# Continuous fitness landscape analysis using a chaos-based random walk algorithm

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## Abstract

Extensive research on heuristic algorithms has proved their potential in solving complex optimization problems. However, it is not easy to choose the best heuristic technique for solving a particular problem. Fitness landscape analysis is used for understanding the problem characteristics based on which the best-suited algorithm for the problem can be chosen. Compared to the literature on discrete search spaces, only a few significant works have been undertaken on landscape analysis in continuous search spaces. Random walk (RW) algorithm has been used for generating sample points in the search space, and fitness landscape is created based on the relative fitness of the neighboring sample points. This paper proposes a chaos-based random walk algorithm, called as the chaotic random walk (CRW), applied in continuous search space to generate the landscape structure for a problem. The chaotic map is used to generate the chaotic pseudorandom numbers for determining variable scaled step size and direction of the proposed RW algorithm. Histogram analysis demonstrates better coverage of search space by the CRW algorithm compared to the simple and progressive random walk algorithms. In addition, we test the efficiency of the proposed method by quantifying the ruggedness and deception of a problem using entropy and fitness distance correlation measures. Experiments are conducted on the IEEE Congers on Evolutionary Computing 2013 benchmark functions in continuous search space having different levels of complexity. Extensive experiments indicate the capability for generating landscape structure on the continuous search space and efficiency of the proposed method to investigate the structural features of fitness landscapes.

## Keywords

Random walk algorithm Fitness landscape analysis Continuous random walk algorithm Chaotic map Entropy measure Fitness distance correlation## Notes

### Compliance with ethical standards

### Conflict of interest

No potential conflict of interest to be declared.

## References

- Blum C, Li X (2008) Swarm intelligence in optimization. In: Blum C, Merkle D (eds) Swarm intelligence. Natural computing series. Springer, Berlin, pp 43–85Google Scholar
- Caraffini F, Neri F, Picinali L (2014) An analysis on separability for memetic computing automatic design. Inf Sci 265:1–22MathSciNetCrossRefGoogle Scholar
- Huang SY, Zou XW, Jin ZZ (2002) Directed random walks in continuous space. Phys Rev E 65:052105. doi: 10.1103/PhysRevE.65.052105 CrossRefGoogle Scholar
- Iba T, Shimonishi K (2011) The origin of diversity: thinking with chaotic walk. In: Proceedings of the eighth international conference on complex systems, pp 447–461Google Scholar
- Jones T, Forrest S (1995) Fitness distance correlation as a measure of problem difficulty for genetic algorithms. In: Sixth international conference on genetic algorithms, pp 184–192Google Scholar
- Jong KD (2005) Parameter setting in eas: a 30 year perspective. In: Lobo FG, Lima CF, Michalewicz Z (eds) Parameter setting in evolutionary algorithms. Studies in Computational Intelligence, vol 54. Springer, Berlin, pp 1–18Google Scholar
- Liang JJ, Qu BY, Suganthan PN, Hernandez-Diaz AG (2013) Problem definitions and evaluation criteria for the CEC 2013 special session on real-parameter optimization. Technical Report DAMTP 2000/NA10. Nanyang Technological University. SingaporeGoogle Scholar
- Lu G, Li J, Yao X (2011) Fitness-probability cloud and a measure of problem hardness for evolutionary algorithms. In: 11th European conference on evolutionary computation in combinatorial optimization (EvoCOP’11), pp 108–117Google Scholar
- Lunacek M, Whitley D (2006) The dispersion metric and the cma evolution strategy. In: 8th Annual conference on genetic and evolutionary computation, pp 477–484Google Scholar
- Malan KM, Engelbrecht AP (2009) Quantifying ruggedness of continuous landscapes using entropy. In: IEEE Congress on evolutionary computation (CEC’09), pp 1440–1447Google Scholar
- Malan KM, Engelbrecht AP (2013) A survey of techniques for characterising fitness landscapes and some possible ways forward. Inf Sci 241:148–163CrossRefGoogle Scholar
- Malan KM, Engelbrecht AP (2014a) Fitness landscape analysis for metaheuristic performance prediction. In: Richter H, Engelbrecht AP (eds) Recent advances in the theory and application of fitness landscapes. Emergence, Complexity and Computation, vol 6. Springer, Berlin, pp 103–132Google Scholar
- Malan KM, Engelbrecht AP (2014b) A progressive random walk algorithm for sampling continuous fitness landscapes. In: IEEE congress on evolutionary computation (CEC’14), pp 2507–2514Google Scholar
- May RM (1976) Simple mathematical models with very complicated dynamics. Nature 261:459–467Google Scholar
- Mersmann O, Bischl B, Trautmann H, Preuss M, Weihs C, Rudolph G (2011) Exploratory landscape analysis. In: 13th Annual conference on genetic and evolutionary computation (GECCO’11), pp 829–836Google Scholar
- Merz P, Freisleben B (2000) Fitness landscape analysis and memetic algorithms for the quadratic assignment problem. IEEE Trans Evol Comput 4:337–352CrossRefGoogle Scholar
- Mohammad HTN, Bennett AP (2014) On the landscape of combinatorial optimization problems. IEEE Trans Evol Comput 18:420–434CrossRefGoogle Scholar
- Morgan R, Gallagher M (2012) Length scale for characterising continuous optimization problems. In: 12th International conference on parallel problem solving from nature—part I, pp 407–416Google Scholar
- Morgan R, Gallagher M (2014) Sampling techniques and distance metrics in high dimensional continuous landscape analysis: Limitations and improvements. IEEE Trans Evol Comput 18:456–461CrossRefGoogle Scholar
- Munoz M, Kirley M, Halgamuge S (2014) Exploratory landscape analysis of continuous space optimization problems using information content. IEEE Trans Evol Comput. doi: 10.1109/TEVC.2014.2302006
- Munoz MA, Kirley M, Halgamuge S (2012) Landscape characterization of numerical optimization problems using biased scattered data. In: IEEE congress on evolutionary computation (CEC’12), pp 1–8Google Scholar
- Naudts B, Kallel L (2000) A comparison of predictive measures of problem difficulty in evolutionary algorithms. IEEE Trans Evol Comput 4:1–15CrossRefGoogle Scholar
- Pearson K (1905) The problem of the random walk. Nature 72:294, 318, 342zbMATHGoogle Scholar
- Peitgen H, Jurgens H, Saupe D (1992) Chaos and fractals. Springer, BerlinCrossRefzbMATHGoogle Scholar
- Reeves CR, Rowe JE (2002) Genetic algorithms—principles and perspectives: a guide to GA theory. Kluwer Academic Publishers, NorwellzbMATHGoogle Scholar
- Reidys CM, Stadler PF (2001) Neutrality in fitness landscapes. Appl Math Comput 117:321–350MathSciNetzbMATHGoogle Scholar
- Rose H, Ebeling W, Asselmeyer T (1996) The density of states—a measure of the difficulty of optimisation problems. In: 4th International conference on parallel problem solving from nature, pp 208–217Google Scholar
- Steer K, Wirth A, Halgamuge S (2008) Information theoretic classification of problems for metaheuristics. In: Simulated evolution and learning, pp 319–328Google Scholar
- Tavares J, Pereira FB, Costa E (2008) Multidimensional knapsack problem: a fitness landscape analysis. IEEE Trans Syst Man Cybern Part B Cybern 38:604–616CrossRefGoogle Scholar
- Tavazoei MS, Haeri M (2007) Comparison of different one-dimensional maps as chaotic search pattern in chaos optimization algorithms. Appl Math Comput 187:1076–1085MathSciNetzbMATHGoogle Scholar
- Vanneschi L, Clergue M, Collard P, Tomassini M, Verel S (2004) Fitness clouds and problem hardness in genetic programming. In: Genetic and evolutionary computation (GECCO’04), pp 690–701Google Scholar
- Vassilev VK, Fogarty TC, Miller JF (2000) Information characteristics and the structure of landscapes. Evol Comput 8:31–60CrossRefGoogle Scholar
- Vassilev VK, Fogarty TC, Miller JF (2003) Smoothness, ruggedness and neutrality of fitness landscapes: from theory to application. In: Advances in evolutionary computing. Natural Computing series. Springer, Berlin, pp 3–44Google Scholar
- Venkatesan A et al (2013) Computational approach for protein structure prediction. Healthc Inf Res 19:137–147CrossRefGoogle Scholar
- Wolpert D, Macready W (1997) No free lunch theorems for optimization. IEEE Trans Evol Comput 1:67–82CrossRefGoogle Scholar
- Yuan X, Dai X, Wu L (2015) A mutative-scale pseudo-parallel chaos optimization algorithm. Soft Comput 19:1215–1227CrossRefGoogle Scholar