Investigation and comparison of iteration curves of optimization algorithms
DOI:
https://doi.org/10.32972/dms.2023.020Keywords:
Iteration history curves, sigmoid curves, saturation curves, comparison of algorithms, group achievements.Abstract
The iteration history curve of optimization algorithms is a saturation- type development curve or sigmoid shape curve. After an overview of several different sigmoid curves, the iteration history curve of the RVA (Random Virus Algorithm) is analysed in order to find its best settings for a given optimization problem. The analysis of the characteristics and numerical parameters of the iteration history curve provides the possibility to discover the speed and efficiency of the algorithm without the necessity to wait throughout the whole running until its final result, which can speed up numerical experiments during the search for the solution to the optimization problem and while ‘fine tuning’ the algorithm to the given task. Since sigmoid-type curves can be found in many different fields of life (the history of the sport world records, comparison of the achievements of several groups), the results of this analysis can be used in several different domains of life, when the ranking, comparison, evaluation or qualification of several individuals or groups is important.
References
Abraham, A., Hassanien, A.-E., Siarry, P., & Engelbrecht, A. (2009). Foundations of Computa-tional Intelligence (Vol. 3.). Springer.
Andrews, L. (1998). Special Functions of Mathematics for Engineers. Bellingham, USA: SPIE Optical Engineering Press.
Bihari, J., & Sarka, F. (2018). Human-electric hybrid drives in medium-sized cities by daily traffic. In K. Jármai, & B. Bolló (Ed.), Vehicle and Automotive Engineering 2. VAE 2018. Lecture Notes in Mechanical Engineering. Springer. https://doi.org/10.1007/978-3-319-75677-6_5
Cramer , E., Dennis, Jr., J., Frank, P., Lewis, R., & Shubin, G. (1994). Problem Formulation for Multidisciplinary Optimization. SIAM Journal on Optimization, 4(4), 754-776. https://doi.org/10.1137/0804044
Das, S., Dasgupta, S., Biswas, A., Abraham, A., & Konar, A. (2009). On Stability of the Chemotactic Dynamics in Bacterial-Foraging Optimization Algorithm. IEEE Transactions on Systems, Man, and Cybernetics, Part A: Systems and Humans, 39(3), 670-679. https://doi.org/10.1109/TSMCA.2008.2011474
Deb, K. (2007). Current trends in evolutionary multi-objective optimization. International Journal for Simulation and Multidisciplinary Design Optimization, 1(1), 1-8. https://doi.org/10.1051/ijsmdo:2007001
Eberhart, R., & Kennedy, J. (1995). A new optimizer using particle swarm theory. Proceedings of the Sixth International Symposium on Micro Machine and Human Science, pp. 39-43. Nagoya, Japan: IEEE. https://doi.org/10.1109/MHS.1995.494215
Fisher, J., & Pry, R. (1971). A Simple Substitution Model of Technological Change. Technological Forecasting and Social Change, 3, 75-88. https://doi.org/10.1016/S0040-1625(71)80005-7
Fogel, L. (1999.). Intelligence through Simulated Evolution: Forty Years of Evolutionary Programming (1. ed.). Chichester,: John Wiley,.
Fokasz, N. (2006). Növekedési függvények, társadalmi diffúzió, társadalmi változás. Szociológiai szemle, 16(3).
Gao, F., Hongwei, L., Zhao, Q., & Cui, G. (2006). Virus-Evolutionary Particle Swarm Optimization Algorithm. In L. Jiao, L. Wang, F. Wu, X. Gao, & J. Liu (Ed.), ICNC 2006. II., pp. 156-165. Berlin Heidelberg: Springer. https://doi.org/10.1007/11881223_20
Goldberg, D. (1989). Genetic Algorithms in Search, Optimization and Machine Learning (1. ed.). Massachusetts, USA: Addison-Wesley Longman Publishing.
Herskovits, J., Mappa, P., Goulart, E., & Mota Soares, C. (2005). Mathematical Programming Models and Algorithms for Engineering Design Optimization. Computer Methods in Applied Mechanics and Engineering, 194(30-33), 3244-3268. https://doi.org/10.1016/j.cma.2004.12.017
Jang, S., Dai, S., & Sung, S. (2005). The pattern and Externality Effect of Diffusion of Mobile Telecommunications: the Case of OECD and Taiwan. Information Economics and Policy, 17(2), 133-148. https://doi.org/10.1016/j.infoecopol.2004.05.001
Kozuko, F., & Bajzer, Z. (2003). Combining Gompertzian Growth and CellPopulation Dynamics 185, 153- 167. Mathematical Biosciences, 185(2), 153-167. https://doi.org/10.1016/S0025-5564(03)00094-4
Lorentz, M. (1905). Methods of Measuring the Concentration of Wealth. Publications of the American Statistical Association, 9(70), 209-219. https://doi.org/10.2307/2276207
Malthus, T. (1798). An Essay on the Principle of Population. London.
Mansfield, E. (1961). Technical Change and the Rate of Imitation. Econometrica, 29(4), 741-766. https://doi.org/10.2307/1911817
Martens, D., De Backer, M., Haesen, R., Vanthienen, J., Snoeck, M., & Baesens, B.(2007). Classification with Ant Colony Optimization. IEEE Transactions on Evolutionary Computation, 11(5), 651-665. https://doi.org/10.1109/TEVC.2006.890229
Martins, J., & Lambe, A. (2013). Multidisciplinary design optimization: A Survey of architectures. AIAA Journal, 51(9), 2049-2075. https://doi.org/10.2514/1.J051895
Meyer, W. B., & Turner II, B. L. (1994). Changes in Land Use and Land Cover: A Global Perspective. Vol. 4,. Cambridge: Cambridge University Press.
Mitscherlich, E. (1909). Das gesetz des minimums und das gesetz des abnehmenden bodenertrages. Landwirtsch Jahrbuch, 38, 537-552.
Moore, E. (1965). Cramming more Components onto Integrated Circuits. 38 (8):pp.114- 117. Electronics, 38(8), 114-117.
Nelder, J., & Mead, R. (1965). A Simplex Method for Function Minimization. The Computer Journal, 7(4), 308-313. https://doi.org/10.1093/comjnl/7.4.308
Pang, X., Chen, J., Wang, J., & Hou, Y. (2012). Parametric and Controllable Shape Model of the Water-Lubricated Rubber Journal Bearing. Advanced Materials Research, 455-456, 1468-1473. https://doi.org/10.4028/www.scientific.net/AMR.455-456.1468
Pearl, R., & Reed, L. (1920). On the rate of growth of the population of the United States since 1790 and its mathematical representation. Proceedings of the National Academy of Sciences, 6(6), 275-288. doi:https://doi.org/10.1073/pnas.6.6.275
Rézsó, F. (2020). Szigmoid görbék alkalmazása tanulócsoportok eredményeinek vizsgálatához. Multidiszciplináris Tudományok, 10(3), 195-211. https://doi.org/10.35925/j.multi.2020.3.25
Richards, F. (1959). A Flexible Growth Function for Empirical Use. Journal of Experimental Botany, 10(2), 290-300. https://doi.org/10.1093/jxb/10.2.290
Rogers, M. (1962). Diffusion of Innovations (3. ed.). New York, USA: The Free Press, Macmillan Publishing Co.
Rosenbrock, H. (1960). An automatic method for finding the greatest or least value of a function. The Computer Journal, 3(3), 175-184. https://doi.org/10.1093/comjnl/3.3.175
Sheel, A. (1985). Betrag zur Theorie der Evolutionsstrategie. Dissertation. Berlin:TU Berlin.
Silverberg, G., & Lehnert, D. (2003). Evolúciós káosz: növekedés és fluktuációk az "alkotó rombolás" Schumpeter-féle modelljében. In N. Fokasz, Káosz és nemlineáris dinamika a társadalomtudományokban. Typotex.
Szabó, F. (2008). Multidisciplinary optimization of a structure with temperature dependent material characteristics, subjected to impact loading. Internationa Review of Mechanical Engineering, 2(3), 499-505.
Szabó, F. (2011). Analógia a sport-világcsúcsok története és az evolúciós optimáló algoritmusok iteráció-története között. GÉP, 62(9-10), 28-31.
Szabó, F. (2016). Multidisciplinary Optimization of Journal Bearings, using a RVA evolutionary type optimization algorithm. Acta Polytechnica Hungarica, 13(7), 181-195. https://doi.org/10.12700/APH.13.7.2016.7.10
Szabó, F. (2017). Evolutionary based system for qualification and evaluation of group achievments (EBSYQ). International Journal of Current Research, 9(8), 55507-55516.
Szabó, F. (2018). Optimumkereső algoritmusok iterációtörténetének vizsgálata. GÉP, 69(4), 82-85.
Szabó, F. (2019). Application of sigmoid curves in environmental protection. In K. Tóthné Szita, K. Jármai, & K. Voith (Eds.), Solutions for Sustainable Development (1. ed., p. 7). London: CRC Press.
Szabó, F. (2020). A COVID-19 járvány időbeli alakulásának vizsgálata szigmoid görbékkel. Multidiszciplináris Tudományok, 10(3), 294-306. https://doi.org/10.35925/j.multi.2020.3.35
Szabó, F. (2021). Analysis of Wear Curves as Sigmoid Functions. In K. Jármai, & K. Voith (Ed.), Vehicle and Automotive Engineering 3. VAE 2020. Lecture Notes in Mechanical Engineering (pp. 273-281). Singapore: Springer. https://doi.org/10.1007/978-981-15-9529-5_24
Törnquist, L. (1936, 10). The Bank of Finland's Consumption Price Index. Bank of Finland Monthly, 1-8.
Törnquist, L. (1981). Collected Scientific Papers of Leo Törnquist. Research Institute of the Finnish Economy.
Vajna, S. (2020). Integrated Design Engineering. Interdisciplinary and holistic product development (1. ed.). Switzerland: Springer Nature.
Vanderplaats, G. (2007). Multidiscipline Design Optimization. Vanderplaats R&D, Inc.
Verhulst, P.-F. (1847). Deuxieme mémoire sur la loi d’accroissement de la population. Mémoires de l’Académie Royale des Sciences, des Lettres et des Beaux-Arts de Belgique, 20, 1-32.
Von Bertalanffy, L. (1960). Principles and theory of growth. In W. Nowinski, Fundamental Aspects of Normal and Malignant Growth (pp. 137-259). Amsterdam: Elsevier.
Zhang, Z., Zhou, J., Zhou, N., Wang, X., & Zhang, L. (2005). Shape Optimization Using Reproducing Kernel Practice Method and an Enriched Genetic Algorithm. Computer Methods in Applied Mechanics and Engineering,, 194(39-41), 4048-4070. https://doi.org/10.1016/j.cma.2004.10.004
