# A new real-coded Bayesian optimization algorithm based on a team of learning automata for continuous optimization

## Abstract

Estimation of distribution algorithms have evolved as a technique for estimating population distribution in evolutionary algorithms. They estimate the distribution of the candidate solutions and then sample the next generation from the estimated distribution. Bayesian optimization algorithm is an estimation of distribution algorithm, which uses a Bayesian network to estimate the distribution of candidate solutions and then generates the next generation by sampling from the constructed network. The experimental results show that the Bayesian optimization algorithms are capable of identifying correct linkage between the variables of optimization problems. Since the problem of finding the optimal Bayesian network belongs to the class of NP-hard problems, typically Bayesian optimization algorithms use greedy algorithms to build the Bayesian network. This paper proposes a new real-coded Bayesian optimization algorithm for solving continuous optimization problems that uses a team of learning automata to build the Bayesian network. This team of learning automata tries to learn the optimal Bayesian network structure during the execution of the algorithm. The use of learning automaton leads to an algorithm with lower computation time for building the Bayesian network. The experimental results reported here show the preference of the proposed algorithm on both uni-modal and multi-modal optimization problems.

### Keywords

Estimation of distribution algorithms Bayesian optimization algorithm Learning automata### References

- 1.D.E. Goldberg,
*Genetic Algorithms in Search, Optimization and Machine Learning*(Addison-Wesley, Reading, 1989)MATHGoogle Scholar - 2.D.E. Goldberg, K. Sastry,
*Genetic Algorithms: The Design of Innovation*, 2nd edn. (Springer, Berlin, 2002)Google Scholar - 3.J.A. Lozano, P. Larranaga, I. Inza, E. Bengoetxea (eds.),
*Towards a New Evolutionary Computation: Advances on Estimation of Distribution Algorithms*(Springer, Berlin, 2006)Google Scholar - 4.M. Pelikan, K. Sastry, E. Cantu-Paz (eds.),
*Scalable Optimization Via Probabilistic Modeling: From Algorithms to Applications*(Springer, Berlin, 2006)Google Scholar - 5.M. Pelikan,
*Bayesian Optimization Algorithm: From Single Level to Hierarchy*, Ph.D. Dissertation (University of Illinois at Urbana-Champaign, Urbana, 2002)Google Scholar - 6.G. Harik,
*Linkage Learning in Via Probabilistic Modeling in the ECGA*(University of Illinois at Urbana-Champaign, Urbana, 1999)Google Scholar - 7.H. Muhlenbein, T. Mahnig, FDA—a scalable evolutionary algorithm for the optimization of additively decomposed functions. Evol. Comput.
**7**(4), 353–376 (1999)CrossRefGoogle Scholar - 8.M. Pelikan,
*BOA: The Bayesian Optimization Algorithm*(Springer, Berlin, 2005)Google Scholar - 9.P. Larranaga, R. Etxeberria, J.A. Lozano, J.M. Pena,
*Proceedings of the 2000 Genetic and Evolutionary Computation Conference Workshop Program*. Optimization in continuous domains by learning and simulation of Gaussian networks (2000), pp. 201–204Google Scholar - 10.J. Ocenasek, J. Schwarz, in
*Proceedings of the 2nd Euro-International Symposium on Computational Intelligence*. Estimation of distribution algorithm for mixed continuous discrete optimization problems (2002), pp. 227–232Google Scholar - 11.P.A.N. Bosman,
*Design and Application of Iterated Density-Estimation Evolutionary Algorithms*, Ph.D. Dissertation (Utrecht University, Utrecht, 2003)Google Scholar - 12.C.W. Ahn, R.S. Ramakrishna, D.E. Goldberg,
*Real-Coded Bayesian Optimization Algorithm: Bringing the Strength of BOA into the Continuous World,*Lecture Notes in Computer Science, vol. 3102 (Springer, Berlin, 2004), pp. 840–851Google Scholar - 13.X. Wei, In
*Proceedings of the IEEE 10th International Conference of Signal Processing (ICSP)*. Evolutionary continuous optimization by Bayesian networks and Gaussian mixture model (2010), pp. 1437–1440Google Scholar - 14.B. Moradabadi, H. Beigy, C.W. Ahn, In
*Proceedings of IEEE Congress on Evolutionary Algorithm*. An improved real-coded Bayesian optimization algorithm (2011)Google Scholar - 15.B. Moradabadi, H. Beigy, C.W. Ahn, An improved real-coded Bayesian optimization algorithm for continuous global optimization. Int. J. Innov. Comput. Inf. Control
**9**(6), 2505–2519 (2013)Google Scholar - 16.M.A.L. Thathachar, P.S. Sastry, Varieties of learning automata: an overview. IEEE Trans. Syst. Man Cybern. B Cybern.
**32**, 711–722 (2002)CrossRefGoogle Scholar - 17.D.M. Chickering, D. Geiger, D. Heckerman,
*Learning Bayesian Network is NP-hard*, Technical Report MSR-TR-94-17 (1994)Google Scholar - 18.D. Heckerman, D. Geiger, D.M. Chickering,
*A Tutorial on Learning with Bayesian Networks, Innovations in Bayesian Networks, Chapter 3*(Springer, Berlin, 2008), pp. 33–82Google Scholar - 19.X.M. Hu, J. Zhang, H. Chung, Y. Li, O. Liu, Sam-ACO: variable sampling ant colony optimization algorithm. IEEE Trans. Syst. Man Cybern. B Cybern.
**40**, 1555–1566 (2010)CrossRefGoogle Scholar - 20.Q. Pan, P.N. Suganthan, M.F. Tasgetiren, J.J. Liang, A self-adaptive global best harmony search algorithm for continuous. Appl. Math. Comput.
**216**, 830–848 (2010)CrossRefMATHMathSciNetGoogle Scholar - 21.M. Gallagher, I. Wood, J. Keith, In
*Proceedings of the IEEE Congress on Evolutionary Computation*. Bayesian inference in estimation of distribution algorithms (2007), pp. 127–133Google Scholar - 22.M. Li, D.E. Goldberg, K. Sastry, T.L. Yu, Real-coded ECGA for solving decomposable real-valued optimization problems. Link. Evol. Comput.
**157**, 61–86 (2008)Google Scholar - 23.C.W. Ahn, R.S. Ramakrishna, On the scalability of real-coded Bayesian optimization algorithm. IEEE Trans. Evol. Comput.
**12**(3), 307–322 (2008)CrossRefGoogle Scholar - 24.D. Heckerman, D. Geiger, D.M. Chickering,
*Learning Bayesian Networks: The Combination of Knowledge and Statistical Data*, Technical Report MSR-TR-94-09 (Microsoft Research, Redmond, 1995)Google Scholar - 25.C.W. Ahn,
*Advances in Evolutionary Algorithms: Theory, Design and Practice, Studies in Computational Intelligence*(Springer, Berlin, 2006)Google Scholar - 26.H. Beigy, M.R. Meybodi, A new continuous action-set learning automata for function optimization. J. Franklin Inst.
**343**, 27 (2006)CrossRefMATHMathSciNetGoogle Scholar - 27.G. Santharam, P.S. Sastry, M.A.L. Thathachar, Continuous action set learning automata for stochastic optimization. J. Franklin Inst.
**331B**(5), 607–628 (1994)CrossRefMATHMathSciNetGoogle Scholar - 28.B.J. Oommen, T.D. Roberts, Continuous learning automata solutions to the capacity assignment problem. IEEE Trans. Comput.
**49**, 608–620 (2000)CrossRefGoogle Scholar - 29.M.S. Obaidat, G.I. Papadimitriou, A.S. Pomportsis, H.S. Laskaridis, Learning automata-based bus arbitration for shared-medium ATM switches. IEEE Trans. Syst. Man Cybern. B Cybern.
**32**, 815–820 (2002)CrossRefGoogle Scholar - 30.G.I. Papadimitriou, M.S. Obaidat, A.S. Pomportsis, On the use of learning automata in the control of broad-cast networks: a methodology. IEEE Trans. Syst. Man Cybern. B Cybern.
**32**, 781–790 (2002)CrossRefGoogle Scholar - 31.H. Beigy, M.R. Meybodi, Cellular learning automata based dynamic channel assignment algorithms. Int. J. Comput. Intell. Appl.
**8**(3), 287–314 (2009)CrossRefMATHGoogle Scholar - 32.H. Beigy, M.R. Meybodi, Utilizing distributed learning automata to solve stochastic shortest path problems. Int. J. Uncertain Fuzz Knowl. Based Syst.
**14**, 591–615 (2006)CrossRefMATHMathSciNetGoogle Scholar - 33.O.C. Granmo, B.J. Oommen, S.A. Myrer, M.G. Olsen, Learning automata-based solutions to the non-linear fractional knapsack problem with applications to optimal resource allocation. IEEE Trans. Syst. Man Cybern. B Cybern.
**37**, 166–175 (2007)CrossRefGoogle Scholar - 34.H. Beigy, M.R. Meybodi, Adaptive limited fractional guard channel algorithms a learning automata approach. Int. J. Uncertain. Fuzziness Knowl. Based Syst.
**17**(6), 881–913 (2009)CrossRefGoogle Scholar - 35.H. Beigy, M.R. Meybodi, A learning automata-based algorithm for determination of the number of hidden units for three layer neural networks. Int. J. Syst. Sci.
**40**, 101–118 (2009)CrossRefMATHGoogle Scholar - 36.P.S. Sastry, G.D. Nagendra, N. Manwani, A team of continuous-action learning automata for noise-tolerant learning of half-spaces. IEEE Trans. Syst. Man Cybern. B Cybern.
**40**, 19–28 (2010)CrossRefGoogle Scholar - 37.H. Beigy, M.R. Meybodi, Learning automata based dynamic guard channel algorithms. J. Comput. Electr. Eng.
**37**(4), 601–613 (2011)CrossRefMATHGoogle Scholar - 38.B.J. Oommen, M.K. Hashem, Modeling a student classroom interaction in a tutorial-like system using learning automata. IEEE Trans. Syst. Man Cybern. B Cybern.
**40**, 29–42 (2010)CrossRefGoogle Scholar - 39.S. Narendra, K.S. Thathachar,
*Learning Automata: An Introduction*(Prentice-Hall, New York, 1989)Google Scholar - 40.K.R. Zãlik, An efficient k’-means clustering algorithm. Pattern Recogn. Lett.
**29**, 1385–1391 (2008)CrossRefGoogle Scholar - 41.Function definitions and performance criteria for the special session on real-parameter optimization at CEC2005 (2005). http://www.ntu.edu.sg/home/EPNSugan. Accessed Apr 2013
- 42.Problem Definitions and Evaluation Criteria for CEC 2011 Competition on Testing Evolutionary Algorithms on Real World Optimization Problems. http://www.ntu.edu.sg/home/EPNSugan. Accessed Apr 2013
- 43.D.H. Wolpert, W.G. Macready, No free lunch theorems for search. IEEE Trans. Evol. Comput.
**5**(3), 295–296 (1997)Google Scholar - 44.Competition on Testing Evolutionary Algorithms on Real-world Numerical Optimization Problems. http://www.ntu.edu.sg/home/EPNSugan. Accessed Apr 2013