Premature Convergence in Constrained Continuous Search Spaces

The optimum of numerical problems quite often lies on the constraint boundary or even in a vertex of the feasible search space. In such cases the evolutionary algorithm (EA) frequently suffers from premature convergence because of a low success probability near the constraint boundaries. We analyze premature fitness stagnation and the success rates experimentally for an EA using self-adaptive step size control. For a (1+1)-EA with a Rechenberg-like step control mechanism we prove premature step size reduction at the constraint boundary. The proof is based on a success rate analysis considering a simplified mutation distribution model. From the success rates and the possible state transitions, the expected step size change can be derived at each step. We validate the theoretical model with an experimental analysis.

[1]  Ko-Hsin Liang,et al.  An Experimental Investigation of Self-Adaptation in Evolutionary Programming , 1998, Evolutionary Programming.

[2]  Hans-Paul Schwefel,et al.  Evolution and optimum seeking , 1995, Sixth-generation computer technology series.

[3]  Zbigniew Michalewicz,et al.  Parameter Setting in Evolutionary Algorithms , 2007, Studies in Computational Intelligence.

[4]  A. E. Eiben,et al.  Evolutionary Programming VII , 1998, Lecture Notes in Computer Science.

[5]  Hans-Paul Schwefel,et al.  Evolution and Optimum Seeking: The Sixth Generation , 1993 .

[6]  Christopher R. Houck,et al.  On the use of non-stationary penalty functions to solve nonlinear constrained optimization problems with GA's , 1994, Proceedings of the First IEEE Conference on Evolutionary Computation. IEEE World Congress on Computational Intelligence.

[7]  Oliver Kramer,et al.  On three new approaches to handle constraints within evolution strategies , 2006, Natural Computing.

[8]  Christopher Stone,et al.  Strategy Parameter Variety In Self-adaptation Of Mutation Rates , 2002, GECCO.

[9]  Hans-Paul Schwefel,et al.  Evolution strategies – A comprehensive introduction , 2002, Natural Computing.

[10]  Nikolaus Hansen,et al.  An Analysis of Mutative -Self-Adaptation on Linear Fitness Functions , 2006, Evolutionary Computation.

[11]  Günter Rudolph,et al.  Self-adaptive mutations may lead to premature convergence , 2001, IEEE Trans. Evol. Comput..

[12]  Hans-Georg Beyer,et al.  Self-Adaptation in Evolutionary Algorithms , 2007, Parameter Setting in Evolutionary Algorithms.

[13]  Oliver Kramer,et al.  A new mutation operator for evolution strategies for constrained problems , 2005, 2005 IEEE Congress on Evolutionary Computation.