Identifying the number of niches in multimodal optimization is vital to enhancement of efficiency of algorithms. This paper presents a genetic algorithm (GA)-based clustering method for multiple optimal determinations. The approach uses self-organizing map (SOM) neural networks to detect clusters in GA population. After clustering all population and recognizing the number of niches, the phenotypic space is partitioned. Within each partition, a simple GA is independently running to evolve to the actual optima. Before the SOM starts, we allow GA to run several generations until the borders of clusters are identified. Our proposed algorithm is easy to implement, and does not require any prior knowledge about the fitness function. The algorithm was tested for seven multimodal functions and four constrained engineering optimization functions, and the results have been compared with the other related algorithms based on three performance criteria. We found that the present algorithm has acceptable diversification and function evaluation number.
A.D.Belegundu, A study of mathematical programming methods for structural optimization, Ph.D. thesis, Department of Civil and Environmental Engineering, University of Iowa, USA, 1982.
2.
N.Shabani and T.Sowlati, A mixed integer non-linear programming model for tactical value chain optimization of a wood biomass power plant, Appl Energy104 (2013), 353–361.
3.
R.G.Garroppo, S.Giordano, G.Nencioni and M.G.Scutella, Mixed integer non-linear programming models for green network design, Comput OperRes40(1) (2013), 273–281.
4.
T.Kompas and L.Chu, Comparing approximation techniques to continuous-time stochastic dynamic programming problems: Applications to natural resource modelling, Environ Model Softw38 (2012), 1–12.
5.
D.Liu, D.Wang and X.Yang, An iterative adaptive dynamic programming algorithm for optimal control of unknown discrete-time nonlinear systems with constrained inputs, Inform Sci220 (2013), 331–342.
6.
J.Si, A.G.Barto, W.B.Powell and D.C.Wunsch, Handbook of Learning and Approximate Dynamic Programming, Wiley, New York, 2004.
7.
F.Samadi, A.Mirzazadeh and M.M.Pedram, Fuzzy pricing, marketing and service planning in a fuzzy inventory model: A geometric programming approach, Appl Math Model37 (2013), 6683–6694.
8.
R.Yelchuru and S.Skogestad, Convex formulations for optimal selection of controlled variables and measurements using Mixed Integer Quadratic Programming, J Process Contr22(6) (2012), 995–1007.
9.
Y.M.Zhang, G.H.Huang, Q.G.Lin and H.W.Lu, Integer fuzzy credibility constrained programming for power system management, Energy38(1) (2012), 398–405.
10.
S.Burer and A.N.Letchford, Non-convex mixed-integer nonlinear programming: A survey, Surv Oper Res Manage Sci17(2) (2012), 97–106.
11.
W.M.Jenkins, Towards structural optimization via the genetic algorithm, Comput Struct40(5) (1991), 1321–1327.
12.
R.Poli, Analysis of the publications on the applications of particle swarm optimization, JAEA10 (2008), 1–10.
13.
A.Kaveh and S.Talatahari, An improved ant colony optimization for constrained engineering design problems, Eng Comput27(1) (2010), 155–182.
14.
F.O.Sonmez, Structural Optimization using Simulated Annealing, InTech, 2008.
15.
M.Sonmez, Artificial bee colony algorithm for optimization of truss structures, Appl Soft Comput11(2) (2011), 2406–2418.
16.
K.S.Lee and Z.W.Geem, A new meta-heuristic algorithm for continuous engineering optimization: Harmony search theory and practice, Comput Method Appl M194 (2005), 3902–3933.
17.
H.Yi, Q.Duan and T.W.Liao, Three improved hybrid meta-heuristic algorithms for engineering design optimization, Appl Soft Comput13(5) (2013), 2433–2444.
18.
A.Kaveh and S.Talatahari, A novel heuristic optimization method: Charged system search, Acta Mech213(3) (2010), 267–289.
19.
B.Y.Qu, J.J.Liang and P.N.Suganthan, Niching particle swarm optimization with local search for multi-modal optimization, Inform Sci197 (2012), 131–143.
20.
J.Yao, N.Kharma and P.Grogono, Bi-objective multipopulation genetic algorithm for multimodal function optimization, Evol Comput14(1) (2010), 80–102.
21.
J.M.Pena, J.A.Lozano and P.Larranaga, Benefits of data clustering in multimodal function optimization via EDAs, in: P. Larranaga and J.A. Lozano (eds.), Estimation ofDis-tribution Algorithms, Springer, 2002, pp. 101–127.
22.
L.Costa and P.Oliveira, An adaptive sharing elitist evolution strategy for multiobjective optimization, Evol Comput11(4) (2003), 417–438.
23.
D.E.Goldberg and L.Wang, Adaptive niching via coevo-lutionary sharing, ILLiGAL Technical Report No. 97007, Illinois Genetic Algorithms Laboratory, University of Illi-noise, Urbana, IL, 1997.
24.
R.K.Ursem, Multinational evolutionary algorithms, Proceedings of the IEEE Int. Conference on Evolutionary Computation, Washington, USA, 1999, pp. 1633–1640.
25.
A.E.Imrani, A.Bouroumi, Z.E.Abidine, M.Limouri and A.Essaid, A fuzzy clustering-based niching approach to multimodal function optimization, Cogn Syst Res1 (2) (2000), 119–133.
26.
J.Alami, L.Benameur and A.El.Imrani, A fuzzy clustering based PSO for multimodal optimization, Int J Comput Intell Res (2009), 96–107.
27.
J.Alami, A.El.Imrani and A.Bouroumi, A multipopulation cultural algorithm using fuzzy clustering, Appl Soft Comput7(2) (2007), 506–519.
28.
J.Kennedy, Stereotyping: Improving particle swarm performance with cluster analysis, Proceedings of the Congress on Evolutionary Computation, California, USA, 2000, pp. 1507–1512.
29.
Ch.Magele, O.Aichholzer, F.Aurenhammer, B.Brandstat-ter, H.Krasser, M.Muhlmann and W.Renhart, Evolution strategy and hierarchical clustering, Proceeding of 13th COMPUMAG Conference, Lyon-Evian, France, 2001, pp.
30.
A.Passaro and A.Starita, Particle swarm optimization for multimodal functions: A clustering approach, J Artif Evol Appl2008(2008), 1–15.
31.
S.Ando, J.Sakuma and S.Kobayashi, Adaptive isolation model using data clustering for multimodal function optimization, Proc Genet Evol Comput Conf (2005), 1417–1424.
32.
P.J.Ballester and J.N.Carter, An algorithm to identify clusters of solutions in multimodal optimisation, in: C.C. Ribeiro and S.L. Martins (eds.),Experimental and Efficient Algorithms.Lecture Notes in Computer Science, Springer Verlag, 2004, pp. 42–56.
33.
G.Zhang, L.Yu, Q.Shao and Y.Feng, A clustering based GA for multimodal optimization in uneven search space, Proceeding of 6th World Congress on Intelligent Control and Automation, China, 2006, pp. 3134–3138.
34.
R.Duda and P.Hart, Pattern Classification and Scene Analysis, Wiley, New York, 1973.
35.
R.Bellman, Adaptive Control Processes: A Guided Tour, Princeton University Press, 1961.
36.
H.Wang, I.Moon, S.Yang and D.Wang, A memetic particle swarm optimization algorithm for multimodal optimization problems, Inform Sci197 (2012), 38–52.
37.
K.C.Wong, C.H.Wu, R.K.P.Mok, C.Peng and Z.Zhang, Evolutionary multimodal optimization using the principle of locality, Inform Sci194 (2012), 138–170.
38.
T.Otani, R.Suzuki and T.Arita, DE/isolated/1: A new mutation operator for multimodal optimization with differential evolution, J Mach Learn Cybern4(2) (2013), 99–105.
39.
J.Joinnes and C.Houck, On the use of non-stationary penalty function to solve nonlinear constrained optimization problems with GAs, Proceedings of Evolutionary Computation, Florida, USA,1994, pp. 579–584.
40.
E.Sandgren, Nonlinear integer and discrete programming in mechanical design, Proceedings of the ASME Design Technology Conference, Kissimme, FL,1988, pp. 95–105.
41.
B.K.Kannan and S.N.Kramer, An augmented Lagrange multiplier based method for mixed integer discrete continuous optimization and its applications to mechanical design, JMechDes116(2) (2008), 405–411.
42.
K.Deb and M.Goyal, A combined genetic adaptive search (GeneAS) for engineering design, Comput Sci Inform26(4) (1988), 1–20.
43.
D.F.W.Yap, S.P.Koh, S.K.Tiong and S.K.Prajindra, A swarm-based artificial immune system for solving multimodal functions, Appl Artif Intell25(8) (2011), 693–707.
44.
J.S.Arora, Introduction to Optimum Design, McGraw-Hill, New York, 1989.
45.
C.A.C.Coello, Use of a self-adaptive penalty approach for engineering optimization problems, Comput Ind41(2) (2000), 113–127.
46.
C.A.C.Coello and E.M.Montes, Constraint-handling in genetic algorithms through the use of dominance-based tournament selection, Adv Eng Inform16(3) (2002), 193–203.
47.
Q.He and L.Wang, An effective co-evolutionary particle swarm optimization for constrained engineering design problems, Eng Appl Artif Intel20(1) (2007), 89–99.
48.
E.M.Montes and C.A.C.Coello, An empirical study about the usefulness of evolution strategies to solve constrained optimization problems, Int J Gen Syst37(4) (2008), 443–473.
49.
K.Deb, Optimal design of a welded beam via genetic algorithms, Astronaut Aeronaut29(11) (1991), 2013–2015.
50.
K.M.Ragsdell and D.T.Phillips, Optimal design of a class of welded structures using geometric programming, J Manuf Sci E-T ASME98(3) (1976), 1021–1025.
