## An effective co-evolutionary differential evolution for constrained optimization.(English)Zbl 1114.65061

Summary: Many practical problems can be formulated as constrained optimization problems. Due to the simple concept and easy implementation, the penalty function method has been one of the most common techniques to handle constraints. However, the performance of this technique greatly relies on the setting of penalty factors, which are usually determined by manual trial and error, and the suitable penalty factors are often problem-dependent and difficult to set.
In this paper, a differential evolution approach based on a co-evolution mechanism is proposed to solve the constrained problems. First, a special penalty function is designed to handle the constraints. Second, a co-evolution model is presented and differential evolution is employed to perform evolutionary search in spaces of both solutions and penalty factors. Thus, the solutions and penalty factors evolve interactively and self-adaptively, and both the satisfactory solutions and suitable penalty factors can be obtained simultaneously.
Simulation results based on several benchmark functions and three well-known constrained design problems as well as comparisons with some existed methods demonstrate the effectiveness, efficiency and robustness of the proposed method.

### MSC:

 65K05 Numerical mathematical programming methods 90C30 Nonlinear programming

### Keywords:

co-evolution; penalty function; numerical examples
Full Text:

### References:

 [1] Wang, L., Intelligent optimization algorithms with application, (2001), Tsinghua University & Springer Press Beijing [2] Coello, C.A.C., Theoretical and numerical constraint handling techniques used with evolutionary algorithms: a survey of the state of the art, Computer methods in applied mechanics and engineering, 191, 11-12, 1245-1287, (2002) · Zbl 1026.74056 [3] Michalewicz, Z., A survey of constraint handling techniques in evolutionary computation methods, (), 135-155 [4] Coello, C.A.C., Use of a self-adaptive penalty approach for engineering optimization problems, Computers in industry, 41, 113-127, (2000) [5] Hamida, S.B.; Schoenauer, M., ASCHEA: new results using adaptive segregational constraint handling, (), 884-889 [6] He, Q.; Wang, L., An effective co-evolutionary particle swarm optimization for constrained engineering design problems, Engineering applications of artificial intelligence, 19, 7, (2006) [7] Runarsson, T.P.; Yao, X., Stochastic ranking for constrained evolutionary optimization, IEEE transactions on evolutionary computation, 4, 3, 284-294, (2000) [8] Coello, C.A.C.; Montes, E.M., Constraint-handling in genetic algorithms through the use of dominance-based tournament selection, Advanced engineering informatics, 16, 193-203, (2002) [9] Horn, J.; Nafpliotis, N.; Goldberg, D.E., A niched Pareto genetic algorithm for multiobjective optimization, (), 82-87 [10] Montes, E.M.; Coello, C.A.C., A simple multimembered evolution strategy to solve constrained optimization problems, IEEE transactions on evolutionary computation, 9, 1, 1-17, (2005) [11] Koziel, S.; Michalewicz, Z., Evolutionary, algorithms, homomorphous, mappings, and constrained parameter optimization, Evolutionary computation, 7, 1, 19-44, (1999) [12] Storn, R.; Price, K., Differential evolution-a simple and efficient heuristic for global optimization over continuous spaces, Journal of global optimization, 11, 341-359, (1997) · Zbl 0888.90135 [13] K.V. Price, Differential evolution: a fast and simple numerical optimizer, in: Proceedings of the North American Fuzzy Information Processing Society, 1996, pp. 524-527. [14] Storn, R., Differential evolution design of an IIR-filter with requirements for magnitude and group delay, (1995), International Computer Science Institute [15] B.V. Babu, A.S. Chaurasia, Optimization of pyrolysis of biomass using differential evolution approach, in: The Second International Conference on Computational Intelligence, Robotics, and Autonomous Systems, Singapore, 2003. [16] J. Lampinen, A constraint handling approach for the differential evolution algorithm, in: Proceedings of the Congress on Evolutionary Computation, 2002, pp. 1468-1473. [17] Becerra, R.L.; Coello, C.A.C., A cultural algorithm with differential evolution to solve constrained optimization problems, Lecture notes in artificial intelligence, 3315, 881-890, (2004) [18] R. Storn, On the usage of differential evolution for function optimization, in: Biennial Conference of the North American Fuzzy Information Processing Society, Berkeley, 1996, pp. 519-523. [19] Michalewicz, Z.; Attia, N., Evolutionary optimization of constrained problems, (), 98-108 [20] Richardson, J.T.; Palmer, M.R.; Liepins, G.; Hilliard, M., Some guidelines for genetic algorithms with penalty functions, (), 191-197 [21] Arora, J.S., Introduction to optimum design, (1989), McGraw-Hill New York [22] A.D. Belegundu, A study of mathematical programming methods for structural optimization. Department of Civil and Environmental Engineering, University of Iowa, Iowa, 1982. [23] Deb, K.; Gene, A.S., A robust optimal design technique for mechanical component design, (), 497-514 [24] Kannan, B.K.; Kramer, S.N., An augmented Lagrange multiplier based method for mixed integer discrete continuous optimization and its applications to mechanical design, Transactions of the ASME, journal of mechanical design, 116, 318-320, (1994) [25] E. Sandgren, Nonlinear integer and discrete programming in mechanical design, in: Proceedings of the ASME Design Technology Conference, Kissimine, FL, 1988, pp. 95-105.
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