Skip to main content

Analysis of different MMAS ACO algorithms on unimodal functions and plateaus

Abstract

Recently, the first rigorous runtime analyses of ACO algorithms appeared, covering variants of the MAX–MIN ant system and their runtime on pseudo-Boolean functions. Interestingly, a variant called 1-ANT is very sensitive to the evaporation factor while Gutjahr and Sebastiani proved partly opposite results for their variant MMASbs. These algorithms differ in their pheromone update mechanisms and, moreover, 1-ANT accepts equally fit solutions in contrast to MMASbs.

By analyzing variants of MMASbs, we prove that the different behavior of 1-ANT and MMASbs results from the different pheromone update mechanisms. Building upon results by Gutjahr and Sebastiani, we extend their analyses of MMASbs to the class of unimodal functions and show improved results for test functions using new and specialized techniques; in particular, we present new lower bounds. Finally, we compare MMASbs with a variant that also accepts equally fit solutions as this enables the exploration of plateaus. For well-known plateau functions we prove that this drastically reduces the optimization time. Our findings are complemented by experiments that support our asymptotic analyses and yield additional insights.

References

  • Brémaud, P. (1998). Markov chains: Gibbs fields, Monte Carlo simulation, and queues. New York: Springer.

    Google Scholar 

  • Doerr, B., & Johannsen, D. (2007). Refined runtime analysis of a basic ant colony optimization algorithm. In Proceedings of the congress on evolutionary computation (CEC ’07) (pp. 501–507). New York: IEEE Press.

    Chapter  Google Scholar 

  • Doerr, B., Hebbinghaus, N., & Neumann, F. (2006). Speeding up evolutionary algorithms through restricted mutation operators. In LNCS : Vol. 4193. Parallel problem solving from nature (PPSN ’06) (pp. 978–987). Berlin: Springer.

    Chapter  Google Scholar 

  • Doerr, B., Neumann, F., Sudholt, D., & Witt, C. (2007). On the runtime analysis of the 1-ANT ACO algorithm. In Proceedings of the genetic and evolutionary computation conference (GECCO ’07) (pp. 33–40). New York: ACM.

    Chapter  Google Scholar 

  • Dorigo, M., & Blum, C. (2005). Ant colony optimization theory: A survey. Theoretical Computer Science, 344, 243–278.

    MATH  Article  MathSciNet  Google Scholar 

  • Dorigo, M., & Stützle, T. (2004). Ant colony optimization. Cambridge: MIT Press.

    MATH  Google Scholar 

  • Dorigo, M., Maniezzo, V., & Colorni, A. (1996). Ant system: Optimization by a colony of cooperating agents. IEEE Transactions on Systems, Man, and Cybernetics Part B, 26(1), 29–41.

    Article  Google Scholar 

  • Droste, S., Jansen, T., & Wegener, I. (2002). On the analysis of the (1+1) evolutionary algorithm. Theoretical Computer Science, 276, 51–81.

    MATH  Article  MathSciNet  Google Scholar 

  • Droste, S., Jansen, T., & Wegener, I. (2006). Upper and lower bounds for randomized search heuristics in black-box optimization. Theory of Computing Systems, 39(4), 525–544.

    MATH  Article  MathSciNet  Google Scholar 

  • Eiben, A., & Smith, J. (2007). Introduction to evolutionary computing (2nd ed.). Berlin: Springer.

    Google Scholar 

  • Garnier, J., Kallel, L., & Schoenauer, M. (1999). Rigorous hitting times for binary mutations. Evolutionary Computation, 7(2), 173–203.

    Article  Google Scholar 

  • Giel, O., & Wegener, I. (2003). Evolutionary algorithms and the maximum matching problem. In LNCS : Vol. 2607. Proceedings of the 20th annual symposium on theoretical aspects of computer science (STACS ’03) (pp. 415–426). Berlin: Springer.

    Google Scholar 

  • Gutjahr, W. J. (2002). ACO algorithms with guaranteed convergence to the optimal solution. Information Processing Letters, 82(3), 145–153.

    MATH  Article  MathSciNet  Google Scholar 

  • Gutjahr, W. J. (2006). On the finite-time dynamics of ant colony optimization. Methodology and Computing in Applied Probability, 8(1), 105–133.

    MATH  Article  MathSciNet  Google Scholar 

  • Gutjahr, W. J. (2007). Mathematical runtime analysis of ACO algorithms: Survey on an emerging issue. Swarm Intelligence, 1, 59–79.

    Article  Google Scholar 

  • Gutjahr, W. J. (2008). First steps to the runtime complexity analysis of ant colony optimization. Computers and Operations Research, 35(9), 2711–2727.

    MATH  Article  Google Scholar 

  • Gutjahr, W. J., & Sebastiani, G. (2008). Runtime analysis of ant colony optimization with best-so-far reinforcement. Methodology and Computing in Applied Probability, 10, 409–433.

    Article  MathSciNet  Google Scholar 

  • Jansen, T. Sudholt, D. (2009, to appear). Analysis of an asymmetric mutation operator. Evolutionary Computation.

  • Jansen, T., & Wegener, I. (2001). Evolutionary algorithms—how to cope with plateaus of constant fitness and when to reject strings of the same fitness. IEEE Transactions on Evolutionary Computation, 5(6), 589–599.

    Article  Google Scholar 

  • Mitzenmacher, M., & Upfal, E. (2005). Probability and computing—randomized algorithms and probabilistic analysis. Cambridge: Cambridge University Press.

    MATH  Google Scholar 

  • Motwani, R., & Raghavan, P. (1995). Randomized algorithms. Cambridge: Cambridge University Press.

    MATH  Google Scholar 

  • Neumann, F. (2004). Expected runtimes of evolutionary algorithms for the Eulerian cycle problem. In Proceedings of the congress on evolutionary computation (CEC ’04) (Vol. 1, pp. 904–910). New York: IEEE Press.

    Google Scholar 

  • Neumann, F. (2007). Expected runtimes of a simple evolutionary algorithm for the multi-objective minimum spanning tree problem. European Journal of Operational Research, 181(3), 1620–1629.

    MATH  Article  Google Scholar 

  • Neumann, F., & Wegener, I. (2006). Minimum spanning trees made easier via multi-objective optimization. Natural Computing, 5(3), 305–319.

    MATH  Article  MathSciNet  Google Scholar 

  • Neumann, F., & Wegener, I. (2007). Randomized local search, evolutionary algorithms, and the minimum spanning tree problem. Theoretical Computer Science, 378(1), 32–40.

    MATH  Article  MathSciNet  Google Scholar 

  • Neumann, F., & Witt, C. (2006). Runtime analysis of a simple ant colony optimization algorithm. In LNCS : Vol. 4288. Proceedings of the 17th international symposium on algorithms and computation (ISAAC ’06) (pp. 618–627). Berlin: Springer. Extended version to appear in Algorithmica.

    Google Scholar 

  • Neumann, F., Sudholt, D., & Witt, C. (2007). Comparing variants of MMAS ACO algorithms on pseudo-Boolean functions. In LNCS : Vol. 4638. Proceedings of engineering stochastic local search algorithms (SLS ’07) (pp. 61–75). Berlin: Springer.

    Google Scholar 

  • Rudolph, G. (1997). Convergence properties of evolutionary algorithms. Hamburg: Kovač.

    Google Scholar 

  • Scharnow, J., Tinnefeld, K., & Wegener, I. (2004). The analysis of evolutionary algorithms on sorting and shortest paths problems. Journal of Mathematical Modelling and Algorithms, 3(4), 349–366.

    MATH  Article  MathSciNet  Google Scholar 

  • Stützle, T., & Hoos, H. H. (2000). MAX-MIN ant system. Future Generations Computer Systems, 16, 889–914.

    Article  Google Scholar 

  • Wegener, I. (2002). Methods for the analysis of evolutionary algorithms on pseudo-Boolean functions. In R. Sarker, X. Yao, & M. Mohammadian (Eds.), Evolutionary optimization (pp. 349–369). Dordrecht: Kluwer Academic.

    Google Scholar 

  • Wegener, I., & Witt, C. (2005). On the optimization of monotone polynomials by simple randomized search heuristics. Combinatorics, Probability and Computing, 14, 225–247.

    MATH  Article  MathSciNet  Google Scholar 

  • Witt, C. (2005). Worst-case and average-case approximations by simple randomized search heuristics. In LNCS : Vol. 3404. Proceedings of the 22nd symposium on theoretical aspects of computer science (STACS ’05) (pp. 44–56). Berlin: Springer.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Frank Neumann.

Additional information

A conference version appeared in SLS 2007 (Neumann et al. 2007).

Rights and permissions

Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

Reprints and Permissions

About this article

Cite this article

Neumann, F., Sudholt, D. & Witt, C. Analysis of different MMAS ACO algorithms on unimodal functions and plateaus. Swarm Intell 3, 35–68 (2009). https://doi.org/10.1007/s11721-008-0023-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11721-008-0023-3

Keywords

  • Ant colony optimization
  • MMAS
  • Runtime analysis
  • Theory