Filled Fonksiyonunda Yeni Bir Stokastik Arama Yöntemi

Abstract

Bu çalışmada, yeni bir stokastik arama yaklaşımı, klasik filled fonksiyon arama stratejisine daha hızlı ve daha verimli bir alternatif olarak sunulmuştur. Deterministik bir yöntem olan L tipi filled fonksiyonunu hızlandırmak için stokastik bir yöntem olan kümeleme ve parabolik yaklaşım tabanlı kısıtsız global optimizasyon yöntemi (GOBC-PA) kullanılmıştır. Filled fonksiyonun havza bölgelerinin aranması GOBC-PA tarafından gerçekleştirilmiştir. Bu çalışmada kullanılan yöntemler popülerlikleri, hızları ve gürbüzlükleri nedeniyle tercih edilmişlerdir. Stokastik yöntemin amaç fonksiyonunu, havza bölgesinin yerini belirleyen gradyanın epsilon değeri oluşturmaktadır. Bu nedenle, stokastik yöntemin tüm amacı global optimumu bulmak değil, havza bölgesini bulmaktır. Global minimumun bulunma rolü deterministik yönteme bırakılmıştır. Geliştirilen yöntem, 11 kıyaslama fonksiyonu kullanılarak klasik filled fonksiyona karşı test edilip bu işlem 10 kez tekrarlanmıştır. Elde edilen sonuçlar incelendiğinde, stokastik arama yaklaşımının ortalama hata, standart sapma ve geçen süre değerlerinde klasik yaklaşıma göre üstünlüğü görülmektedir. Bu sonuçlar, deterministik ve stokastik yöntemlerin kombinasyonunun, klasik deterministik yönteme karşı küresel minimumun bulunmasında daha başarılı olabileceğini göstermektedir.

Keywords

• Stokastik arama,GOBC-PA,L tipi filled fonksiyonu,global optimizasyon

A New Stochastic Search Method for Filled Function

Abstract

In this study, a new stochastic search approach is presented as a faster and more efficient alternative to classic filled function search strategy. An unconstrained global optimization method based on clustering and parabolic approximation (GOBC-PA) has been used as a stochastic method for accelerating the L type filled function as a deterministic method. Searching the basin regions of the filled function is performed by GOBC-PA. The methods used in this study are preferred due to their popularity, speed and robustness. The objective function of the stochastic method is the epsilon value of the gradient that gives the location of basin region. Therefore, the whole purpose of the stochastic method is not to find the global optimum but to find the basin region. The role of finding the global minimum has been left to the deterministic method. The developed method has been tested against classical filled function using 11 benchmark functions and process repeated 10 times. When the obtained results are examined, it is seen that the stochastic search approach has superiority over the mean error, standard deviation and elapsed time values according to the classical approach. These results show that the combination of deterministic and stochastic methods can be more successful in finding the global minimum against the classic deterministic method.

Keywords

• Stochastic search,GOBC-PA,L type filled function,global optimization

References

1. Marseglia, G.R., Scott, J.K., Magni, L., Braatz, R.D., Raimondo, D.M., “A hybrid stochastic-deterministic approach for active fault diagnosis using scenario optimization”, in Proceedings of the 19th World Congress on the International Federation of Automatic Control, Cape Town, South Africa, August 24-29, 1102-1107, (2014).
2. Ma, S., Yang, Y., Liu, H., “A parameter free filled function for unconstrained global optimization”, Appl Math Comput, 2010, 215 (10): 3610-3619.
3. Branin, F.H., “Widely convergent method for finding multiple solutions of simultaneous nonlinear equations”, IBM J Res Dev, 1972, 16 (5): 504-522.
4. Levy, A.V., Montalvo, A., “The tunneling algorithm for the global minimization of functions”, SIAM J Sci Stat Comp, 1985, 6 (1): 15-29.
5. Basso, P., “Iterative methods for the localization of the global maximum”, SIAM J Numer Anal, 1982, 19 (4): 781-792.
6. Dang, C., Ma, W., Liang, J., “A deterministic annealing algorithm for approximating a solution of the min-bisection problem”, Neural Networks, 2009, 22 (1): 58-66.
7. Zaki, M.R., Jaleh, V., Milad, F., “Preparation of agar nanospheres: comparison of response surface and artificial neural network modeling by a genetic algorithm approach”, Carbohyd Polym, 2015, 122: 314-320.
8. Karaboğa, D., “An idea based on honey bee swarm for numerical optimization”, Erciyes University Engineering Faculty Computer Engineering Department (Technical Report,TR06), Kayseri, (2005).

9. Kennedy, J., Eberhart, R., “Particle swarm optimization”, in IEEE 1995 International Conference on Neural Networks, Piscataway, November 27-December 1, 1942-1948, (1995).
10. Pence, I., Cesmeli, M.S., Senel, F.A., Cetisli, B., “A new unconstrained global optimization method based on clustering and parabolic approximation”, Expert Syst Appl, 2016, 55: 493-507.
11. Wei, F., Wang, Y., Lin, H., “A new filled function method with two parameters for global optimization”, J Optimiz Theory App, 2014, 163 (2): 510-527.
12. Kim, W., Lee, K.M., “A hybrid approach for MRF optimization problems: combination of stochastic sampling and deterministic algorithms”, Comput Vis Image Und, 2011, 115 (12): 1623-1637.
13. Tsitsiklis, J., Bertsekas, D., Athans, M., “Distributed asynchronous deterministic and stochastic gradient optimization algorithms”, IEEE T Automat Contr, 1986, 31 (9): 803-812.
14. Vasudeva, K., Bhalla, U.S., “Adaptive stochastic-deterministic chemical kinetic simulations”, Bioinformatics, 2004, 20 (1): 78-84.
15. Balsa-Canto, E., Alonso, A.A., Banga, J.R., “Dynamic optimization of bioprocesses: deterministic and stochastic strategies”, in International Symposium on Automatic Control of Food and Biological Processes (ACoFop IV), Göteborg, Sweden, September 21-23, (1998).
16. Carrasco, E.F., Banga, J.R., “A hybrid method for the optimal control of chemical processes”, in IEEE 1998 Conference Publication of Institution of Electrical Engineers, Swansea, UK, September 1-4, 925-930, (1998).
17. Balsa-Canto, E., Vassiliadis, V.S., Banga, J.R., “Dynamic optimization of single-and multi-stage systems using a hybrid stochastic-deterministic method”, Ind Eng Chem Res, 2005, 44 (5): 1514-1523.
18. Hendriks, R.C., Heusdens, R., Jensen, J., “An MMSE estimator for speech enhancement under a combined stochastic–deterministic speech model”, IEEE T Audio Speech, 2007, 15 (2): 406-415.
19. Arjona, M.A., Cisneros-González, M., Hernandez, C., “Parameter estimation of a synchronous generator using a sine cardinal perturbation and mixed stochastic–deterministic algorithms”, IEEE T Ind Electron, 2011, 58 (2): 486-493.
20. Cottereau, R., Clouteau, D., Dhia, H.B., Zaccardi, C., “A stochastic-deterministic coupling method for continuum mechanics”, Comput Method Appl M, 2011, 200 (47): 3280-3288.
21. Alotto, P., “A hybrid multiobjective differential evolution method for electromagnetic device optimization”, Compel, 2011, 30 (6): 1815-1828.
22. Scott, J.K., Marseglia, G.R., Magni, L., Braatz, R.D., Raimondo, D.M., “A hybrid stochastic deterministic input design method for active fault diagnosis”, in IEEE 2013 Conference on Decision and Control, Florence, Italy, December 10-13, 5656-5661, (2013).
23. Andretta, M., Birgin, E.G., “Deterministic and stochastic global optimization techniques for planar covering with ellipses problems”, Eur J Oper Res, 2013, 224 (1): 23-40.
24. Chen, C.T., Peng, S.T., Chen, C.L., “A hybrid global optimization scheme that combines deterministic and stochastic techniques for process design”, J Chin Inst Eng, 2013, 36 (6): 787-796.
25. Collins, R.T., Carr, P., “Hybrid stochastic/deterministic optimization for tracking sports players and pedestrians”, in European Conference on Computer Vision, Zurich, Switzerland, September 6-12, 298-313, (2014).
26. Custódio, A.L., Madeira, J.A., Vaz, A.I.F., Vicente, L.N., “Direct multisearch for multiobjective optimization”, SIAM J Optimiz, 2011, 21 (3): 1109-1140.
27. Alotto, P., Capasso, G., “A deterministic multiobjective optimizer”, Compel, 34 (5): 1351-1363.
28. Alb, M., Alotto, P., Capasso, G., Guarnieri, M., Magele, C., Renhart, W., “Real-Time pose detection for magnetic-assisted medical applications by means of a hybrid deterministic/stochastic optimization method”, IEEE T Magn, 2016, 52 (3): 1-4.
29. Ho, C.Y.F., Ling, B.W.K., “Design of multi-layer perceptrons via joint filled function and genetic algorithm approach for video forensics”, in IEEE 2016 International Conference in Consumer Electronics, Guangzhou, China, December 19-21, 1-4, (2016).
30. Tan, W.S., Shaaban, M., “A hybrid stochastic/deterministic unit commitment based on projected disjunctive milp reformulation”, IEEE T Power Syst, 2016, 31 (6): 5200-5201.
31. Külahcioğlu, T., Bulut, H., “On scalable RDFS reasoning using a hybrid approach”, Turk J Elec Eng & Comp Sci, 2016, 24 (3): 1208-1222.
32. Megel, O., Mathieu, J.L., Andersson, G., “Hybrid stochastic-deterministic multi-period DC optimal power flow”, IEEE T Power Syst, 2017, 32 (5): 3934-3945.
33. Inclan, E., Geohegan, D., Yoon, M., “A hybrid optimization algorithm to explore atomic configurations of TiO 2 nanoparticles”, Comp Mater Sci, 2018, 141: 1-9.
34. Santos, R., Borges, G., Santos, A., Silva, M., Sales, C., Costa, J.C., “A semi-autonomous particle swarm optimizer based on gradient information and diversity control for global optimization”, Appl Soft Comput, 2018, 69: 330-343.
35. Ge, R.P., Qin, Y.F., “A class of filled functions for finding global minimizers of a function of several variables”, J Optimiz Theory App, 1987, 54 (2): 241-252.
36. Renpu, G.E., “A filled function method for finding a global minimizer of a function of several variables”, Math Program, 1990, 46 (1-3): 191-204.
37. Wu, Z.Y., Bai, F.S., Lee, H.W.J., Yang, Y.J., “A filled function method for constrained global optimization”, J Global Optim, 2007, 39 (4): 495-507.
38. Şahiner, A., Yilmaz, N., Demirozer, O., “Mathematical modeling and an application of the filled function method in entomology”, Int J Pest Manage, 2014, 60 (3): 232-237.
39. Liu, X., “A computable filled function used for global minimization”, Appl Math Comput, 2002, 126 (2-3): 271-278.