECH REPUBLIC varacha@fai.utb.cz Abstract: - SOMA - Self-Organizing Migration Algorithm is highly effective tool of evolutionary optimization as already proven on many real life problems. This article introduces new innovative strategy how to set one of the most important SOMA control parameter, which makes the algorithm even more efficient E6 single evaluations have been calculated for 1 different test functions to statistically prove impact of this improvement. Application of the strategy proposed by this article can bring better results in about 5% of optimized functions and possibly even achieve breakthrough in cases, which standard SOMA setting does not perform well. Key-Words: - SOMA, optimization, evolutionary algorithm,, Vector 1 Introduction In recent years, a broad class of algorithms has been developed for stochastic optimization, i.e. for optimizing systems where the functional relationship between the independent input variables x and output (objective function) y of a system S is not known. Using stochastic optimization algorithms such as Genetic Algorithms, Simulated Annealing, Differential Evolution and Particle Swarm Optimization, a system is confronted with the random input vector and its response is measured. This response is then used by the algorithm to tune the input vector in such a way that the system produces the desire output or target value in an iterative process. [1] SOMA Self-Organizing Migration Algorithm, is based on the self-organizing behavior of groups of individuals in a social environment. It can also be classified as an evolutionary algorithm, despite the fact that no new generations of individuals are created during the search (based on philosophy of this algorithm). Only the positions of the individuals in the search space are changed during a generation, called a migration loop (ML). The algorithm was published in journals, book and presented at international conferences, symposiums, as well as in various invitational presentations, for example [2, 3, 4]. A disadvantage of SOMA, as well as of others evolutional algorithms, is that it has slight dependence on the control parameter setting. During various tests was found that SOMA is even more sensitive on the parameter setting than others algorithms. Although there was found the setting, which is almost universal and perform well in most simulations and experiments [5], this article introduces breakthrough strategy on how to set SOMA control parameters to achieve even better results. 2 SOMA All-to-One Several different versions of SOMA exist, nevertheless, this article is aimed on most common All-to-One version. All basic All-to-One SOMA principles important for correct understanding of executed experiment are described below. 1) Parameter definition Before starting the algorithm, SOMA s parameters: Step, PathLength, PopSize, and the Cost Function needs to be defined. The Cost Function is simply the function which returns a scalar that can directly serve as a measure of fitness. 2) Creation of Population Population of individuals is randomly generated. Each parameter for each individual has to be chosen randomly from the given range <Low, High>

3) Migration loop Each individual from population (PopSize) is evaluated by Cost Function and the Leader (individual with the highest fitness) is chosen for the current migration loop.

2 3) Migration loop Each individual from population (PopSize) is evaluated by Cost Function and the Leader (individual with the highest fitness) is chosen for the current migration loop. Then all other individuals begin to jump, (according to the Step definition) towards the Leader. Each individual is evaluated after each jump using the Cost Function. The jumping continues, until a new position defined by the PathLength has been reached. The new position x i,j after each jump is calculated by (1). This is shown graphically in Fig. 1. The individual returns then to that position where it found the best fitness on its trajectory. (1) 4) Test for stopping condition If the maximum number of migration loops has been reached, stop and recall the best solution(s) found during the search. where t <, by Step to, PathLegth> and ML is actual migration loop Before an individual begins jumping towards the Leader, a random number rnd is generated (for each individual s component), and then compared with. If the generated random number is larger than, then the associated component of the individual is set to by means of the Vector if rnd j < then Vector j = else 1 where rnd <, 1> and j = 1, n param j rnd j Vector ,545 3,865 4,12 1 (2) Table 1, An example of Vector for 4 parameters individual with =.3 Hence, the individual moves in the N-k dimensional subspace, which is perpendicular to the original space. This fact establishes a higher robustness of the algorithm. Earlier experiments have demonstrated that, without the use of, SOMA tends to determine a local optimum rather than the global one. [5] Fig. 1, Vector and its action on individual movement [5] 3 SOMA recommended settings Based on huge number of experiments author of SOMA (prof. Zelinka) recommended optimal setting for the algorithm s control parameters. Parameter name Recommended range PathLenght <1.1 ;3> Step <.11, PathLength> <> PopSize <1, up to user> Table 2, SOMA parameters and their recommended domain As can been seen on Fig. 2, published in [5], parameter was tested within the range <.1;.9> and performs the best for <.1;.3>. By contrast, this article explores SOMA s behavior for <.5,.1>. Reasons why this possibility has never been explored before are described in the next chapter

3 Fig. 2, SOMA dependence on size 4 Null Vector problem definition All the experiments mentioned in chapter 3 (see also chapter 5) were performed on Cost Functions with 1 parameters. Naturally Vector s length (L) was also 1. Probability P o that generated Vector is a null vector (vector which contains nulls only, see also (2)) is very low for <.1;.3>. (3) P = (1 ) L P,5,6577,1 6632,3,47553,5,5921,7,75 2,66E-5 2,4E-1 3,23E-16 Table 3, Probability of null Vector for L = 1 P,6,5 Fig 3, Probability of null Vector for L = 1 However P increase dramatically if L or value decreases. P,5,886654,1,785678, , , ,79766,4722,192 Table 4, Probability of null Vector for L = 25 P,6,5 Fig 4, Probability of null Vector for L = 25 If null Vector is generated, an individual does not move during actual migration loop and Cost Function is always evaluated with the very same parameters. For example 27 evaluations are wasted if Step =.11 and PathLength = 3. This waste of computation time is very improbable for L = 1 and also very low if theoretical test functions (see chapter 5) are computed. Consider real life problem of heating-plant parameters optimization. [6] L = 24, one parameter for every hour during the day has to be optimized. If =.1, P =.79, almost 8% of Cost Function evaluations are wasted. In doing so one evaluation of Cost Function is very time demanding (even in range of minutes [7]) as waste database has to be processed. Such conditions approve institution of simple null Vector repair mechanism: (4) If Vector is null vector, new Vector is generated instead. Consequently P is always. Instead of P, probability P 1 of Vector which contains 1 only ones can be considered. (5) P 1 = (1 ) L (L - 1) + L * * (1 - )

4 (7) P 1 1,8,6 P 1,5,91178,1,735762, ,5,3781,7,613,322 5,3E-9 1,42E-14 Table 5, P 1 for L = 1 Fig 5, P 1 for L = 1 (8) (9) (1) (11) (12) (13) (14) Application of (4) into SOMA allows parameter to be set within the range (;.1> which was previously unreachable due to high values of P. Following experiment is designed to explore SOMA efficiency for (;.1> and compare it with results obtained for <.1;.3>. In other words the experiment measures dependence of P 1 on SOMA behavior. 5 Experiment set up Ten different test functions were chosen for the experiments. All these functions as well as other SOMA control parameter settings were based on [5] and used in the same way as prof. Zelinka used them for initial testing of SOMA. PopSize = 6, PathLength = 3, Step =.11 and number of parameters = 1 are constant for all these functions. Number of migration loops and borders of the function s parameters vary in accordance with table 6. (6) Function ML Low Hight Ackley (6) EggHolder (7) Griewangk (8) Masters (9) Michalewicz (1) 2 3,1415 Rana (11) Rastrigin (12) 4-5,12 5,12 Rosenbrock (13) 125-2,48-2,48 Schwefel (14) SineWave (15) Table 6, Test functions, ML and borders (15) 6 Results For every test function, optimization (search for global minimum) via SOMA was repeated 1 times for different = {.5,.1,.3,.5,

5 .7,.1,.2,.3}. Overall 8 repetitions were made (test functions * variants * 1). Table 7 shows obtained average results. In total * 1 6 evaluations of Cost Function were computed (Round(PathLength/Step) * ML * PopSize * 1 * test functions). (4) was applied in all cases. Final results were normalized in Table 8. Best case for given test function is set as (base) and all other cases are expressed as percent divergence. Fig. 6 and 7 graphically show values from table 8 describing SOMA behavior based on various test functions and settings.,5,1,3,5,7 Ackley (6) 3895, , , , , , , ,19 EggHolder (7) , ,3-5893,7-5881,4-5565, , , Griewangk (8) 25, , , ,191591,972961,872625, , Masters (9) -55, , ,631-77, , ,4593 Michalewicz (1) -67, , , , ,976-97, ,88-89,7736 Rana (11) , ,3-2733, , , ,1 Rastrigin (12) Rosenbrock (13) 1584,54 797, , , ,84 335, ,733 Schwefel (14) -3561,2-4778, , , ,1-4531, SineWave (15) -69, ,28-614, , , , , ,679 Table 7, Average results for different cost functions and settings,5,1,3,5,7 Ackley (6) 5711,39351,1247,1243,392,14783,4943 EggHolder (7) 6321,9597,9232, Griewangk (8) 27,8232 8, Masters (9) ,9544,95872,7875, Michalewicz (1) ,1485,1549,9949,49385,92976 Rana (11) ,1778 Rastrigin (12) 3775,38366,324,181, Rosenbrock (13) 59, ,3784 4, ,54253, Schwefel (14) 62988,2657 1,16E-5,1753, SineWave (15),38291,3992, Table 8, Average results for different cost functions and settings Norm. average 5 5,5 Ackley Michalewicz Rastrigin Schwefel SineWave Fig 6, Test functions performing better for <.5;.7>

6 Norm. average 5 5 5,5 EggHolder Griewangk Masters Rana Rosenbrock Fig 7, Test functions performing better for <.1;.3> 7 Conclusion From 1 test functions considered in the experiment 5 (see fig. 6) proved better results for <.5;.7> and other 5 (see fig. 7) for <.1;.3>. This conclusion represents significant breakthrough in setting strategy. Formally recommended range <.1;.3> can be extended to <.1;.3>, furthermore around 5% of functions can be optimized by SOMA more effectively if <.1;.7>. Increasing value of P 1 can positively influence obtained results. However SOMA efficiency always decline if P 1 >.74. Based on this conclusion final recommendation can be formulated: If SOMA does not perform satisfactory for <.1;.3>, apply (4) and lower control parameter to obtain higher P 1. This arrangement has very good probability to bring better results. 8 Acknowledgment The work was performed with financial support of research project NPVII-2C67, by the Ministry of Education of the Czech Republic. References: [1] E. Král, V. Dolinay, L. Vašek, P. Vařacha, Usage of PSO Algorithm for Parameters Identification of District Heating Network Simulation Model. In 14th WSEAS International Conference on Systems. Latest Trands on Systems.Volume II, Rhodes, WSEAS Press (GR), 21. p ISBN/ISSN: [2] M. Červenka, I. Zelinka, Application of Evolutionary Algorithm on Aerodynamic Wing Optimisation. In Proceedings of the 2nd European Computing Conference, Venice, WSEAS Press (IT), 28, ISBN/ISSN: [3] Z. Oplatková, I. Zelinka, Investigation on Shannon - Kotelnik Theorem Impact on SOMA Algorithm Performance. In European Simulation Multiconference, 25, Riga, ESM, 25. p ISBN/ISSN: [4] R. Šenkeřík, I. Zelinka, Optimization and Evolutionary Control of Chemical Reactor. In 1th International Research/Expert Conference Trends in the Development of Machinery and Associated Technology, TMT, Zenica, Bosna and Hercegovina, 26, p ISBN/ISSN: [5] I. Zelinka, Studies in Fuzziness and Soft Computing, New York : Springer-Verlag, 24. [6] P. Vařacha, Impact of Weather Inputs on Heating Plant - Agglomeration Modeling. In Proceedings of the 1th WSEAS Ing. Conf. on Neural Networks, Athens, WSEAS World Science and Engineering Academy and Science, 29. p ISBN/ISSN: [7] B. Chramcov, Forecast of heat demand according the Box-Jenkins methodology for specific locality. In Latest Trends on Systems (Volume I), Rhodes, WSEAS Press (GR), 21, p , ISBN/ISSN:

A Computation Time Comparison of Self-Organising Migrating Algorithm in Java and C#

A Computation Time Comparison of Self-Organising Migrating Algorithm in Java and C# JAN KOLEK 1, PAVEL VAŘACHA 2, ROMAN JAŠEK 3 Tomas Bata University in Zlin Faculty of Applied Informatics nam. T.G..Masaryka