Local function approximation in evolutionary algorithms for the optimization of costly functions
Local function approximation in evolutionary algorithms for the optimization of costly functions
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DOI：
10.1109/tevc.2004.835247
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发表时间：
2004-10
影响因子：
14.3
通讯作者：
R. Regis;C. Shoemaker
中科院分区：
文献类型：
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作者：
R. Regis;C. Shoemaker
我们开发了一种连续昂贵函数的优化方法，该方法使用空间填充实验设计和局部函数逼近来减少进化算法中函数评估的次数。我们的方法是通过在k最近的先前评估点上拟合函数近似模型来估计后代的目标函数值，其中k=(d+1)(d+2)/2， d是问题的维度。估计的函数值用于筛选后代，以确定最有希望进行功能评估的后代。为了拟合函数逼近模型，采用对称拉丁超立方体设计（SLHD）确定函数求值的初始点。我们比较了局部二次逼近进化策略（ES）、局部三次径向基函数（RBF）插值进化策略（ES）、初始亲本群体来自SLHD的进化策略（ES）和传统进化策略（ES）的性能。将这些算法应用于一个涉及复杂非线性有限元模拟模型的12维地下水生物修复问题。在Dixon-Szego测试函数和十维（10-D） Rastrigin和Ackley测试函数上比较了这些算法的性能。所有的比较都涉及方差分析（ANOVA）和同时置信区间的计算。结果表明，除Goldstein-Price外，局部近似ES算法在所有Dixon-Szego测试函数上均显著优于传统ES算法和slhd初始化ES算法。然而，对于更困难的10-D和12-D函数，只有三次RBF方法能够成功地提高ES的性能。此外，结果还表明，三次RBF方法在所有测试函数上都优于二次逼近方法，并且对于维数为d/spl /4的所有测试函数，性能差异具有统计学意义。
We develop an approach for the optimization of continuous costly functions that uses a space-filling experimental design and local function approximation to reduce the number of function evaluations in an evolutionary algorithm. Our approach is to estimate the objective function value of an offspring by fitting a function approximation model over the k nearest previously evaluated points, where k=(d+1)(d+2)/2 and d is the dimension of the problem. The estimated function values are used to screen offspring to identify the most promising ones for function evaluation. To fit function approximation models, a symmetric Latin hypercube design (SLHD) is used to determine initial points for function evaluation. We compared the performance of an evolution strategy (ES) with local quadratic approximation, an ES with local cubic radial basis function (RBF) interpolation, an ES whose initial parent population comes from an SLHD, and a conventional ES. These algorithms were applied to a twelve-dimensional (12-D) groundwater bioremediation problem involving a complex nonlinear finite-element simulation model. The performances of these algorithms were also compared on the Dixon-Szego test functions and on the ten-dimensional (10-D) Rastrigin and Ackley test functions. All comparisons involve analysis of variance (ANOVA) and the computation of simultaneous confidence intervals. The results indicate that ES algorithms with local approximation were significantly better than conventional ES algorithms and ES algorithms initialized by SLHDs on all Dixon-Szego test functions except for Goldstein-Price. However, for the more difficult 10-D and 12-D functions, only the cubic RBF approach was successful in improving the performance of an ES. Moreover, the results also suggest that the cubic RBF approach is superior to the quadratic approximation approach on all test functions and the difference in performance is statistically significant for all test functions with dimension d/spl ges/4.