Evolutionary methods are well-known techniques for solving nonlinear constrained optimization problems. Due to the exploration power of evolution-based optimizers, population usually converges to a region around global optimum after several generations. Although this convergence can be efficiently used to reduce search space, in most of the existing optimization methods, search is still continued over original space and considerable time is wasted for searching ineffective regions. This paper proposes a simple and general approach based on search space reduction to improve the exploitation power of the existing evolutionary methods without adding any significant computational complexity. After a number of generations when enough exploration is performed, search space is reduced to a small subspace around the best individual, and then search is continued over this reduced space. If the space reduction parameters (red_gen and red_factor) are adjusted properly, reduced space will include global optimum. The proposed scheme can help the existing evolutionary methods to find better near-optimal solutions in a shorter time. To demonstrate the power of the new approach, it is applied to a set of benchmark constrained optimization problems and the results are compared with a previous work in the literature. 1. Introduction A significant part of today’s engineering problems are constrained optimization problems (COP). Although there exist efficient methods like Simplex for solving linear COP, solving nonlinear COP (NCOP) is still open for novel investigations. Different methods have been proposed for solving NCOP. Among them, natural optimization and especially population-based schemes are the most general and promising ones. These methods can be applied to all types of COP including convex and nonconvex, analytical and non-analytical, real-, integer- and mixed-valued problems. One of the most applied techniques for solving NCOP are evolutionary methods. Various techniques have been introduced for handling nonlinear constrains by evolutionary optimization (EO) methods. These approaches can be grouped in four major categories [1, 2]: (1) methods based on penalty functions that are also known as indirect constraint handling, (2) methods based on a search of feasible solutions including repairing unfeasible individuals [3, 4], superiority of feasible points [5], and behavioral memory [6], (3) methods based on preserving feasibility of solutions like preserving feasibility by designing special crossover and mutation operators [7], the GENOCOP system [8], searching
References
[1]
Z. Michalewicz and M. Schoenauer, “Evolutionary algorithms for constrained parameter optimization problems,” Evolutionary Computation, vol. 4, no. 1, pp. 1–32, 1996.
[2]
？. Yeniay, “Penalty function methods for constrained optimization with genetic algorithms,” Mathematical and Computational Applications, vol. 10, no. 1, pp. 45–56, 2005.
[3]
Z. Michalewicz and G. Nazhiyath, “Genocop III: a co-evolutionary algorithm for numerical optimization problems with nonlinear constraints,” in Proceedings of the 2nd IEEE International Conference on Evolutionary Computation, pp. 647–651, December 1995.
[4]
A. Rowhanimanesh, A. Khajekaramodin, and M.-R. Akbarzadeh-T, “Evolutionary constrained design of seismically excited buildings, actuators placement,” in Proceedings of the 1st Joint Congress on Intelligent and Fuzzy Systems (ISFS '07), pp. 297–304, Mashhad, Iran, 2007.
[5]
K. Deb, “An efficient constraint handling method for genetic algorithms,” Computer Methods in Applied Mechanics and Engineering, vol. 186, no. 2–4, pp. 311–338, 2000.
[6]
M. Schouenauer and S. Xanthakis, “Constrained GA optimization,” in Proceedings of the 5th International Conference on Genetic Algorithms, pp. 473–580, 1993.
[7]
A. Rowhanimanesh, A. Khajekaramodin, and M.-R. Akbarzadeh-T, “Evolutionary constrained design of seismically excited buildings: sensor placement,” Applications of Soft Computing, vol. 58, pp. 159–169, 2009.
[8]
Z. Michalewicz and C. Z. Janikow, “Handling constraints in genetic algorithms,” in Proceedings of the 4th International Conference on Genetic Algorithms, pp. 151–157, 1993.
[9]
M. Schoenauer and Z. Michalewicz, “Evolutionary computation at the edge of feasibility,” in Proceedings of the 4rth International Conference on Parallel Problem Solving from Nature, pp. 22–27, 1996.
[10]
S. Koziel and Z. Michalewicz, “Evolutionary algorithms, homomorphous mappings, and constrained parameter optimization,” Evolutionary Computation, vol. 7, no. 1, pp. 19–44, 1999.
[11]
H. Adeli and N. T. Cheng, “Augmented lagrangian genetic algorithm for structural optimization,” Journal of Aerospace Engineering, vol. 7, no. 1, pp. 104–118, 1994.
[12]
B. W. Wah and Y. Chen, “Hybrid constrained simulated annealing and genetic algorithms for nonlinear constrained optimization,” in Proceedings of the Congress on Evolutionary Computation, vol. 2, pp. 925–932, May 2001.
[13]
J. H. Kim and H. Myung, “Evolutionary programming techniques for constrained optimization problems,” IEEE Transactions on Evolutionary Computation, vol. 1, no. 2, pp. 129–140, 1997.
[14]
T. P. Runarsson and X. Yao, “Constrained evolutionary optimization: the penalty function approach,” in Evolutionary Optimization, R. Sarker, M. Mohammadian, and X. Yao, Eds., pp. 87–113, Kluwer Academic Publishers, 2002.
[15]
T. B？ck, F. Hoffmeister, and H. P. Schwell, “A survey of evolution strategies,” in Proceedings of the 4th International Conference on Genetic Algorithms, pp. 2–9, 1991.
[16]
A. Homaifar, S. H. Y. Lai, and X. Qi, “Constrained optimization via genetic algorithms,” Simulation, vol. 62, no. 4, pp. 242–254, 1994.
[17]
M. A. Kuri and C. C. Quezada, “A universal eclectic genetic algorithm for constrained optimization,” in Proceedings of the 6th European Congress on Intelligent Techniques & Soft Computing, pp. 518–522, 1998.
[18]
J. A. Joines and C. R. Houck, “On the use of non-stationary penalty functions to solve nonlinear constrained optimization problems with GA's,” in Proceedings of the 1st IEEE Conference on Evolutionary Computation, pp. 579–584, June 1994.
[19]
S. Kazarlis and V. Petridis, “Varying fitness functions in genetic algorithms: studying the rate of increase in the dynamic penalty terms,” in Proceedings of the 5th International Conference on Parallel Problem Solving from Nature, pp. 211–220, 1998.
[20]
F. Mendivil and R. Shonkwiler, “Annealing a genetic algorithm for constrained optimization,” Journal of Optimization Theory and Applications, vol. 147, no. 2, pp. 395–410, 2010.
[21]
Z. Michalewicz and N. Attia, “Evolutionary optimization of constrained problems,” in Proceedings of the 3rd Annual Conference on Evolutionary Programming, pp. 98–108, 1994.
[22]
H. J. C. Barbosa and A. C. C. Lemonge, “An adaptive penalty scheme in genetic algorithms for constrained optimization problems,” in Proceedings of the Genetic and Evolutionary Computation Conference (GECCO '02), pp. 287–294, 2002.
[23]
M. Gen and R. Cheng, “A survey of penalty techniques in genetic algorithms,” in Proceedings of the IEEE International Conference on Evolutionary Computation, pp. 804–809, May 1996.
[24]
A. Smith and D. Tate, “Genetic optimization using a penalty function,” in Proceedings of the 5th International Conference on Genetic Algorithms, pp. 499–503, 1993.
[25]
R. Le Riche, C. Knopf-Lenior, and R. T. Haftka, “A segregated genetic algorithm for constrained structural optimization,” in Proceedings of the 6th International Conference on Genetic Algorithms, pp. 558–565, 1995.
[26]
C. A. C. Coello, “Use of a self-adaptive penalty approach for engineering optimization problems,” Computers in Industry, vol. 41, no. 2, pp. 113–127, 2000.
[27]
K. Deb and S. Agarwal, “A niched-penalty approach for constraint handling in genetic algorithms,” in Proceedings of the International Conference on Adaptive and Natural Computing Algorithms (ICANNGA '99), Portoroz, Slovenia, 1999.
[28]
T. P. Runarsson and X. Yao, “Stochastic ranking for constrained evolutionary optimization,” IEEE Transactions on Evolutionary Computation, vol. 4, no. 3, pp. 284–294, 2000.
[29]
K. Deb, “An efficient constraint handling method for genetic algorithms,” Computer Methods in Applied Mechanics and Engineering, vol. 186, no. 2–4, pp. 311–338, 2000.
[30]
R. L. Haupt and S. E. Haupt, Practical Genetic Algorithms, John Wiley & Sons, 2004.
