Local function approximation in evolutionary algorithms for the optimization of costly functions
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 offsprin...
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| Veröffentlicht in: | IEEE transactions on evolutionary computation Jg. 8; H. 5; S. 490 - 505 |
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| Hauptverfasser: | , |
| Format: | Journal Article |
| Sprache: | Englisch |
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New York, NY
IEEE
01.10.2004
Institute of Electrical and Electronics Engineers The Institute of Electrical and Electronics Engineers, Inc. (IEEE) |
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| ISSN: | 1089-778X, 1941-0026 |
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| Abstract | 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. |
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| AbstractList | 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. 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/ges/4. 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 greater than or equal to 4. |
| Author | Regis, R.G. Shoemaker, C.A. |
| Author_xml | – sequence: 1 givenname: R.G. surname: Regis fullname: Regis, R.G. organization: Sch. of Oper.s Res. & Ind. Eng., Cornell Univ., Ithaca, NY, USA – sequence: 2 givenname: C.A. surname: Shoemaker fullname: Shoemaker, C.A. |
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| Cites_doi | 10.1023/A:1008306431147 10.1093/oso/9780195099713.001.0001 10.1061/(ASCE)0733-9496(1999)125:1(54) 10.1029/91wr02826 10.1145/321127.321128 10.1002/0471749214 10.1109/20.582647 10.1214/ss/1177012413 10.1109/ICEC.1996.542329 10.1007/978-1-4613-8122-8 10.1109/CEC.1999.785531 10.2307/1268522 10.1023/A:1011584207202 10.2514/6.1996-4019 10.2514/6.2000-4841 10.1007/978-3-0348-5927-1 10.2172/7192422 10.2514/6.2000-4923 10.1109/CEC.2002.1006266 10.2514/6.1997-1230 10.2514/6.1996-714 10.1109/CEC.2001.934445 10.1109/TEVC.2002.800884 10.1007/978-1-4613-1997-9 10.1109/CEC.2000.870752 10.1093/oso/9780198534396.003.0003 10.1023/A:1012771025575 10.1016/S0378-3758(00)00105-1 10.1109/20.767363 10.1007/978-3-0348-8696-3_14 10.1023/A:1011255519438 |
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| Keywords | Statistical analysis Evolutionary algorithm evolutionary algorithm (EA) Numerical method Nonlinear problems Symmetric function Approximation algorithm quadratic regression Variance analysis Modeling Optimization radial basis function (RBF) Radial basis function Finite element method Algorithm performance Genetic algorithm function approximation Approximation by function Quadratic approximation Cost function Latin hypercube Objective function Costly function |
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| SubjectTerms | Algorithms Analysis of variance Applied sciences Approximation Approximation algorithms Artificial intelligence Computer science; control theory; systems Design engineering Design for experiments Design optimization Evolutionary algorithms Evolutionary computation Exact sciences and technology Finite element methods Function approximation Hypercubes Interpolation Mathematical analysis Mathematical models Mathematical programming Operational research and scientific management Operational research. Management science Optimization Problem solving, game playing Testing |
| Title | Local function approximation in evolutionary algorithms for the optimization of costly functions |
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