Toward an Estimation of Nadir Objective Vector Using a Hybrid of Evolutionary and Local Search Approaches

A nadir objective vector is constructed from the worst Pareto-optimal objective values in a multiobjective optimization problem and is an important entity to compute because of its significance in estimating the range of objective values in the Pareto-optimal front and also in executing a number of interactive multiobjective optimization techniques. Along with the ideal objective vector, it is also needed for the purpose of normalizing different objectives, so as to facilitate a comparison and agglomeration of the objectives. However, the task of estimating the nadir objective vector necessitates information about the complete Pareto-optimal front and has been reported to be a difficult task, and importantly an unsolved and open research issue. In this paper, we propose certain modifications to an existing evolutionary multiobjective optimization procedure to focus its search toward the extreme objective values and combine it with a reference-point based local search approach to constitute a couple of hybrid procedures for a reliable estimation of the nadir objective vector. With up to 20-objective optimization test problems and on a three-objective engineering design optimization problem, one of the proposed procedures is found to be capable of finding the nadir objective vector reliably. The study clearly shows the significance of an evolutionary computing based search procedure in assisting to solve an age-old important task in the field of multiobjective optimization.

[1]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[2]  R. Benayoun,et al.  Linear programming with multiple objective functions: Step method (stem) , 1971, Math. Program..

[3]  H. P. Benson,et al.  Existence of efficient solutions for vector maximization problems , 1978 .

[4]  Andrzej P. Wierzbicki,et al.  The Use of Reference Objectives in Multiobjective Optimization , 1979 .

[5]  M. Dessouky,et al.  Estimates of the minimum nondominated criterion values in multiple-criteria decision-making , 1986 .

[6]  Ralph E. Steuer,et al.  Computational experience concerning payoff tables and minimum criterion values over the efficient set , 1988 .

[7]  Kalyanmoy Deb,et al.  An Investigation of Niche and Species Formation in Genetic Function Optimization , 1989, ICGA.

[8]  Singiresu S Rao,et al.  Genetic algorithmic approach for multiobjective optimization of structures , 1993 .

[9]  K. Miettinen,et al.  Interactive bundle-based method for nondifferentiable multiobjeective optimization: nimbus § , 1995 .

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

[11]  Ralph E. Steuer,et al.  A Heuristic for Estimating Nadir Criterion Values in Multiple Objective Linear Programming , 1997, Oper. Res..

[12]  John Buchanan,et al.  A naïve approach for solving MCDM problems: the GUESS method , 1997 .

[13]  Kalyanmoy Deb,et al.  A flexible optimization procedure for mechanical component design based on genetic adaptive search , 1998 .

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

[15]  Kalyanmoy Deb,et al.  Multi-objective Genetic Algorithms: Problem Difficulties and Construction of Test Problems , 1999, Evolutionary Computation.

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

[17]  Martin J. Oates,et al.  The Pareto Envelope-Based Selection Algorithm for Multi-objective Optimisation , 2000, PPSN.

[18]  Kalyanmoy Deb,et al.  Multi-objective optimization using evolutionary algorithms , 2001, Wiley-Interscience series in systems and optimization.

[19]  Marco Laumanns,et al.  Combining Convergence and Diversity in Evolutionary Multiobjective Optimization , 2002, Evolutionary Computation.

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

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

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

[23]  Matthias Ehrgott,et al.  Computation of ideal and Nadir values and implications for their use in MCDM methods , 2003, Eur. J. Oper. Res..

[24]  A. P. Wierzbicki,et al.  Application of multiple criteria evolutionary algorithms to vector optimisation, decision support and reference point approaches , 2003 .

[25]  Xin Yao,et al.  Performance Scaling of Multi-objective Evolutionary Algorithms , 2003, EMO.

[26]  Kaisa Miettinen.  Graphical Illustration of Pareto Optimal Solutions , 2003 .

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

[28]  Kalyanmoy Deb,et al.  I-MODE: An Interactive Multi-objective Optimization and Decision-Making Using Evolutionary Methods , 2007, EMO.

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

[30]  Aravind Srinivasan,et al.  Innovization: innovating design principles through optimization , 2006, GECCO.

[31]  Kaisa Miettinen,et al.  Synchronous approach in interactive multiobjective optimization , 2006, Eur. J. Oper. Res..

[32]  Kalyanmoy Deb,et al.  Towards estimating nadir objective vector using evolutionary approaches , 2006, GECCO.

[33]  Kaisa Miettinen,et al.  Experiments with classification-based scalarizing functions in interactive multiobjective optimization , 2006, Eur. J. Oper. Res..

[34]  Eckart Zitzler,et al.  Dimensionality Reduction in Multiobjective Optimization: The Minimum Objective Subset Problem , 2006, OR.

[35]  A. Ravindran,et al.  Engineering Optimization: Methods and Applications , 2006 .

[36]  David W. Corne,et al.  Techniques for highly multiobjective optimisation: some nondominated points are better than others , 2007, GECCO '07.

[37]  Carlos A. Coello Coello,et al.  Objective reduction using a feature selection technique , 2008, GECCO '08.

[38]  Kalyanmoy Deb,et al.  Nadir Point Estimation Using Evolutionary Approaches: Better Accuracy and Computational Speed Through Focused Search , 2010, MCDM.

[39]  Kalyanmoy Deb,et al.  A Local Search Based Evolutionary Multi-objective Optimization Approach for Fast and Accurate Convergence , 2008, PPSN.

[40]  Kalyanmoy Deb,et al.  Multiobjective optimization , 1997 .

[41]  Kathrin Klamroth,et al.  Integrating Approximation and Interactive Decision Making in Multicriteria Optimization , 2008, Oper. Res..

[42]  K. Miettinen,et al.  Incorporating preference information in interactive reference point methods for multiobjective optimization , 2009 .

[43]  Lothar Thiele,et al.  A Preference-Based Evolutionary Algorithm for Multi-Objective Optimization , 2009, Evolutionary Computation.

[44]  Kalyanmoy Deb,et al.  A Hybrid Integrated Multi-Objective Optimization Procedure for Estimating Nadir Point , 2009, EMO.

[45]  Kathrin Klamroth,et al.  Pareto navigator for interactive nonlinear multiobjective optimization , 2010, OR Spectr..

[46]  Kaisa Miettinen,et al.  A Preference Based Interactive Evolutionary Algorithm for Multi-objective Optimization: PIE , 2011, EMO.

[47]  W. Marsden.  I and J , 2012 .