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Online Performance Measures for Metaheuristic Optimization

  • Kay Hamacher
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8457)

Abstract

(Global) optimization is one of the fundamental challenges in scientific computing. Frequently, one encounters objective functions or search space topologies that do not fulfill necessary requirements for well understood and efficient procedures like, e.g., linear programming. This methodological gap is filled by metaheuristic optimization approaches. Their search dynamics in high dimensional search spaces and for complicated objective functions is not well understood at present. In particular, the choice of parameters driving the procedures is a demanding task. In this contribution we show how insight from time series analysis help to investigate – on a pure empirical basis – metaheuristic schemes. Rather than deriving analytical results on convergence behavior, ex ante, we propose online observation of the search and optimization progress. To this end, we use the Detrended Fluctuation Analysis – a method from time series analysis – to investigate the search dynamics of metaheuristics as stochastic processes. We apply the proposed method to two different metaheuristic, namely differential evolution and basin hopping.

Keywords

Spin Glass Detrended Fluctuation Analysis Iterate Local Search Travel Salesperson Problem Search Dynamic 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Bäck, T., Schwefel, H.: An overview of evolutionary algorithms for parameter optimization. Evolutionary Computation 1(1), 1–23 (1993), http://dx.doi.org/10.1162/evco.1993.1.1.1 CrossRefGoogle Scholar
  2. 2.
    Binder, K., Heermann, D.: Monte Carlo Simulation in Statistical Physics, 3rd edn. Springer, Berlin (1997)CrossRefzbMATHGoogle Scholar
  3. 3.
    Binder, K., Young, A.: Spin glasses: Experimental facts, theoretical concepts, and open questions. Rev. Mod. Phys. 58(4), 801–976 (1986)CrossRefGoogle Scholar
  4. 4.
    Birattari, M.: Tuning Metaheuristics. SCI, vol. 197. Springer, Heidelberg (2009)zbMATHGoogle Scholar
  5. 5.
    Bunde, A., Kantelhardt, J.: Langzeitkorrelationen in der natur: von klima, erbgut und herzrhythmus. Phys. Bl. 57(5), 49–54 (2001)CrossRefGoogle Scholar
  6. 6.
    Chou, C., Hand, R., Li, S., Lee, T.: Guided simulated annealing method for optimization problems. Phys. Rev. E 67, 66704 (2003)CrossRefGoogle Scholar
  7. 7.
    Das, S., Suganthan, P.: Differential evolution: A survey of the state-of-the-art. IEEE Transactions on Evolutionary Computation 15(1), 4–31 (2011)CrossRefGoogle Scholar
  8. 8.
    Doye, J., Wales, D.: Saddle points and dynamics of Lennard-Jones clusters, solids, and supercooled liquids. J. Chem. Phys. 116(9), 3777–3788 (2002)CrossRefGoogle Scholar
  9. 9.
    Friedrich, T., Kroeger, T., Neumann, F.: Weighted preferences in evolutionary multi-objective optimization. In: Wang, D., Reynolds, M. (eds.) AI 2011. LNCS, vol. 7106, pp. 291–300. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  10. 10.
    Friedrich, T., Sauerwald, T.: The cover time of deterministic random walks. In: Thai, M.T., Sahni, S. (eds.) COCOON 2010. LNCS, vol. 6196, pp. 130–139. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  11. 11.
    Hamacher, K.: On stochastic global optimization of one-dimensional functions. Physica A 354, 547–557 (2005)CrossRefGoogle Scholar
  12. 12.
    Hamacher, K.: Adaptation in stochastic tunneling global optimization of complex potential energy landscapes. Europhys. Lett. 74(6), 944–950 (2006)CrossRefGoogle Scholar
  13. 13.
    Hamacher, K.: Adaptive extremal optimization by detrended fluctuation analysis. J. Comp. Phys. 227(2), 1500–1509 (2007)CrossRefzbMATHGoogle Scholar
  14. 14.
    Hamacher, K., Wenzel, W.: The scaling behaviour of stochastic minimization algorithms in a perfect funnel landscape. Phys. Rev. E 59(1), 938–941 (1999)CrossRefGoogle Scholar
  15. 15.
    Hansmann, U., Wille, L.T.: Global Optimization by Energy Landscape Paving. Phys. Rev. Lett. 88(23), 68105 (2002)CrossRefGoogle Scholar
  16. 16.
    Hoos, H., Stützle, T.: On the empirical evaluation of Las Vegas algorithms (1998)Google Scholar
  17. 17.
    Hu, K., Ivanov, P.C., Chen, Z., Carpena, P., Eugene Stanley, H.: Effect of trends on detrended fluctuation analysis. Phys. Rev. E 64(1), 011114 (2001)Google Scholar
  18. 18.
    Jack, W., Rogers, J., Donnelly, R.A.: Potential transformation methods for large-scale global optimization. SIAM Journal on Optimization 5(4), 871–891 (1995), http://link.aip.org/link/?SJE/5/871/1 CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Kirkpatrick, S., Gelatt, C., Vecchi, M.: Optimization by simulated annealing. Science 220, 671–680 (1983)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H., Teller, E.: Equation of state calculations by fast computing machines. J. Chem. Phys. 21(6), 1087–1092 (1953)CrossRefGoogle Scholar
  21. 21.
    Panos, M., Pardalos, D.S., Xue, G. (eds.): Global Minimization of Nonconvex Energy Functions: Molecular Conformation and Protein Folding. dIMACS workshop, March 20-21. DIMACS – Series in Discrete Mathematics and Theoretical Computer Science, vol. 23 (1995)Google Scholar
  22. 22.
    Pardalos, P.M., Shalloway, D., Xue, G.: Optimization methods for computing global minima of nonvoncex potential energy functions. J. Glob. Opt. 4, 117–133 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Pardalos, P., Romeijn, E., Tuy, H.: Recent developments and trends in global optimization. J. Comp. Appl. Math. 124(1-2), 209–228 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Pellegrini, P., Stützle, T., Birattari, M.: Off-line vs. on-line tuning: A study on MAX-MIN ant system for the TSP, pp. 239–250 (2010)Google Scholar
  25. 25.
    Peng, C.K., Buldyrev, S., Havlin, S., Simons, M., Stanley, H., Goldberger, A.: Mosaic organization of dna nucleotides. Phys. Rev. E 49, 1685 (1994)CrossRefGoogle Scholar
  26. 26.
    Ratschek, H., Rokne, J.G.: Efficiency of a global optimization algorithm. SIAM Journal on Numerical Analysis 24(5), 1191–1201 (1987), http://link.aip.org/link/?SNA/24/1191/1 CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Schelstraete, S., Schepens, W., Verschelde, H.: Energy minimization by smoothing techniques: a survey. In: Balbuena, P., Seminario, J. (eds.) Molecular Dynamics: From Classical to Quantum Methods, Amsterdam, pp. 129–185 (1999)Google Scholar
  28. 28.
    Schöbel, A., Scholz, D.: The theoretical and empirical rate of convergence for geometric branch-and-bound methods. J. Global Optimization 48(3), 473–495 (2010)CrossRefzbMATHGoogle Scholar
  29. 29.
    Shi, Y.-j., Teng, H.-f., Li, Z.-q.: Cooperative co-evolutionary differential evolution for function optimization. In: Wang, L., Chen, K., S. Ong, Y. (eds.) ICNC 2005. LNCS, vol. 3611, pp. 1080–1088. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  30. 30.
    Shubert, B.O.: A sequential method seeking the global maximum of a function. SIAM J. Numer. Anal. 9(3), 379–388 (1972)CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    Simone, C., Diehl, M., Jünger, M., Mutzel, P., Reinelt, G.: Exact ground states of ising spin glasses: New experimental results with a branch-and-cut algorithm. J. Stat. Phys. 80, 487 (1995)CrossRefzbMATHGoogle Scholar
  32. 32.
    Storn, R.: On the usage of differential evolution for function optimization. In: 1996 Biennial Conference of the North American Fuzzy Information Processing Society (1996)Google Scholar
  33. 33.
    Storn, R., Price, K.: Differential evolution – a simple and efficient heuristic for global optimization over continuous spaces. J. Glob. Opt. 11(4), 341–359 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  34. 34.
    Stützle, T.: Iterated local search for the quadratic assignment problem. European Journal of Operational Research 174(3), 1519–1539 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  35. 35.
    Sttzle, T., Hoos, H.H.: Analyzing the run-time behaviour of iterated local search for the TSP. In: III Metaheuristics International Conference. Kluwer Academic Publishers (1999)Google Scholar
  36. 36.
    Sutton, A.M., Neumann, F.: A parameterized runtime analysis of evolutionary algorithms for the euclidean traveling salesperson problem. In: Hoffmann, J., Selman, B. (eds.) AAAI, AAAI Press (2012)Google Scholar
  37. 37.
    Törn, A., Žilinskas, A.: Global Optimization. LNCS, vol. 350. Springer, Heidelberg (1989)zbMATHGoogle Scholar
  38. 38.
    Wales, D.J., Scheraga, H.A.: Global Optimization of Clusters, Crystals, and Biomolecules. Science 285(5432), 1368–1372 (1999), http://www.sciencemag.org/cgi/content/abstract/285/5432/1368 CrossRefGoogle Scholar
  39. 39.
    Wenzel, W., Hamacher, K.: A Stochastic tunneling approach for global minimization. Phys. Rev. Lett. 82(15), 3003–3007 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  40. 40.
    Wolpert, D.H., Macready, W.G.: No free lunch theorems for optimization. IEEE Transactions on Evolutionary Computation 1(1), 67–82 (1997)CrossRefGoogle Scholar
  41. 41.
    Yang, X.S.: Metaheuristic optimization: Algorithm analysis and open problems. In: Pardalos, P.M., Rebennack, S. (eds.) SEA 2011. LNCS, vol. 6630, pp. 21–32. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  42. 42.
    Zemel, E.: Measuring the quality of approximate solutions to zero-one programming problems. Mathematics of Operations Research 6(3), 319–332 (1981)CrossRefzbMATHMathSciNetGoogle Scholar
  43. 43.
    Zlochin, M., Dorigo, M.: Model-based search for combinatorial optimization: A comparative study. In: Guervós, J.J.M., Adamidis, P.A., Beyer, H.-G., Fernández-Villacañas, J.-L., Schwefel, H.-P. (eds.) PPSN 2002. LNCS, vol. 2439, pp. 651–661. Springer, Heidelberg (2002)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Kay Hamacher
    • 1
  1. 1.Dept. of Computer Science, Dept. of Physics & Dept. of BiologyTechnical University DarmstadtDarmstadtGermany

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