A survey on handling computationally expensive multiobjective optimization problems with evolutionary algorithms

Evolutionary algorithms are widely used for solving multiobjective optimization problems but are often criticized because of a large number of function evaluations needed. Approximations, especially function approximations, also referred to as surrogates or metamodels are commonly used in the literature to reduce the computation time. This paper presents a survey of 45 different recent algorithms proposed in the literature between 2008 and 2016 to handle computationally expensive multiobjective optimization problems. Several algorithms are discussed based on what kind of an approximation such as problem, function or fitness approximation they use. Most emphasis is given to function approximation-based algorithms. We also compare these algorithms based on different criteria such as metamodeling technique and evolutionary algorithm used, type and dimensions of the problem solved, handling constraints, training time and the type of evolution control. Furthermore, we identify and discuss some promising elements and major issues among algorithms in the literature related to using an approximation and numerical settings used. In addition, we discuss selecting an algorithm to solve a given computationally expensive multiobjective optimization problem based on the dimensions in both objective and decision spaces and the computation budget available.

[1]  Sung-Bae Cho,et al.  An efficient genetic algorithm with less fitness evaluation by clustering , 2001, Proceedings of the 2001 Congress on Evolutionary Computation (IEEE Cat. No.01TH8546).

[2]  Kyriakos C. Giannakoglou,et al.  A multilevel approach to single- and multiobjective aerodynamic optimization , 2008 .

[3]  Yaochu Jin,et al.  A comprehensive survey of fitness approximation in evolutionary computation , 2005, Soft Comput..

[4]  Bernd Bischl,et al.  Model-Based Multi-objective Optimization: Taxonomy, Multi-Point Proposal, Toolbox and Benchmark , 2015, EMO.

[5]  Sonja Kuhnt,et al.  Design and analysis of computer experiments , 2010 .

[6]  W. Fink,et al.  Comparison of Multi-Objective Genetic Algorithms in Optimizing Q-Law Low-Thrust Orbit Transfers , 2005 .

[7]  G. G. Wang,et al.  Adaptive Response Surface Method Using Inherited Latin Hypercube Design Points , 2003 .

[8]  Andy J. Keane,et al.  Recent advances in surrogate-based optimization , 2009 .

[9]  Hirotaka Nakayama,et al.  Meta-Modeling in Multiobjective Optimization , 2008, Multiobjective Optimization.

[10]  Roman Neruda,et al.  Improving many-objective optimizers with aggregate meta-models , 2011, 2011 11th International Conference on Hybrid Intelligent Systems (HIS).

[11]  Eckart Zitzler,et al.  Evolutionary algorithms for multiobjective optimization: methods and applications , 1999 .

[12]  Tapabrata Ray,et al.  Infeasibility Driven Evolutionary Algorithm for Constrained Optimization , 2009 .

[13]  Kalyanmoy Deb,et al.  An Evolutionary Many-Objective Optimization Algorithm Using Reference-Point Based Nondominated Sorting Approach, Part II: Handling Constraints and Extending to an Adaptive Approach , 2014, IEEE Transactions on Evolutionary Computation.

[14]  Nikolaus Hansen,et al.  Completely Derandomized Self-Adaptation in Evolution Strategies , 2001, Evolutionary Computation.

[15]  Kalyanmoy Deb,et al.  On the performance of classification algorithms for learning Pareto-dominance relations , 2014, 2014 IEEE Congress on Evolutionary Computation (CEC).

[16]  Jasper A Vrugt,et al.  Improved evolutionary optimization from genetically adaptive multimethod search , 2007, Proceedings of the National Academy of Sciences.

[17]  Kalyanmoy Deb,et al.  High dimensional model representation for solving expensive multi-objective optimization problems , 2016, 2016 IEEE Congress on Evolutionary Computation (CEC).

[18]  Taimoor Akhtar,et al.  Multi objective optimization of computationally expensive multi-modal functions with RBF surrogates and multi-rule selection , 2016, J. Glob. Optim..

[19]  Kalyanmoy Deb,et al.  Improving convergence of evolutionary multi-objective optimization with local search: a concurrent-hybrid algorithm , 2011, Natural Computing.

[20]  Bernhard Sendhoff,et al.  On Evolutionary Optimization with Approximate Fitness Functions , 2000, GECCO.

[21]  Pramudita Satria Palar,et al.  A comparative study of local search within a surrogate-assisted multi-objective memetic algorithm framework for expensive problems , 2016, Appl. Soft Comput..

[22]  Geoffrey T. Parks,et al.  Comparison of scalarization functions within a local surrogate assisted multi-objective memetic algorithm framework for expensive problems , 2015, 2015 IEEE Congress on Evolutionary Computation (CEC).

[23]  Lamjed Ben Said,et al.  Steady state IBEA assisted by MLP neural networks for expensive multi-objective optimization problems , 2014, GECCO.

[24]  M. Sasena,et al.  Exploration of Metamodeling Sampling Criteria for Constrained Global Optimization , 2002 .

[25]  Gary B. Lamont,et al.  Multiobjective evolutionary algorithms: classifications, analyses, and new innovations , 1999 .

[26]  Andy J. Keane,et al.  Statistical Improvement Criteria for Use in Multiobjective Design Optimization , 2006 .

[27]  Roman Neruda,et al.  Hypervolume-based local search in multi-objective evolutionary optimization , 2014, GECCO.

[28]  Kalyanmoy Deb,et al.  Study of the approximation of the fitness landscape and the ranking process of scalarizing functions for many-objective problems , 2016, 2016 IEEE Congress on Evolutionary Computation (CEC).

[29]  Nicola Beume,et al.  An EMO Algorithm Using the Hypervolume Measure as Selection Criterion , 2005, EMO.

[30]  J. K. Lenstra,et al.  Local Search in Combinatorial Optimisation. , 1997 .

[31]  Frank Kursawe,et al.  A Variant of Evolution Strategies for Vector Optimization , 1990, PPSN.

[32]  Carlos A. Coello Coello,et al.  A study of fitness inheritance and approximation techniques for multi-objective particle swarm optimization , 2005, 2005 IEEE Congress on Evolutionary Computation.

[33]  Ujjwal Maulik,et al.  A Simulated Annealing-Based Multiobjective Optimization Algorithm: AMOSA , 2008, IEEE Transactions on Evolutionary Computation.

[34]  Kalyanmoy Deb,et al.  An Evolutionary Multi-objective Adaptive Meta-modeling Procedure Using Artificial Neural Networks , 2007, Evolutionary Computation in Dynamic and Uncertain Environments.

[35]  Kishalay Mitra,et al.  Successive approximate model based multi-objective optimization for an industrial straight grate iron ore induration process using evolutionary algorithm , 2011 .

[36]  Tapabrata Ray,et al.  Surrogate assisted Simulated Annealing (SASA) for constrained multi-objective optimization , 2010, IEEE Congress on Evolutionary Computation.

[37]  T. Simpson,et al.  Efficient Pareto Frontier Exploration using Surrogate Approximations , 2000 .

[38]  Anil K. Jain,et al.  Data clustering: a review , 1999, CSUR.

[39]  Carlos A. Coello Coello,et al.  A Micro-Genetic Algorithm for Multiobjective Optimization , 2001, EMO.

[40]  Marco Laumanns,et al.  Scalable Test Problems for Evolutionary Multiobjective Optimization , 2005, Evolutionary Multiobjective Optimization.

[41]  Kota Sridhar,et al.  コンプライアンス及び剛性楕円体を活用したコンプライアンス機構の概念的シンセシスのビルディングブロック手法 | 文献情報 | J-GLOBAL 科学技術総合リンクセンター , 2008 .

[42]  Kaisa Miettinen,et al.  Nonlinear multiobjective optimization , 1998, International series in operations research and management science.

[43]  Qingfu Zhang,et al.  The performance of a new version of MOEA/D on CEC09 unconstrained MOP test instances , 2009, 2009 IEEE Congress on Evolutionary Computation.

[44]  Martin J. Oates,et al.  PESA-II: region-based selection in evolutionary multiobjective optimization , 2001 .

[45]  Kalyanmoy Deb,et al.  A fast and elitist multiobjective genetic algorithm: NSGA-II , 2002, IEEE Trans. Evol. Comput..

[46]  B. Julstrom,et al.  Design of vector quantization codebooks using a genetic algorithm , 1997, Proceedings of 1997 IEEE International Conference on Evolutionary Computation (ICEC '97).

[47]  Bernard De Baets,et al.  Is Fitness Inheritance Useful for Real-World Applications? , 2003, EMO.

[48]  Hirotaka Nakayama,et al.  Approximate Optimization Using Computaional Intelligence and its Application to Reinforcement of Cable-stayed Bridges , 2006, Integrated Intelligent Systems for Engineering Design.

[49]  Bernhard Sendhoff,et al.  Reducing Fitness Evaluations Using Clustering Techniques and Neural Network Ensembles , 2004, GECCO.

[50]  Beom-Seon Jang,et al.  Managing approximation models in multiobjective optimization , 2003 .

[51]  Kwang-Yong Kim,et al.  Enhanced multi-objective optimization of a microchannel heat sink through evolutionary algorithm coupled with multiple surrogate models , 2010 .

[52]  Piet Demeester,et al.  A Surrogate Modeling and Adaptive Sampling Toolbox for Computer Based Design , 2010, J. Mach. Learn. Res..

[53]  Enrique Alba,et al.  The jMetal framework for multi-objective optimization: Design and architecture , 2010, IEEE Congress on Evolutionary Computation.

[54]  Carolina P. de Almeida,et al.  Extreme Learning Surrogate Models in Multi-objective Optimization based on Decomposition , 2016, Neurocomputing.

[55]  M. Herrera,et al.  Metamodel-assisted optimization based on multiple kernel regression for mixed variables , 2014, Structural and Multidisciplinary Optimization.

[56]  Kalyanmoy Deb,et al.  Toward an Estimation of Nadir Objective Vector Using a Hybrid of Evolutionary and Local Search Approaches , 2010, IEEE Transactions on Evolutionary Computation.

[57]  G. P. Liu,et al.  A novel multi-objective optimization method based on an approximation model management technique , 2008 .

[58]  Luis F. Gonzalez,et al.  Robust design optimisation using multi-objectiveevolutionary algorithms , 2008 .

[59]  Kyriakos C. Giannakoglou,et al.  Multilevel Optimization Algorithms Based on Metamodel- and Fitness Inheritance-Assisted Evolutionary Algorithms , 2010 .

[60]  Carlos A. Coello Coello,et al.  A Review of Techniques for Handling Expensive Functions in Evolutionary Multi-Objective Optimization , 2010 .

[61]  Shigeru Obayashi,et al.  Kriging model based many-objective optimization with efficient calculation of expected hypervolume improvement , 2014, 2014 IEEE Congress on Evolutionary Computation (CEC).

[62]  Thomas Bäck,et al.  Metamodel-Assisted Evolution Strategies , 2002, PPSN.

[63]  R. Lyndon While,et al.  A Scalable Multi-objective Test Problem Toolkit , 2005, EMO.

[64]  Beom-Seon Jang,et al.  Adaptive approximation in multi-objective optimization for full stochastic fatigue design problem , 2009 .

[65]  Donald R. Jones,et al.  Efficient Global Optimization of Expensive Black-Box Functions , 1998, J. Glob. Optim..

[66]  Carlos A. Coello Coello,et al.  Multi-objective airfoil shape optimization using a multiple-surrogate approach , 2012, 2012 IEEE Congress on Evolutionary Computation.

[67]  Rommel G. Regis,et al.  Multi-objective constrained black-box optimization using radial basis function surrogates , 2016, J. Comput. Sci..

[68]  Joshua D. Knowles Closed-loop evolutionary multiobjective optimization , 2009, IEEE Computational Intelligence Magazine.

[69]  Saúl Zapotecas Martínez,et al.  MOEA/D assisted by rbf networks for expensive multi-objective optimization problems , 2013, GECCO '13.

[70]  Hisao Ishibuchi,et al.  How to Choose Solutions for Local Search in Multiobjective Combinatorial Memetic Algorithms , 2010, PPSN.

[71]  Luís N. Vicente,et al.  Direct Multisearch for Multiobjective Optimization , 2011, SIAM J. Optim..

[72]  Rituparna Datta,et al.  A surrogate-assisted evolution strategy for constrained multi-objective optimization , 2016, Expert Syst. Appl..

[73]  Roman Neruda,et al.  ASM-MOMA: Multiobjective memetic algorithm with aggregate surrogate model , 2011, 2011 IEEE Congress of Evolutionary Computation (CEC).

[74]  Tea Tusar,et al.  GP-DEMO: Differential Evolution for Multiobjective Optimization based on Gaussian Process models , 2015, Eur. J. Oper. Res..

[75]  Serpil Sayin,et al.  Using support vector machines to learn the efficient set in multiple objective discrete optimization , 2009, Eur. J. Oper. Res..

[76]  D. Ackley A connectionist machine for genetic hillclimbing , 1987 .

[77]  Kalyanmoy Deb,et al.  A Multi-Objective Optimization Procedure with Successive Approximate Models , .

[78]  Guang-Bin Huang,et al.  Extreme learning machine: a new learning scheme of feedforward neural networks , 2004, 2004 IEEE International Joint Conference on Neural Networks (IEEE Cat. No.04CH37541).

[79]  Stefan Roth,et al.  Covariance Matrix Adaptation for Multi-objective Optimization , 2007, Evolutionary Computation.

[80]  Bernhard Sendhoff,et al.  Generalizing Surrogate-Assisted Evolutionary Computation , 2010, IEEE Transactions on Evolutionary Computation.

[81]  Qingfu Zhang,et al.  Expensive Multiobjective Optimization by MOEA/D With Gaussian Process Model , 2010, IEEE Transactions on Evolutionary Computation.

[82]  Kerstin Vogler,et al.  Applications Of Multi Objective Evolutionary Algorithms , 2016 .

[83]  Jürgen Branke,et al.  Faster convergence by means of fitness estimation , 2005, Soft Comput..

[84]  Bernhard Sendhoff,et al.  A framework for evolutionary optimization with approximate fitness functions , 2002, IEEE Trans. Evol. Comput..

[85]  Kishalay Mitra,et al.  Multi-Objective Optimization of Bulk Vinyl Acetate Polymerization with Branching , 2014 .

[86]  David B. Fogel,et al.  Evolutionary algorithms in theory and practice , 1997, Complex.

[87]  Qingfu Zhang,et al.  MOEA/D: A Multiobjective Evolutionary Algorithm Based on Decomposition , 2007, IEEE Transactions on Evolutionary Computation.

[88]  Wolfgang Ponweiser,et al.  On Expected-Improvement Criteria for Model-based Multi-objective Optimization , 2010, PPSN.

[89]  R. K. Ursem Multi-objective Optimization using Evolutionary Algorithms , 2009 .

[90]  Thomas J. Santner,et al.  The Design and Analysis of Computer Experiments , 2003, Springer Series in Statistics.

[91]  Lothar Thiele,et al.  Comparison of Multiobjective Evolutionary Algorithms: Empirical Results , 2000, Evolutionary Computation.

[92]  Carolina P. de Almeida,et al.  ELMOEA/D-DE: Extreme Learning Surrogate Models in Multi-objective Optimization Based on Decomposition and Differential Evolution , 2014, 2014 Brazilian Conference on Intelligent Systems.

[93]  M. E. Johnson,et al.  Minimax and maximin distance designs , 1990 .

[94]  Yan Liu,et al.  Improving surrogate-assisted variable fidelity multi-objective optimization using a clustering algorithm , 2014, Appl. Soft Comput..

[95]  Robert Ivor John,et al.  A parallel surrogate-assisted multi-objective evolutionary algorithm for computationally expensive optimization problems , 2008, 2008 IEEE Congress on Evolutionary Computation (IEEE World Congress on Computational Intelligence).

[96]  Chao Jiang,et al.  An efficient multi-objective optimization method for black-box functions using sequential approximate technique , 2012, Appl. Soft Comput..

[97]  Bernhard Sendhoff,et al.  A Reference Vector Guided Evolutionary Algorithm for Many-Objective Optimization , 2016, IEEE Transactions on Evolutionary Computation.

[98]  Valerio Lattarulo,et al.  Optimization of a supersonic airfoil using the multi-objective alliance algorithm , 2013, GECCO '13.

[99]  Wahid Ghaly,et al.  Aerodynamic optimization of turbomachinery blades using evolutionary methods and ANN-based surrogate models , 2008 .

[100]  George Kourakos,et al.  Development of a multi-objective optimization algorithm using surrogate models for coastal aquifer management. , 2013 .

[101]  Bernhard Sendhoff,et al.  Individual-based Management of Meta-models for Evolutionary Optimization with Application to Three-Dimensional Blade Optimization , 2007, Evolutionary Computation in Dynamic and Uncertain Environments.

[102]  Yves Crama,et al.  Local Search in Combinatorial Optimization , 2018, Artificial Neural Networks.

[103]  Peter J. Fleming,et al.  An Overview of Evolutionary Algorithms in Multiobjective Optimization , 1995, Evolutionary Computation.

[104]  Richard J. Beckman,et al.  A Comparison of Three Methods for Selecting Values of Input Variables in the Analysis of Output From a Computer Code , 2000, Technometrics.

[105]  N. M. Alexandrov,et al.  A trust-region framework for managing the use of approximation models in optimization , 1997 .

[106]  Qingfu Zhang,et al.  Multiobjective Optimization Problems With Complicated Pareto Sets, MOEA/D and NSGA-II , 2009, IEEE Transactions on Evolutionary Computation.

[107]  Eckart Zitzler,et al.  Indicator-Based Selection in Multiobjective Search , 2004, PPSN.

[108]  George Kourakos,et al.  Pumping optimization of coastal aquifers based on evolutionary algorithms and surrogate modular neural network models , 2009 .

[109]  Michèle Sebag,et al.  A mono surrogate for multiobjective optimization , 2010, GECCO '10.

[110]  Yoel Tenne,et al.  Metamodel accuracy assessment in evolutionary optimization , 2008, 2008 IEEE Congress on Evolutionary Computation (IEEE World Congress on Computational Intelligence).

[111]  M. Hansen,et al.  Evaluating the quality of approximations to the non-dominated set , 1998 .

[112]  Hisao Ishibuchi,et al.  Use of Heuristic Local Search for Single-Objective Optimization in Multiobjective Memetic Algorithms , 2008, PPSN.

[113]  Robert E. Smith,et al.  Fitness inheritance in genetic algorithms , 1995, SAC '95.

[114]  Akira Oyama,et al.  Aerodynamic multiobjective design exploration of a flapping airfoil using a Navier-Stokes solver , 2009 .

[115]  Michèle Sebag,et al.  Dominance-Based Pareto-Surrogate for Multi-Objective Optimization , 2010, SEAL.

[116]  Joshua D. Knowles,et al.  ParEGO: a hybrid algorithm with on-line landscape approximation for expensive multiobjective optimization problems , 2006, IEEE Transactions on Evolutionary Computation.

[117]  Alessandro Turco,et al.  MetaHybrid: Combining Metamodels and Gradient-Based Techniques in a Hybrid Multi-Objective Genetic Algorithm , 2011, LION.

[118]  C. Currin,et al.  A Bayesian Approach to the Design and Analysis of Computer Experiments , 1988 .

[119]  Ivor W. Tsang,et al.  Pareto Rank Learning in Multi-objective Evolutionary Algorithms , 2012, 2012 IEEE Congress on Evolutionary Computation.

[120]  Michael T. M. Emmerich,et al.  Single- and multiobjective evolutionary optimization assisted by Gaussian random field metamodels , 2006, IEEE Transactions on Evolutionary Computation.

[121]  D. Goldberg,et al.  Don't evaluate, inherit , 2001 .

[122]  Qingfu Zhang,et al.  Multiobjective optimization Test Instances for the CEC 2009 Special Session and Competition , 2009 .

[123]  Xin Yao,et al.  A Large Population Size Can Be Unhelpful in Evolutionary Algorithms a Large Population Size Can Be Unhelpful in Evolutionary Algorithms , 2022 .

[124]  Wolfgang Ponweiser,et al.  Multiobjective Optimization on a Limited Budget of Evaluations Using Model-Assisted -Metric Selection , 2008, PPSN.

[125]  Kirsten Schmieder,et al.  Registration of CT and Intraoperative 3-D Ultrasound Images of the Spine Using Evolutionary and Gradient-Based Methods , 2008, IEEE Transactions on Evolutionary Computation.

[126]  Kalyanmoy Deb,et al.  A Hybrid Framework for Evolutionary Multi-Objective Optimization , 2013, IEEE Transactions on Evolutionary Computation.

[127]  Ren Yuan,et al.  Comparison of Neural Network and Kriging Method for Creating Simulation-Optimization Metamodels , 2009, 2009 Eighth IEEE International Conference on Dependable, Autonomic and Secure Computing.

[128]  Bogdan Filipic,et al.  DEMO: Differential Evolution for Multiobjective Optimization , 2005, EMO.

[129]  Handing Wang,et al.  Data-Driven Surrogate-Assisted Multiobjective Evolutionary Optimization of a Trauma System , 2016, IEEE Transactions on Evolutionary Computation.

[130]  J. Dennis,et al.  MANAGING APPROXIMATION MODELS IN OPTIMIZATION , 2007 .

[131]  Marco Laumanns,et al.  SPEA2: Improving the Strength Pareto Evolutionary Algorithm For Multiobjective Optimization , 2002 .

[132]  I. Sobola,et al.  Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates , 2001 .

[133]  Carlos A. Coello Coello,et al.  MODE-LD+SS: A novel Differential Evolution algorithm incorporating local dominance and scalar selection mechanisms for multi-objective optimization , 2010, IEEE Congress on Evolutionary Computation.

[134]  Anil K. Jain,et al.  Algorithms for Clustering Data , 1988 .

[135]  S. Azarm,et al.  Improving multi-objective genetic algorithms with adaptive design of experiments and online metamodeling , 2009 .

[136]  Matthew Collette,et al.  A multi-objective variable-fidelity optimization method for genetic algorithms , 2013 .

[137]  Shigeru Obayashi,et al.  Efficient global optimization (EGO) for multi-objective problem and data mining , 2005, 2005 IEEE Congress on Evolutionary Computation.

[138]  Michael T. M. Emmerich,et al.  Hypervolume-based expected improvement: Monotonicity properties and exact computation , 2011, 2011 IEEE Congress of Evolutionary Computation (CEC).

[139]  Wolfgang Ponweiser,et al.  Clustered multiple generalized expected improvement: A novel infill sampling criterion for surrogate models , 2008, 2008 IEEE Congress on Evolutionary Computation (IEEE World Congress on Computational Intelligence).

[140]  Marco Laumanns,et al.  Performance assessment of multiobjective optimizers: an analysis and review , 2003, IEEE Trans. Evol. Comput..

[141]  Yaochu Jin,et al.  Surrogate-assisted evolutionary computation: Recent advances and future challenges , 2011, Swarm Evol. Comput..

[142]  A. Osyczka,et al.  A new method to solve generalized multicriteria optimization problems using the simple genetic algorithm , 1995 .

[143]  Andreas Zell,et al.  Evolution strategies assisted by Gaussian processes with improved preselection criterion , 2003, The 2003 Congress on Evolutionary Computation, 2003. CEC '03..

[144]  Siti Zaiton Mohd Hashim,et al.  Memetic multiobjective particle swarm optimization-based radial basis function network for classification problems , 2013, Inf. Sci..

[145]  Kaisa Miettinen,et al.  A Surrogate-Assisted Reference Vector Guided Evolutionary Algorithm for Computationally Expensive Many-Objective Optimization , 2018, IEEE Transactions on Evolutionary Computation.

[146]  Tom Dhaene,et al.  Fast calculation of multiobjective probability of improvement and expected improvement criteria for Pareto optimization , 2014, J. Glob. Optim..

[147]  C. Poloni,et al.  Hybridization of a multi-objective genetic algorithm, a neural network and a classical optimizer for a complex design problem in fluid dynamics , 2000 .

[148]  Lothar Thiele,et al.  Multiobjective Optimization Using Evolutionary Algorithms - A Comparative Case Study , 1998, PPSN.

[149]  Kaisa Miettinen,et al.  On Constraint Handling in Surrogate-Assisted Evolutionary Many-Objective Optimization , 2016, PPSN.

[150]  J. D. Schaffer,et al.  Some experiments in machine learning using vector evaluated genetic algorithms (artificial intelligence, optimization, adaptation, pattern recognition) , 1984 .

[151]  Shigeru Obayashi,et al.  Updating Kriging Surrogate Models Based on the Hypervolume Indicator in Multi-Objective Optimization , 2013 .

[152]  Tom Dhaene,et al.  A constrained multi-objective surrogate-based optimization algorithm , 2014, 2014 IEEE Congress on Evolutionary Computation (CEC).

[153]  Shapour Azarm,et al.  A Kriging Metamodel Assisted Multi-Objective Genetic Algorithm for Design Optimization , 2008, DAC 2006.

[154]  Shawn E. Gano,et al.  Update strategies for kriging models used in variable fidelity optimization , 2006 .

[155]  W. Peizhuang Pattern Recognition with Fuzzy Objective Function Algorithms (James C. Bezdek) , 1983 .

[156]  James C. Bezdek,et al.  Pattern Recognition with Fuzzy Objective Function Algorithms , 1981, Advanced Applications in Pattern Recognition.

[157]  Gary B. Lamont,et al.  Evolutionary Algorithms for Solving Multi-Objective Problems , 2002, Genetic Algorithms and Evolutionary Computation.

[158]  Gary B. Lamont,et al.  Multiobjective evolutionary algorithm test suites , 1999, SAC '99.

[159]  David A. Van Veldhuizen,et al.  Evolutionary Computation and Convergence to a Pareto Front , 1998 .