Abstract
Reducing the number of evaluations of expensive fitness functions is one of the main concerns in evolutionary algorithms, especially when working with instances of contemporary engineering problems. As an alternative to this efficiency constraint, surrogate-based methods are grounded in the construction of approximate models that estimate the solutions’ fitness by modeling the relationships between solution variables and their performance. This paper proposes a methodology based on granular computing for the construction of surrogate models for evolutionary algorithms. Under the proposed method, granules are associated with representative solutions of the problem under analysis. New solutions are evaluated with the expensive (original) fitness function only if they are not already covered by an existing granule. The parameters defining granules are periodically adapted as the search goes on using a neuro-fuzzy network that does not only reduce the number of fitness function evaluations, but also provides better convergence capabilities. The proposed method is evaluated on classical benchmark functions and on a recent benchmark created to test large-scale optimization models. Our results show that the proposed method considerably reduces the actual number of fitness function evaluations without significantly degrading the quality of solutions.
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Notes
One should note that although we have chosen genetic algorithms for the implementation of the proposed method, the same approach can be implemented with other population-based search strategies (e.g., particle swarm optimization or differential evolution).
In Akbarzadeh et al. (2008), the radius of the Gaussian similarity function was proposed as:
that is, in inverse proportion to the exponential of the fitness value of the granule’s center, relating fitness function values and similarity measures in the similarity function. Since similarity values and fitness values do not necessarily lie in the same scale, it makes no sense to use the definition for \(\sigma _{k}\) from Eq. (3), see Cruz-Vega and Escalante (2015) for details. Instead, in our work, \(\sigma _{k}\) is only related with measures in the same scale, that is distances in the variables space, see Eq. (2). In this way, we avoid the construction of granules that could have undefined values.
We considered the GA implementation of the global optimization toolbox of Matlab (Goldberg and Holland 1988).
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Acknowledgments
The first author was supported by CONACyT under a postdoctoral scholarship (CVU No. 162347).
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Communicated by V. Loia.
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Cruz-Vega, I., Escalante, H.J., Reyes, C.A. et al. Surrogate modeling based on an adaptive network and granular computing. Soft Comput 20, 1549–1563 (2016). https://doi.org/10.1007/s00500-015-1605-9
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DOI: https://doi.org/10.1007/s00500-015-1605-9