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Refined EM Method for Solving Linearly Constrained Global Optimization Problems

  • Lu YuEmail author
  • Shu-Cherng Fang
Chapter
  • 1.8k Downloads
Part of the International Series in Operations Research & Management Science book series (ISOR, volume 200)

Abstract

The Electromagnetism-like Mechanism (EM) has been widely used for solving global optimization problems with box-constrained variables. It is a population-based stochastic search method. Since the method uses function evaluations only at each step, it does not require any special information or structure of the objective function. In this article, we extend the original EM for solving optimization problems with linear constraints. The proposed method mimics the behavior of electrically charged particles that are restricted in the feasible region formed by the linear constraints. The underlying idea is to direct the sample points to some attractive regions of the feasible domain. In refined EM, the major steps of the original EM are redesigned to handle the explicit linear constraints in an efficient manner to find global optimal solutions. The proposed method is evaluated using many known test problems and is compared with some existing methods. Computational results show that without using the higher-order information, refined EM converges rapidly (in terms of the number of functions evaluations) to the global optimal solutions and produces better results than other known methods in solving problems of a varying degree of difficulty.

Keywords

Global optimization Stochastic search Heuristic method. 

References

  1. Birbil, Ş.I. (2002). Stochastic global optimization techniques. PhD Thesis. Raleigh: North Carolina State University.Google Scholar
  2. Birbil, Ş.I., & Fang, S.-C. (2002). An electromagnetism-like mechanism for global optimization. Journal of Global Optimisation, 25(3), 263–282.CrossRefGoogle Scholar
  3. Birbil, Ş.I., Fang, S.-C., & Sheu, R.L. (2004). On the convergence of a population based global optimization algorithm. Journal of Global Optimisation, 30(2), 301–318.CrossRefGoogle Scholar
  4. Bonnans, J.F., Gilbert, J.C., Lemaréchal, C., & Sagastizábal, C.A. (2006). Numerical optimization: theoretical and practical aspects. Berlin: Springer-Verlag.Google Scholar
  5. Cowan, E.W. (1968). Basic electromagnetism. New York: Academic Press.Google Scholar
  6. Dolan, E.D., & Moré, J.J. (2002). Benchmarking optimization software with performance profiles. Math Program, 91, 201–213.CrossRefGoogle Scholar
  7. Eberhart R.C., & Kennedy, J. (1995). A new optimizer using particle swarm theory. Proceedings of the Sixth International Symposium on Micromachine and Human Science, Nagoya, Japan, 39-43Google Scholar
  8. Floudas, C.A., & Pardalos, P.M. (1990). A collection of test problems for constrained global optimization algorithms. Lecture notes in computer science 455. Berlin: Springer.Google Scholar
  9. Goldberg, D.E. (1989). Genetic algorithms in search, optimization & machine learning. Boston: Addison-Wesley Longman Publishing Company, Inc.Google Scholar
  10. Gould, N.I.M., Orban, D., & Toint Ph.L. (2013). CUTEr, a constrained and unconstrained test environment, revisited. http://cuter.rl.ac.uk.
  11. Hart, W.E. (1994). Adaptive global optimization with local search, PhD Thesis. San Diego: University of California.Google Scholar
  12. Ingber, L. (1994). Simulated annealing: practice versus theory. Journal of Mathematical Computation Modeling, 18, 29–57.CrossRefGoogle Scholar
  13. Ji, Y., Zhang, K.-C., & Qu, S.-J. (2007). A deterministic global optimization algorithm. Applied Mathematics and Computation, 185, 382–387.CrossRefGoogle Scholar
  14. Kan, A.H.G.R., & Timmer, G.T. (1987). Stochastic global optimization methods Part II: Multi level methods. Math Program, 39, 57–78.CrossRefGoogle Scholar
  15. Kennedy, J., & Eberhart, R.C. (1995). Particle swarm optimization. Proceedings of IEEE International Conference on Neural Networks, IV, 1942-1948. Piscataway, NJ, IEEE Service Center.Google Scholar
  16. Kolda, T.G., Lewis, R.M., & Torczon, V. (2003) Optimization by direct search: new perspectives on some classical and modern methods. SIAM Review, 45, 385–482.CrossRefGoogle Scholar
  17. Michalewicz, Z. (1994). Evolutionary computation techniques for nonlinear programming problems. International Transactions in Operational Research, 1:223–240.CrossRefGoogle Scholar
  18. Michalewicz, Z. (1996). Genetic Algorithms + Data Structures = Evolution Programs, 3rd edition. Berlin: Springer.CrossRefGoogle Scholar
  19. Runarsson, T.P., & Yao, X. (2000). Stochastic ranking for constrained evolutionary optimization. IEEE Transactions on Evolutionary Computation, 4, 284–294.CrossRefGoogle Scholar
  20. Vanderbei, R.J. (2013). Benchmarks for nonlinear optimization. http://www.princeton.edu/~rvdb/bench.html.
  21. Vaz, A.I.F., & Vicente, L.N. (2009). PSwarm: a hybrid solver for linearly constrained global derivative-free optimization. Optimization Methods Software, 24(4-5), 669–685.CrossRefGoogle Scholar
  22. Wood, G.R. (1991). Multidimensional bisection and global optimization. Computers & Mathematics with Applications, 21, 161–172CrossRefGoogle Scholar
  23. Zhang, Y., & Gao, L. (2001). On numerical solution of the maximum volume ellipsoid problem. Society for Industrial and Applied Mathematics, 14, 53–76.Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Edward P. Fitts Department of Industrial and Systems EngineeringNorth Carolina State UniversityRaleighNorth Carolina

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