Abstract
We consider the (1+1) Evolution Strategy, a simple evolutionary algorithm for continuous optimization problems, using so-called Gaussian mutations and the 1/5-rule for the adaptation of the mutation strength. Here, the function \(f\colon\mathbb{R}^{n}\to\mathbb{R}\) to be minimized is given by a quadratic form f(x) = x ⊤ Qx, where Q ∈ ℝn×n is a positive definite diagonal matrix and x denotes the current search point. This is a natural extension of the well-known Sphere-function (Q=I). Thus, very simple unconstrained quadratic programs are investigated, and the question is addressed how Q effects the runtime. For this purpose, quadratic forms
with ξ=ω(1), i. e. 1/ξ→0 as n→∞, and ξ=poly(n) are investigated exemplarily. It is proved that the optimization very quickly stabilizes and that, subsequently, the runtime (defined as the number of f-evaluations) to halve the approximation error is Θ(ξ
n). Though ξ
n=poly(n), this result actually shows that the evolving search point indeed creeps along the “gentlest descent” of the ellipsoidal fitness landscape.
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Beyer, H.-G.: Towards a theory of evolution strategies: Progress rates and quality gain for (1+, λ)-strategies on (nearly) arbitrary fitness functions. In: Davidor, Y., Männer, R., Schwefel, H.-P. (eds.) PPSN 1994. LNCS, vol. 866, pp. 58–67. Springer, Heidelberg (1994)
Beyer, H.-G.: On the performance of (1, λ)-evolution strategies for the ridge function class. IEEE Transactions on Evolutionary Computation 5(3), 218–235 (2001a)
Beyer, H.-G.: The Theory of Evolution Strategies. Springer, Heidelberg (2001b)
Bienvenue, A., Francois, O.: Global convergence for evolution strategies in spherical problems: Some simple proofs and difficulties. Theoretical Computer Science 306, 269–289 (2003)
Droste, S., Jansen, T., Wegener, I.: On the analysis of the (1+1) evolutionary algorithm. Theoretical Computer Science 276, 51–82 (2002)
Giel, O., Wegener, I.: Evolutionary algorithms and the maximum matching problem. In: Alt, H., Habib, M. (eds.) STACS 2003. LNCS, vol. 2607, pp. 415–426. Springer, Heidelberg (2003)
Greenwood, G.W., Zhu, Q.J.: Convergence in evolutionary programs with self-adaptation. Evolutionary Computation 9(2), 147–157 (2001)
Hansen, N., Ostermeier, A.: Adapting arbitrary normal mutation distributions in evolution strategies: The covariance matrix adaptation. In: Proc. of the IEEE Int’l Conference on Evolutionary Computation (ICEC), pp. 312–317 (1996)
Hoeffding, W.: Probability inequalities for sums of bounded random variables. American Statistical Association Journal 58(301), 13–30 (1963)
Hofri, M.: Probabilistic Analysis of Algorithms. Springer, Heidelberg (1987)
Jägersküpper, J.: Analysis of a simple evolutionary algorithm for the minimization in euclidean spaces. Technical Report CI-140/02, Univ. Dortmund, SFB 531 (2002), http://sfbci.uni-dortmund.de →Publications→Tech-Reports.
Jägersküpper, J.: Analysis of a simple evolutionary algorithm for minimization in Euclidean spaces. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds.) ICALP 2003. LNCS, vol. 2719, pp. 1068–1079. Springer, Heidelberg (2003)
Neumann, F., Wegener, I.: Randomized local search, evolutionary algorithms, and the minimum spanning tree problem. In: Deb, K., et al. (eds.) GECCO 2004. LNCS, vol. 3102, pp. 713–724. Springer, Heidelberg (2004)
Rechenberg, I.: Evolutionsstrategie. Frommann-Holzboog, Stuttgart (1973)
Rudolph, G.: Convergence Properties of Evolutionary Algorithms. Verlag Dr. Kovač, Hamburg (1997)
Scharnow, J., Tinnefeld, K., Wegener, I.: Fitness landscapes based on sorting and shortest paths problems. 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. 54–63. Springer, Heidelberg (2002)
Schwefel, H.-P.: Evolution and Optimum Seeking. In: Fülöp, Z., Gecseg, F. (eds.) ICALP 1995. LNCS, vol. 944, pp. 64–78. Springer, Heidelberg (1995)
Wegener, I.: Theoretical aspects of evolutionary algorithms. In: Orejas, F., Spirakis, P.G., van Leeuwen, J. (eds.) ICALP 2001. LNCS, vol. 2076, pp. 64–78. Springer, Heidelberg (2001)
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Jägersküpper, J. (2005). Rigorous Runtime Analysis of the (1+1) ES: 1/5-Rule and Ellipsoidal Fitness Landscapes. In: Wright, A.H., Vose, M.D., De Jong, K.A., Schmitt, L.M. (eds) Foundations of Genetic Algorithms. FOGA 2005. Lecture Notes in Computer Science, vol 3469. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11513575_14
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DOI: https://doi.org/10.1007/11513575_14
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