Abstract
An aspect that often causes difficulties when using Genetic Algorithms for optimization is that these algorithms operate as unconstrained search procedures and most of the real-world problems have constraints of different types. There is a lack of efficient constraint handling technique to bias the search in constrained search spaces toward the feasible regions. We propose a novel methodology to be coupled with a Genetic Algorithm to solve optimization problems with inequality constraints. This methodology can be seen as a local search operator that uses quadratic and linear approximations for both objective function and constraints. In the local search phase, these approximations define an associated problem with a quadratic objective function and quadratic and/or linear constraints that is solved using an LMI (linear matrix inequality) formulation. The solution of this associated problems is then re-introduced in the GA population.We test the proposed methodology with a set of analytical function and the results show that the hybrid algorithm has a better performancewhen compared to the same Genetic Algorithmwithout the proposed local search operator. The tests also suggest that the proposed methodology is at least equivalent, and sometimes better than other methods that have been reported recently in literature.
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References
Alotto, P., Kuntsevitch, A.V., Magele, C., Molinari, G., Paul, C., Preis, K., Repetto, M., Richter, K.R.: Multiobjective optimization in magnetostatics: a proposal for benchmark problems. IEEE Transactions on Magnetics, Part 1 32(3), 1238–1241 (1996)
Back, T.: Proceedings of the Seventh International Conference on Genetic Algorithms. Morgan Kaufmann, San Mateo (1997)
Bazaraa, M.S., Sherali, H.D., Shetty, C.M.: Nonlinear Programming: Theory and Algorithms. John Wiley and sons, Inc., Chichester (1979)
Boyd, S.P., El Ghaoui, L., Feron, E., Balakrishnan, V.: Linear Matrix Inequalities in System and Control Theory. Siam Studies in Applied Mathematics, vol. 15. Society for Industrial & Applied Mathematics (1997)
Coello Coello, C.A.: Theoretical and numerical constraint-handling techniques used with evolutionary algorithms: A survey of the state of the art. In: Coello Coello, C.A. (ed.) Computer Methods in Applied Mechanics and Engineering, 2002, vol. 191, pp. 1245–1287 (2002)
Davis, L.: Handbook of genetic algorithms. Van Nostrand Reinhold (1991)
Dawkins, R.: The selfish gene. Oxford Press University, Oxford (1976)
Deb, K.: An efficient constraint handling method for genetic algorithms. Computer Methods in Applied Mechanics and Engineering 186, 311–338 (2000)
Deb, K., Pratap, A., Agarwal, S., Meyarivan, T.: A Fast and Elitist Multi-Objective Genetic Algorithm: NSGA–II, KanGAL Report No. 200001, Kanpur Genetic Algorithm Laboratory, Kanpur (2000)
Fonseca, C., Fleming, P.: Multiobjective optimization and multiple constraint handling with evolutionary algorithms I: a unified formulation. IEEE Transactions on Systems, Man and Cybernetics - Part A 28(1), 26–37 (1998)
Gen, M., Cheng, R.: Genetic Algorithms and Engineering Design. Wiley, New York (1997)
Goldberg, D.E.: Genetic algorithms in Search. In: Optimization and Machine Learning. Addison Wesley, Reading (1989)
den Hertog, D., de Klerk, E., Roos, K.: On convex quadratic approximation. Statistica Neerlandica 56(3), 376–385 (2002)
Himmelblau, D.M.: Applied Nonlinear Programming. McGraw-Hill, New York (1972)
Ishibuchi, H., Murata, T.: Multi-objective genetic local search algorithm. In: Proceedings of the IEEE International Conference on Evolutionary Computation, May 20-22, 1996, pp. 119–124 (1996)
Knowles, J.: Local-search and hybrid evolutionary algorithms for Pareto optimization. PhD thesis, Department of Computer Science, The University of Reading, UK (2002)
Knowles, J., Corne, D.: Recent Advances in Memetic Algorithms. In: Krasnogor, N., Smith, J.E., Hart, W.E. (eds.) Memetic algorithms for multiobjective optimization: issues, methods and prospects, pp. 313–352. Springer, Heidelberg (2004)
Lozano, M., Herrera, F., Krasnogor, N., Molina, D.: Real-coded memetic algorithms with crossover hill-climbing. Evolutionary Computation Journal 12(3), 273–302 (2004)
Mezura-Montes, E., Coello-Coello, C.A.: A Simple Multimembered Evolution Strategy to Solve Constrained Optimization Problem. IEEE Transactions on Evolutionary Computation 9(1), 1–17 (2005)
Michalewicz, Z.: Genetic Algorithms, Numerical Optimization and Constraints. In: Proceedings of the 6th International Conference on Genetic Algorithms, 1995, pp. 151–158. Morgan Kaufmann, San Francisco (1995)
Michalewicz, Z., Schoenauer, M.: Evolutionary algorithms for constrained parameter optimization problem. Evolutionary Computation 4(1), 1–32 (1996)
Moscato, P.: On Evolution, Search, Optimization, Genetic Algorithms and Martial Arts: Towards Memetic Algorithms. Caltech Concurrent Computation Program, C3P Report 826 (1989)
Moscato, P.: New Ideas in Optimization. In: Corne, D., Dorigo, M., Glover, F. (eds.) Memetic algorithms: a short introduction, pp. 219–234. McGraw-Hill, New York (1999)
Ramos, R.M., Saldanha, R.R., Takahashi, R.H.C., Moreira, F.J.S.: The real-biased multiobjective genetic algorithm and its application to design of wire antennas. IEEE Transactions on Magnetics 29(3), 1329–1332 (2003)
Richardson, J.T., Palmer, M.R., Liepins, G., Hilliard, M.: Some guidelines for genetic algorithm with penalty function. In: Proceedings of the Third International Conference on Genetic Algorithms, 1989, pp. 191–197. Morgan Kaufmann, San Francisco (1989)
Sturm, J.F.: Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optimization Methods and Software 11-12, 625–653 (1999)
Takahashi, R.H.C., Vasconcelos, J.A., Ramirez, J.A., Krahenbuhl, L.: A multiobjective methodology for evaluation genetic operators. IEEE Transactions on Magnetics 39, 1321–1324 (2003)
TEAM Workshop Problem 22: SMES Optimization Benchmark, http://www.igte.tugraz.at/archive/team_new/description.php
Wanner, E.F., Guimaraes, F.G., Saldanha, R.R., Takahashi, R.H.C., Fleming, P.J.: Constraint Quadratic Approximation Operator for Treating Equality Constraints with Genetic Algorithms. In: Proceedings of the IEEE Congress on Evolutionary Computation, 2005. IEEE Press, Los Alamitos (2005)
Wanner, E.F., Guimaraes, F.G., Takahashi, R.H.C., Fleming, P.F.: Local search with quadratic approximation into memetic algorithms for optimization with multiple criteria. In: Evolutionary Computation. MIT Press, Cambridge (to appear)
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Araujo, M.C., Wanner, E.F., Guimarães, F.G., Takahashi, R.H.C. (2009). Constrained Optimization Based on Quadratic Approximations in Genetic Algorithms. In: Mezura-Montes, E. (eds) Constraint-Handling in Evolutionary Optimization. Studies in Computational Intelligence, vol 198. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-00619-7_9
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DOI: https://doi.org/10.1007/978-3-642-00619-7_9
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