Algorithmica

, Volume 59, Issue 3, pp 409–424

# Combining Markov-Chain Analysis and Drift Analysis

The (1+1) Evolutionary Algorithm on Linear Functions Reloaded
• Jens Jägersküpper
Article

## Abstract

In their seminal article Droste, Jansen, and Wegener (Theor. Comput. Sci. 276:51–82, 2002) consider a basic direct-search heuristic with a global search operator, namely the so-called (1+1) Evolutionary Algorithm ((1+1) EA). They present the first theoretical analysis of the (1+1) EA’s expected runtime for the class of linear functions over the search space {0,1} n . In a rather long and involved proof they show that, for any linear function, the expected runtime is O(nlog n), i.e., that there are two constants c and n′ such that, for nn′, the expected number of iterations until a global optimum is generated is bounded above by cnlog 2 n. However, neither c nor n′ are specified—they would be pretty large. Here we reconsider this optimization scenario to demonstrate the potential of an analytical method that makes use of the distribution of the evolving candidate solution over the search space {0,1} n . Actually, an invariance property of this distribution is proved, which is then used to obtain a significantly improved bound on the drift, namely the expected change of a potential function, here the number of bits set correctly. Finally, this better estimate of the drift enables an upper bound on the expected number of iterations of 3.8nlog 2 n+7.6log 2 n for n≥2.

## Keywords

Heuristic optimization Direct search Evolutionary algorithms Genetic algorithms Randomized algorithms Drift analysis Markov-Chain analysis

## References

1. 1.
Droste, S., Jansen, T., Wegener, I.: On the analysis of the (1+1) evolutionary algorithm. Theor. Comput. Sci. 276(1–2), 51–82 (2002)
2. 2.
Giel, O., Wegener, I.: Evolutionary algorithms and the maximum matching problem. In: Proc. 20th Int’l Symposium on Theoretical Aspects of Computer Science (STACS). LNCS, vol. 2607, pp. 415–426. Springer, Berlin (2003) Google Scholar
3. 3.
He, J., Yao, X.: Drift analysis and average time complexity of evolutionary algorithms. Artif. Intell. 127(1), 57–85 (2001)
4. 4.
He, J., Yao, X.: Erratum to: Drift analysis and average time complexity of evolutionary algorithms (He and Xao 2001). Artif. Intell. 140(1/2), 245–248 (2002)
5. 5.
He, J., Yao, X.: A study of drift analysis for estimating computation time of evolutionary algorithms. Nat. Comput. 3(1), 21–35 (2004)
6. 6.
Jägersküpper, J.: Algorithmic analysis of a basic evolutionary algorithm for continuous optimization. Theor. Comput. Sci. 379(3), 329–347 (2007)
7. 7.
Jägersküpper, J.: A blend of Markov-chain and drift analysis. In: Proc. 10th Int’l Conference on Parallel Problem Solving from Nature. Lecture Notes in Computer Science, vol. 5199, pp. 41–51. Springer, Berlin (2008)
8. 8.
Jägersküpper, J., Witt, C.: Rigorous runtime analysis of a (μ+1) ES for the Sphere function. In: Proc. 2005 Genetic and Evolutionary Computation Conference (GECCO), pp. 849–856. ACM, New York (2005)
9. 9.
Rudolph, G.: Finite Markov chain results in evolutionary computation: A tour d’horizon. Fundam. Inform. 35(1–4), 67–89 (1998)
10. 10.
Wegener, I.: Methods for the analysis of evolutionary algorithms on pseudo-Boolean functions. In: Sarker, R., Mohammadian, M., Yao, X. (eds.) Evolutionary Optimization. Int’l Series in Operations Research & Management Science, vol. 48, pp. 349–369. Kluwer Academic, Dordrecht (2003) Google Scholar
11. 11.
Wegener, I., Witt, C.: On the optimization of monotone polynomials by simple randomized search heuristics. Comb. Probab. Comput. 14(1–2), 225–247 (2005)