An estimation of nadir objective vector using a: hybrid evolutionary-cum-local-search procedure

A nadir objective vector is constructed from the worstPareto-optimal objective values in a multi-objectiveoptimization problem and is an important entity tocompute because of its significance in estimating therange of objective values in the Pareto-optimal frontand also in executing a number of interactive multi-objective optimization techniques. Along with theideal objective vector, it is also needed for the purposeof normalizing different objectives, so as to facilitatea comparison and agglomeration of the objectives.However, the task of estimating the nadir objectivevector necessitates information about the completePareto-optimal front and has been reported to be adifficult task, and importantly an unsolved and openresearch issue. In this paper, we propose certain mod-ifications to an existing evolutionary multi-objectiveoptimization procedure to focus its search towardsthe extreme objective values and combine it with areference-point based local search approach to con-stitute a couple of hybrid procedures for a reliableestimation of the nadir objective vector. With upto 20-objective optimization test problems and on athree-objective engineering design optimization prob-lem, one of the proposed procedures is found to becapable of finding the nadir objective vector reliably.The study clearly shows the significance of an evolu-tionary computing based search procedure in assist-ing to solve an age-old important task in the field ofmulti-objective optimization.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[15]  J. Periaux,et al.  Evolutionary Methods for Design, Optimization and Control with Applications to Industrial Problems , 2001 .

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

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

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

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

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

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

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

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

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

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

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

[27]  Kaisa Miettinen,et al.  Three Different Ways for Incorporating Preference Information in Interactive Reference Point Based Methods , 2006 .

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

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

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

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

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

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

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

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

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