Interval-based ranking in noisy evolutionary multi-objective optimization

As one of the most competitive approaches to multi-objective optimization, evolutionary algorithms have been shown to obtain very good results for many real-world multi-objective problems. One of the issues that can affect the performance of these algorithms is the uncertainty in the quality of the solutions which is usually represented with the noise in the objective values. Therefore, handling noisy objectives in evolutionary multi-objective optimization algorithms becomes very important and is gaining more attention in recent years. In this paper we present $$\alpha $$α-degree Pareto dominance relation for ordering the solutions in multi-objective optimization when the values of the objective functions are given as intervals. Based on this dominance relation, we propose an adaptation of the non-dominated sorting algorithm for ranking the solutions. This ranking method is then used in a standard multi-objective evolutionary algorithm and a recently proposed novel multi-objective estimation of distribution algorithm based on joint variable-objective probabilistic modeling, and applied to a set of multi-objective problems with different levels of independent noise. The experimental results show that the use of the proposed method for solution ranking allows to approximate Pareto sets which are considerably better than those obtained when using the dominance probability-based ranking method, which is one of the main methods for noise handling in multi-objective optimization.

[1]  Frederico G. Guimarães,et al.  Interval Robust Multi-Objective Evolutionary Algorithm , 2009, 2009 IEEE Congress on Evolutionary Computation.

[2]  Kay Chen Tan,et al.  An investigation on noise-induced features in robust evolutionary multi-objective optimization , 2010, Expert Syst. Appl..

[3]  Jonathan E. Fieldsend,et al.  Multi-objective optimisation in the presence of uncertainty , 2005, 2005 IEEE Congress on Evolutionary Computation.

[4]  Marco Zaffalon,et al.  Bayesian Networks with Imprecise Probabilities: Theory and Application to Classification , 2012 .

[5]  Takao Terano,et al.  Bayesian optimization algorithm for multi-objective solutions: application to electric equipment configuration problems in a power plant , 2003, The 2003 Congress on Evolutionary Computation, 2003. CEC '03..

[6]  P. Wozniak.  Preferences in Evolutionary Multi-Objective Optimisation with Noisy Fitness Functions : Hardware in the Loop Study , 2007 .

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

[8]  RåHW Fkryd.  Multiobjective Bayesian Optimization Algorithm for Combinatorial Problems : Theory and practice , 2002 .

[9]  G. Schwarz.  Estimating the Dimension of a Model , 1978 .

[10]  L. Jain,et al.  Evolutionary multiobjective optimization : theoretical advances and applications , 2005 .

[11]  Eckart Zitzler,et al.  Handling Uncertainty in Indicator-Based Multiobjective Optimization , 2006 .

[12]  David E. Goldberg,et al.  Multiobjective Estimation of Distribution Algorithms , 2006, Scalable Optimization via Probabilistic Modeling.

[13]  Robert Ivor John,et al.  Evolutionary optimisation of noisy multi-objective problems using confidence-based dynamic resampling , 2010, Eur. J. Oper. Res..

[14]  Kay Chen Tan,et al.  Handling Uncertainties in Evolutionary Multi-Objective Optimization , 2008, WCCI.

[15]  Martin Pelikan,et al.  Scalable Optimization via Probabilistic Modeling: From Algorithms to Applications (Studies in Computational Intelligence) , 2006 .

[16]  Hans-Georg Beyer,et al.  A general noise model and its effects on evolution strategy performance , 2006, IEEE Transactions on Evolutionary Computation.

[17]  Ashutosh Tiwari,et al.  Introducing user preference using Desirability Functions in Multi-Objective Evolutionary Optimisation of noisy processes , 2007, 2007 IEEE Congress on Evolutionary Computation.

[18]  Hamidreza Eskandari,et al.  Evolutionary multiobjective optimization in noisy problem environments , 2009, J. Heuristics.

[19]  Evan J. Hughes,et al.  Evolutionary Multi-objective Ranking with Uncertainty and Noise , 2001, EMO.

[20]  H. Mühlenbein,et al.  From Recombination of Genes to the Estimation of Distributions I. Binary Parameters , 1996, PPSN.

[21]  Jürgen Branke,et al.  Evolutionary optimization in uncertain environments-a survey , 2005, IEEE Transactions on Evolutionary Computation.

[22]  Dirk Thierens,et al.  Multi-objective mixture-based iterated density estimation evolutionary algorithms , 2001 .

[23]  Carlos A. Coello Coello,et al.  GIAA TECHNINAL REPORT GIAA2010E001 On Current Model-Building Methods for Multi-Objective Estimation of Distribution Algorithms: Shortcommings and Directions for Improvement , 2010 .

[24]  Lakhmi C. Jain,et al.  Evolutionary Multiobjective Optimization , 2005, Evolutionary Multiobjective Optimization.

[25]  Kalyanmoy Deb,et al.  A combined genetic adaptive search (GeneAS) for engineering design , 1996 .

[26]  Jason R. Schott.  Fault Tolerant Design Using Single and Multicriteria Genetic Algorithm Optimization. , 1995 .

[27]  Marco Laumanns,et al.  Scalable multi-objective optimization test problems , 2002, Proceedings of the 2002 Congress on Evolutionary Computation. CEC'02 (Cat. No.02TH8600).

[28]  Junichi Suzuki,et al.  A Confidence-Based Dominance Operator in Evolutionary Algorithms for Noisy Multiobjective Optimization Problems , 2009, 2009 21st IEEE International Conference on Tools with Artificial Intelligence.

[29]  Martin Pelikan,et al.  Hierarchical Bayesian optimization algorithm: toward a new generation of evolutionary algorithms , 2010, SICE 2003 Annual Conference (IEEE Cat. No.03TH8734).

[30]  Linda C. van der Gaag,et al.  Inference and Learning in Multi-dimensional Bayesian Network Classifiers , 2007, ECSQARU.

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

[32]  Kwang Ryel Ryu,et al.  Accumulative sampling for noisy evolutionary multi-objective optimization , 2011, GECCO '11.

[33]  Pedro Larrañaga,et al.  Towards a New Evolutionary Computation - Advances in the Estimation of Distribution Algorithms , 2006, Towards a New Evolutionary Computation.

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

[35]  David Greiner,et al.  Robust Design of Noise Attenuation Barriers with Evolutionary Multiobjective Algorithms and the Boundary Element Method , 2009, EMO.

[36]  V. Pareto,et al.  The New Theories of Economics , 1897, Journal of Political Economy.

[37]  Hussein A. Abbass,et al.  Fitness inheritance for noisy evolutionary multi-objective optimization , 2005, GECCO '05.

[38]  David E. Goldberg,et al.  Multiobjective hBOA, clustering, and scalability , 2005, GECCO '05.

[39]  David E. Goldberg,et al.  Multiple-objective Bayesian Optimization Algorithm , 2002, GECCO.

[40]  Chang Wook Ahn,et al.  Multiobjective real-coded bayesian optimization algorithmrevisited: diversity preservation , 2007, GECCO '07.

[41]  David E. Goldberg,et al.  Genetic Algorithms in Search Optimization and Machine Learning , 1988 .

[42]  Concha Bielza,et al.  Multi-dimensional classification with Bayesian networks , 2011, Int. J. Approx. Reason..

[43]  David E. Goldberg,et al.  Multi-objective bayesian optimization algorithm , 2002 .

[44]  Kay Chen Tan,et al.  An Investigation on Noisy Environments in Evolutionary Multiobjective Optimization , 2007, IEEE Transactions on Evolutionary Computation.

[45]  David E. Goldberg,et al.  Hierarchical Bayesian Optimization Algorithm , 2006, Scalable Optimization via Probabilistic Modeling.

[46]  P. Koumoutsakos,et al.  Multiobjective evolutionary algorithm for the optimization of noisy combustion processes , 2002 .

[47]  Jürgen Teich,et al.  Pareto-Front Exploration with Uncertain Objectives , 2001, EMO.

[48]  Hussein A. Abbass,et al.  Localization for Solving Noisy Multi-Objective Optimization Problems , 2009, Evolutionary Computation.

[49]  Wray L. Buntine.  Theory Refinement on Bayesian Networks , 1991, UAI.

[50]  Hajime Kita,et al.  Uncertainty of constraint function in Evolutionary Multi-objective Optimization , 2009, 2009 IEEE Congress on Evolutionary Computation.

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

[52]  Hussein A. Abbass,et al.  Performance analysis of evolutionary multi-objective optimization methods in noisy environments , 2004 .

[53]  Carlos A. Coello Coello,et al.  Solving Multiobjective Optimization Problems Using an Artificial Immune System , 2005, Genetic Programming and Evolvable Machines.

[54]  Max Henrion,et al.  Propagating uncertainty in bayesian networks by probabilistic logic sampling , 1986, UAI.

[55]  M. K. Luhandjula.  Studies in Fuzziness and Soft Computing , 2013 .

[56]  Kalyanmoy Deb,et al.  Simulated Binary Crossover for Continuous Search Space , 1995, Complex Syst..

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

[58]  D. E. Goldberg,et al.  Genetic Algorithms in Search , 1989 .

[59]  Gary B. Lamont,et al.  Evolutionary Algorithms for Solving Multi-Objective Problems (Genetic and Evolutionary Computation) , 2006 .

[60]  Bernhard Sendhoff,et al.  On Test Functions for Evolutionary Multi-objective Optimization , 2004, PPSN.

[61]  Concha Bielza,et al.  Multiobjective Estimation of Distribution Algorithm Based on Joint Modeling of Objectives and Variables , 2014, IEEE Transactions on Evolutionary Computation.

[62]  J. A. Lozano,et al.  Estimation of Distribution Algorithms: A New Tool for Evolutionary Computation , 2001 .

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

[64]  Lothar Thiele,et al.  Multiobjective evolutionary algorithms: a comparative case study and the strength Pareto approach , 1999, IEEE Trans. Evol. Comput..

[65]  J. A. Lozano,et al.  Towards a New Evolutionary Computation: Advances on Estimation of Distribution Algorithms (Studies in Fuzziness and Soft Computing) , 2006 .

[66]  Marco Laumanns,et al.  Bayesian Optimization Algorithms for Multi-objective Optimization , 2002, PPSN.

[67]  R. Lyndon While,et al.  A review of multiobjective test problems and a scalable test problem toolkit , 2006, IEEE Transactions on Evolutionary Computation.