代理模型辅助的昂贵优化算法研究

昂贵优化问题（Expensive Optimization Problems，EOPs）在现实应用中广泛存在。其目标函数通常没有解析表达式，需要依赖复杂的计算模型进行评估。每次评估都耗费大量计算资源。为此，一些特定的优化算法，如代理模型辅助的进化算法（Surrogate-Assisted Evolutionary Algorithms，SAEAs），被提出以在有限的评估次数内找到较好的解。然而，随着决策变量维度的升高、决策变量的多样性等复杂情况的出现，现有算法存在效率低下和难以找到满意解的问题。为应对这些挑战，本研究聚焦于两类EOPs，即连续变量的EOPs 和混合连续与类别变量的EOPs，开展以下研究：

（1） 针对连续变量的EOPs，提出了一种名为IDRCEA 的SAEA。IDRCEA利用了个体-分布搜索策略生成整个种群的子代用于搜索空间的有效探索。通过基于回归-分类预筛选机制挑选高质量的子代进行昂贵的评估。在一系列复杂和高维的基准问题上，验证了IDRCEA 的显著优势。

（2） 为了进一步提高SAEA 在求解连续变量的EOPs 的泛化能力，开发了一种名为AutoSAEA 的SAEA，其具备模型和填充准则的自动配置特性。AutoSAEA将模型和填充准则的选择建模为一个两级多臂赌博问题。第一和第二级协作选择代理模型和填充准则。通过使用两级奖励机制来衡量代理模型和填充准则的效用，两级置信上界方法以在线方式选择模型和填充准则。在许多的复杂基准问题和一个实际油藏生产优化问题上，证实了AutoSAEA 的性能。

（3） 针对连续变量与类别变量混合的EOPs，提出了一种名为CEEO 的SAEA。CEEO 设计了一个定制的搜索算子，以有效地探索混合变量搜索空间。此外，开发了一种专门用于连续和类别变量的相似性度量方法，以提高径向基函数模型的逼近精度。在三组不同的混合变量的基准问题上验证了CEEO 的有效性。

[1] LI G, WANG Z, GONG M. Expensive optimization via surrogate-assisted and model-free evolutionary optimization[J]. IEEE Transactions on Systems, Man, and Cybernetics: Systems,2022, 53(5): 2758-2769.
[2] LIU Y, LIU J, JIN Y. Surrogate-assisted multipopulation particle swarm optimizer for high dimensional expensive optimization[J]. IEEE Transactions on Systems, Man, and Cybernetics: Systems, 2021, 52(7): 4671-4684.
[3] 韩忠华, 许晨舟, 乔建领, 等. 基于代理模型的高效全局气动优化设计方法研究进展[J]. 航空学报, 2020, 41(5): 25-65.
[4] MITCHELL M. An introduction to genetic algorithms[M]. MIT press, 1998.
[5] EBERHART R, KENNEDY J. A new optimizer using particle swarm theory[C]//MHS’95. Proceedings of the Sixth International Symposium on Micro Machine and Human Science. IEEE, 1995: 39-43.
[6] DE BONET J, ISBELL C, VIOLA P. MIMIC: Finding optima by estimating probability densities[J]. Advances in Neural Information Processing Systems, 1996, 9.
[7] WANG X, WANG G G, SONG B, et al. A novel evolutionary sampling assisted optimization method for high-dimensional expensive problems[J]. IEEE Transactions on Evolutionary Computation, 2019, 23(5): 815-827.
[8] LIU J, WANG Y, SUN G, et al. Multisurrogate-assisted ant colony optimization for expensive optimization problems with continuous and categorical variables[J]. IEEE Transactions on Cybernetics, 2021, 52(11): 11348-11361.
[9] LIU Y, WANG H. Surrogate-assisted hybrid evolutionary algorithm with local estimation of distribution for expensive mixed-variable optimization problems[J]. Applied Soft Computing, 2023, 133: 109957.
[10] JIN Y. Surrogate-assisted evolutionary computation: Recent advances and future challenges[J]. Swarm and Evolutionary Computation, 2011, 1(2): 61-70.
[11] 孙超利, 李贞, 金耀初. 模型辅助的计算费时进化高维多目标优化[J]. 自动化学报, 2022, 48(4): 1119-1128.
[12] FRAZIER P I. A tutorial on Bayesian optimization[J]. arXiv preprint arXiv:1807.02811, 2018.
[13] ZHANG Q, LIU W, TSANG E, et al. Expensive multiobjective optimization by MOEA/D with Gaussian process model[J]. IEEE Transactions on Evolutionary Computation, 2009, 14(3): 456-474.
[14] DENNIS J, TORCZON V. Managing approximation models in optimization[J]. Multidisciplinary design optimization: State-of-the-art, 1997, 5: 330-347.
[15] JONES D R, SCHONLAU M, WELCH W J. Efficient global optimization of expensive blackbox functions[J]. Journal of Global Optimization, 1998, 13(4): 455.
[16] LIU B, ZHANG Q, GIELEN G G. A Gaussian process surrogate model assisted evolutionary algorithm for medium scale expensive optimization problems[J]. IEEE Transactions on Evolutionary Computation, 2013, 18(2): 180-192.
[17] KNOWLES J. ParEGO: A hybrid algorithm with on-line landscape approximation for expensive multiobjective optimization problems[J]. IEEE Transactions on Evolutionary Computation, 2006, 10(1): 50-66.
[18] ZHANG Q, LI H. MOEA/D: A multiobjective evolutionary algorithm based on decomposition [J]. IEEE Transactions on Evolutionary Computation, 2007, 11(6): 712-731.
[19] LYU W, YANG F, YAN C, et al. Batch Bayesian optimization via multi-objective acquisition ensemble for automated analog circuit design[C]//International Conference on Machine Learning. PMLR, 2018: 3306-3314.
[20] ZHANG S, YANG F, YAN C, et al. An efficient batch-constrained Bayesian optimization approach for analog circuit synthesis via multiobjective acquisition ensemble[J]. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 2021, 41(1): 1-14.
[21] HOFFMAN M, BROCHU E, DE FREITAS N, et al. Portfolio Allocation for Bayesian Optimization.[C]//UAI. 2011: 327-336.
[22] CHEN J, LUO F, LI G, et al. Batch Bayesian optimization with adaptive batch acquisition functions via multi-objective optimization[J]. Swarm and Evolutionary Computation, 2023, 79: 101293.
[23] LI G, XIE L, WANG Z, et al. Evolutionary algorithm with individual-distribution search strategy and regression-classification surrogates for expensive optimization[J]. Information Sciences, 2023, 634: 423-442.
[24] WANG H, XU H, ZHANG Z. High-Dimensional Multi-Objective Bayesian Optimization With Block Coordinate Updates: Case Studies in Intelligent Transportation System[J]. IEEE Transactions on Intelligent Transportation Systems, 2024, 25(1): 884-895.
[25] GOEL T, HAFKTA R T, SHYY W. Comparing error estimation measures for polynomial and kriging approximation of noise-free functions[J]. Structural and Multidisciplinary Optimization, 2009, 38(5): 429-442.
[26] ZOUHAL L M, DENOEUX T. An evidence-theoretic k-NN rule with parameter optimization[J]. IEEE Transactions on Systems, Man, and Cybernetics, Part C (Applications and Reviews), 1998, 28(2): 263-271.
[27] WEI F F, CHEN W N, YANG Q, et al. A classifier-assisted level-based learning swarm optimizer for expensive optimization[J]. IEEE Transactions on Evolutionary Computation, 2020, 25(2):219-233.
[28] CLARKE S M, GRIEBSCH J H, SIMPSON T W. Analysis of support vector regression for approximation of complex engineering analyses[J]. Journal of Mechanical Design, 2005, 127 (6): 1077-1087.
[29] JIN Y, OLHOFER M, SENDHOFF B. A framework for evolutionary optimization with approximate fitness functions[J]. IEEE Transactions on Evolutionary Computation, 2002, 6(5): 481-494.
[30] TIAN J, TAN Y, ZENG J, et al. Multiobjective infill criterion driven Gaussian process-assisted particle swarm optimization of high-dimensional expensive problems[J]. IEEE Transactions on Evolutionary Computation, 2018, 23(3): 459-472.
[31] LI F, CAI X, GAO L, et al. A surrogate-assisted multiswarm optimization algorithm for high dimensional computationally expensive problems[J]. IEEE Transactions on Cybernetics, 2020, 51(3): 1390-1402.
[32] WANG W, LIU H L, TAN K C. A surrogate-assisted differential evolution algorithm for high dimensional expensive optimization problems[J]. IEEE Transactions on Cybernetics, 2022, 53 (4): 2685-2697.
[33] WANG H, JIN Y, DOHERTY J. Committee-based active learning for surrogate-assisted particle swarm optimization of expensive problems[J]. IEEE Transactions on Cybernetics, 2017, 47(9): 2664-2677.
[34] SONODA T, NAKATA M. Multiple classifiers-assisted evolutionary algorithm based on decomposition for high-dimensional multiobjective problems[J]. IEEE Transactions on Evolutionary Computation, 2022, 26(6): 1581-1595.
[35] WU X, LIN Q, LI J, et al. An ensemble surrogate-based coevolutionary algorithm for solving large-scale expensive optimization problems[J]. IEEE Transactions on Cybernetics, 2023, 53 (9): 5854-5866.
[36] SONG Z, WANG H, HE C, et al. A Kriging-assisted two-archive evolutionary algorithm for expensive many-objective optimization[J]. IEEE Transactions on Evolutionary Computation, 2021, 25(6): 1013-1027.
[37] LIU Z, WANG H, JIN Y. Performance indicator-based adaptive model selection for offline data driven multiobjective evolutionary optimization[J]. IEEE Transactions on Cybernetics, 2023, 53(10): 6263-6276.
[38] ZHEN H, GONG W, WANG L. Evolutionary sampling agent for expensive problems[J]. IEEE Transactions on Evolutionary Computation, 2023, 27(3): 716-727.
[39] ZHANG J, ZHOU A, ZHANG G. A classification and Pareto domination based multiobjective evolutionary algorithm[C]//2015 IEEE Congress on Evolutionary Computation (CEC). IEEE, 2015: 2883-2890.
[40] SUN C, JIN Y, CHENG R, et al. Surrogate-assisted cooperative swarm optimization of high dimensional expensive problems[J]. IEEE Transactions on Evolutionary Computation, 2017, 21(4): 644-660.
[41] YU H, TAN Y, ZENG J, et al. Surrogate-assisted hierarchical particle swarm optimization[J]. Information Sciences, 2018, 454: 59-72.
[42] ZHEN H, GONG W, WANG L, et al. Two-stage data-driven evolutionary optimization for high dimensional expensive problems[J]. IEEE Transactions on Cybernetics, 2023, 53(4): 2368-2379.
[43] CAI X, GAO L, LI X. Efficient generalized surrogate-assisted evolutionary algorithm for high dimensional expensive problems[J]. IEEE Transactions on Evolutionary Computation, 2019, 24(2): 365-379.
[44] WANG Z, ZHEN H L, DENG J, et al. Multiobjective optimization-aided decision-making system for large-scale manufacturing planning[J]. IEEE Transactions on Cybernetics, 2021, 52(8): 8326-8339.
[45] GUO D, JIN Y, DING J, et al. Heterogeneous ensemble-based infill criterion for evolutionary multiobjective optimization of expensive problems[J]. IEEE Transactions on Cybernetics, 2018, 49(3): 1012-1025.
[46] LI J Y, ZHAN Z H, WANG H, et al. Data-driven evolutionary algorithm with perturbation-based ensemble surrogates[J]. IEEE Transactions on Cybernetics, 2020, 51(8): 3925-3937.
[47] YU M, LI X, LIANG J. A dynamic surrogate-assisted evolutionary algorithm framework for expensive structural optimization[J]. Structural and Multidisciplinary Optimization, 2020, 61 (2): 711-729.
[48] LIU Q, CHENG R, JIN Y, et al. Reference vector-assisted adaptive model management for surrogate-assisted many-objective optimization[J]. IEEE Transactions on Systems, Man, and Cybernetics: Systems, 2022, 52(12): 7760-7773.
[49] DESHWAL A, BELAKARIA S, DOPPA J R. Bayesian optimization over hybrid spaces[C]// International Conference on Machine Learning. PMLR, 2021: 2632-2643.
[50] WAN X, NGUYEN V, HA H, et al. Think global and act local: Bayesian optimisation over high-dimensional categorical and mixed search spaces[M]//Proceedings of the 38th International Conference on Machine Learning: volume 139. 2021: 10663-10674.
[51] RU B, ALVI A, NGUYEN V, et al. Bayesian optimisation over multiple continuous and categorical inputs[C]//International Conference on Machine Learning. PMLR, 2020: 8276-8285.
[52] NGUYEN D, GUPTA S, RANA S, et al. Bayesian optimization for categorical and category specific continuous inputs[C]//Proceedings of the AAAI Conference on Artificial Intelligence: volume 34. 2020: 5256-5263.
[53] LIU B, SUN N, ZHANG Q, et al. A surrogate model assisted evolutionary algorithm for computationally expensive design optimization problems with discrete variables[C]//2016 IEEE Congress on Evolutionary Computation (CEC). IEEE, 2016: 1650-1657.
[54] XIE L, LI G, LIN K, et al. Dual-state-driven evolutionary optimization for expensive optimization problems with continuous and categorical variables[C]//2023 5th International Conference on Data-driven Optimization of Complex Systems (DOCS). IEEE, 2023: 1-7.
[55] BERGSTRA J, BARDENET R, BENGIO Y, et al. Algorithms for hyper-parameter optimization [J]. Advances in Neural Information Processing Systems, 2011, 24.
[56] HUTTER F, HOOS H H, LEYTON-BROWN K. Sequential model-based optimization for general algorithm configuration[C]//Learning and Intelligent Optimization: 5th International Conference, LION 5, Rome, Italy, January 17-21, 2011. Selected Papers 5. Springer, 2011: 507-523.
[57] DAULTON S, WAN X, ERIKSSON D, et al. Bayesian optimization over discrete and mixed spaces via probabilistic reparameterization[J]. Advances in Neural Information Processing Systems,2022, 35: 12760-12774.
[58] SHAHRIARI B, SWERSKY K, WANG Z, et al. Taking the human out of the loop: A review of Bayesian optimization[J]. Proceedings of the IEEE, 2015, 104(1): 148-175.
[59] LIAO T, SOCHA K, DE OCA M A M, et al. Ant colony optimization for mixed-variable optimization problems[J]. IEEE Transactions on Evolutionary Computation, 2013, 18(4): 503-518.
[60] MIRINEJAD H, INANC T, ZURADA J M. Radial basis function interpolation and Galerkin projection for direct trajectory optimization and costate estimation[J]. IEEE/CAA Journal of Automatica Sinica, 2021, 8(8): 1380-1388.
[61] SUTTON C D. Classification and regression trees, bagging, and boosting[J]. Handbook of Statistics, 2005, 24: 303-329.
[62] XIE L, LI G, WANG Z, et al. Surrogate-assisted evolutionary algorithm with model and infill criterion auto-configuration[J]. IEEE Transactions on Evolutionary Computation, 2023.
[63] LARRAÑAGA P, LOZANO J A. Estimation of distribution algorithms: A new tool for evolutionary computation: volume 2[M]. Springer Science & Business Media, 2001.
[64] ZHAO F, SHAO Z, WANG J, et al. A hybrid differential evolution and estimation of distribution algorithm based on neighbourhood search for job shop scheduling problems[J]. International Journal of Production Research, 2016, 54(4): 1039-1060.
[65] SUN J, ZHANG Q, TSANG E P. DE/EDA: A new evolutionary algorithm for global optimization[J]. Information Sciences, 2005, 169(3-4): 249-262.
[66] ZHOU A, SUN J, ZHANG Q. An estimation of distribution algorithm with cheap and expensive local search methods[J]. IEEE Transactions on Evolutionary Computation, 2015, 19(6): 807- 822.
[67] 杨启文, 蔡亮, 薛云灿. 差分进化算法综述[J]. 模式识别与人工智能, 2008, 21(4): 506-513.
[68] 王圣尧, 王凌, 方晨, 等. 分布估计算法研究进展[J]. 控制与决策, 2012, 27(7): 961-966.
[69] DONG W, CHEN T, TIŇO P, et al. Scaling up estimation of distribution algorithms for continuous optimization[J]. IEEE Transactions on Evolutionary Computation, 2013, 17(6): 797-822.
[70] 崔佳旭, 杨博. 贝叶斯优化方法和应用综述[J]. 软件学报, 2018, 29(10): 3068-3090.
[71] KARLSSON R, BLIEK L, VERWER S, et al. Continuous surrogate-based optimization algorithms are well-suited for expensive discrete problems[C]//Benelux Conference on Artificial Intelligence. Springer, 2020: 48-63.
[72] BUHMANN M D. Radial basis functions[J]. Acta Numerica, 2000, 9: 1-38.
[73] LI G, ZHANG Q, SUN J, et al. Radial basis function assisted optimization method with batch infill sampling criterion for expensive optimization[C]//2019 IEEE Congress on Evolutionary Computation (CEC). IEEE, 2019: 1664-1671.
[74] HARTIGAN J A, WONG M A. Algorithm AS 136: A k-means clustering algorithm[J]. Journal of the Royal Statistical Society. Series c (Applied Statistics), 1979, 28(1): 100-108.
[75] KHURI A I, MUKHOPADHYAY S. Response surface methodology[J]. Wiley Interdisciplinary Reviews: Computational Statistics, 2010, 2(2): 128-149.
[76] ZHANG J, ZHOU A, ZHANG G. A multiobjective evolutionary algorithm based on decomposition and preselection[C]//Bio-inspired Computing-theories and Applications. Springer, 2015: 631-642.
[77] GAO W, LI G, ZHANG Q, et al. Solving nonlinear equation systems by a two-phase evolutionary algorithm[J]. IEEE Transactions on Systems, Man, and Cybernetics: Systems, 2019, 51(9): 5652-5663.
[78] LI G, ZHANG Q, LIN Q, et al. A three-level radial basis function method for expensive optimization[J]. IEEE Transactions on Cybernetics, 2021, 52(7): 5720-5731.
[79] PAN L, HE C, TIAN Y, et al. A classification-based surrogate-assisted evolutionary algorithm for expensive many-objective optimization[J]. IEEE Transactions on Evolutionary Computation, 2018, 23(1): 74-88.
[80] KOCSIS L, SZEPESVÁRI C. Bandit based monte-carlo planning[C]//European conference on machine learning. Springer, 2006: 282-293.
[81] CARLSSON E, DUBHASHI D, JOHANSSON F D. Thompson sampling for bandits with clustered arms[A]. 2021.
[82] PANDEY S, CHAKRABARTI D, AGARWAL D. Multi-armed bandit problems with dependent arms[C]//Proceedings of the 24th International Conference on Machine Learning. 2007: 721-728.
[83] ZHAO T, LI M, POLOCZEK M. Fast reconfigurable antenna state selection with hierarchical Thompson sampling[C]//ICC 2019-2019 IEEE International Conference on Communications (ICC). IEEE, 2019: 1-6.
[84] SINGH R, LIU F, SUN Y, et al. Multi-armed bandits with dependent arms[J]. Machine Learning, 2024, 113(1): 45-71.
[85] HONG J, KVETON B, ZAHEER M, et al. Hierarchical bayesian bandits[C]//International Conference on Artificial Intelligence and Statistics. PMLR, 2022: 7724-7741.
[86] STANFILL C, WALTZ D. Toward memory-based reasoning[J]. Communications of the ACM, 1986, 29(12): 1213-1228.
[87] COST S, SALZBERG S. A weighted nearest neighbor algorithm for learning with symbolic features[J]. Machine Learning, 1993, 10: 57-78.
[88] ZHOU Z H, YU Y. Ensembling local learners through multimodal perturbation[J]. IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), 2005, 35(4): 725-735.
[89] JIA H, CHEUNG Y M, LIU J. A new distance metric for unsupervised learning of categorical data[J]. IEEE Transactions on Neural Networks and Learning Systems, 2015, 27(5): 1065-1079.
[90] HAUSCHILD M, PELIKAN M. An introduction and survey of estimation of distribution algorithms[J]. Swarm and Evolutionary Computation, 2011, 1(3): 111-128.
[91] LI Z, ZHANG Q, LIN X, et al. Fast covariance matrix adaptation for large-scale black-box optimization[J]. IEEE Transactions on Cybernetics, 2018, 50(5): 2073-2083.
[92] RUBINSTEIN R Y. Optimization of computer simulation models with rare events[J]. European Journal of Operational Research, 1997, 99(1): 89-112.
[93] LIM D, JIN Y, ONG Y S, et al. Generalizing surrogate-assisted evolutionary computation[J]. IEEE Transactions on Evolutionary Computation, 2009, 14(3): 329-355.
[94] HABIB A, SINGH H K, CHUGH T, et al. A multiple surrogate assisted decomposition-based evolutionary algorithm for expensive multi/many-objective optimization[J]. IEEE Transactions on Evolutionary Computation, 2019, 23(6): 1000-1014.
[95] WANG Y, YIN D Q, YANG S, et al. Global and local surrogate-assisted differential evolution for expensive constrained optimization problems with inequality constraints[J]. IEEE Transactions on Cybernetics, 2018, 49(5): 1642-1656.
[96] LU X, TANG K, YAO X. Classification-assisted differential evolution for computationally expensive problems[C]//2011 IEEE Congress of Evolutionary Computation (CEC). IEEE, 2011: 1986-1993.
[97] LI J, WANG P, DONG H, et al. A classification surrogate-assisted multi-objective evolutionary algorithm for expensive optimization[J]. Knowledge-Based Systems, 2022, 242: 108416.
[98] SUN Y, KAMEL M S, WONG A K, et al. Cost-sensitive boosting for classification of imbalanced data[J]. Pattern Recognition, 2007, 40(12): 3358-3378.
[99] SUGANTHAN P N, HANSEN N, LIANG J J, et al. Problem definitions and evaluation criteria for the CEC 2005 special session on real-parameter optimization[J]. KanGAL report, 2005.
[100] ALCALÁ-FDEZ J, SANCHEZ L, GARCIA S, et al. KEEL: a software tool to assess evolutionary algorithms for data mining problems[J]. Soft Computing, 2009, 13(3): 307-318.
[101] LU X F, TANG K. Classification-and regression-assisted differential evolution for computationally expensive problems[J]. Journal of Computer Science and Technology, 2012, 27(5): 1024-1034.
[102] CHEN G, LUO X, JIAO J J, et al. Data-driven evolutionary algorithm for oil reservoir well placement and control optimization[J]. Fuel, 2022, 326: 125125.
[103] JANSEN J D, FONSECA R M, KAHROBAEI S, et al. The egg model–a geological ensemble for reservoir simulation[J]. Geoscience Data Journal, 2014, 1(2): 192-195.
[104] LIE K A. An introduction to reservoir simulation using MATLAB/GNU Octave: User guide for the MATLAB Reservoir Simulation Toolbox (MRST)[M]. Cambridge University Press, 2019.
[105] LIANG J, QU B, SUGANTHAN P, et al. Problem definitions and evaluation criteria for the CEC 2015 competition on learning-based real-parameter single objective optimization[J]. Technical Report201411A, Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou China and Technical Report, Nanyang Technological University, Singapore, 2014, 29: 625-640.
[106] WANG Z, ONG Y S, ISHIBUCHI H. On scalable multiobjective test problems with hardly dominated boundaries[J]. IEEE Transactions on Evolutionary Computation, 2018, 23(2): 217-231.
[107] COX D D, JOHN S. A statistical method for global optimization[C]//[Proceedings] 1992 IEEE International Conference on Systems, Man, and Cybernetics. IEEE, 1992: 1241-1246.
[108] ZHAN D, XING H. A fast kriging-assisted evolutionary algorithm based on incremental learning[J]. IEEE Transactions on Evolutionary Computation, 2021, 25(5): 941-955.
[109] SHIELDS M D, ZHANG J. The generalization of Latin hypercube sampling[J]. Reliability Engineering & System Safety, 2016, 148: 96-108.
[110] LIPOWSKI A, LIPOWSKA D. Roulette-wheel selection via stochastic acceptance[J]. Physica A: Statistical Mechanics and its Applications, 2012, 391(6): 2193-2196.
[111] WANG Z, YAO S, LI G, et al. Multiobjective combinatorial optimization using a single deep reinforcement learning model[J]. IEEE Transactions on Cybernetics, 2024, 54(3): 1984-1996.
[112] LIN Y, LIU Y, CHEN W N, et al. A hybrid differential evolution algorithm for mixed-variable optimization problems[J]. Information Sciences, 2018, 466: 170-188.
[113] WANG F, LI Y, ZHOU A, et al. An estimation of distribution algorithm for mixed-variable newsvendor problems[J]. IEEE Transactions on Evolutionary Computation, 2019, 24(3): 479-493.
[114] WANG F, ZHANG H, ZHOU A. A particle swarm optimization algorithm for mixed-variable optimization problems[J]. Swarm and Evolutionary Computation, 2021, 60: 100808.
[115] LI G, ZHANG Q. Multiple penalties and multiple local surrogates for expensive constrained optimization[J]. IEEE Transactions on Evolutionary Computation, 2021, 25(4): 769-778.
[116] SPARCK JONES K. A statistical interpretation of term specificity and its application in retrieval[J]. Journal of Documentation, 1972, 28(1): 11-21.
[117] BORIAH S, CHANDOLA V, KUMAR V. Similarity measures for categorical data: A comparative evaluation[C]//Proceedings of the 2008 SIAM International Conference on Data Mining. SIAM, 2008: 243-254.
[118] GOODALL D W. A new similarity index based on probability[J]. Biometrics, 1966: 882-907.
[119] LIN D, et al. An information-theoretic definition of similarity[C]//ICML: volume 98. 1998:296-304.
[120] ZHANG Y, CHEUNG Y M, TAN K C. A unified entropy-based distance metric for ordinaland-nominal-attribute data clustering[J]. IEEE Transactions on Neural Networks and Learning Systems, 2019, 31(1): 39-52.
[121] ALAMURI M, SURAMPUDI B R, NEGI A. A survey of distance/similarity measures for categorical data[C]//2014 International Joint Conference on Neural Networks (IJCNN). IEEE, 2014: 1907-1914.