On Finding Pareto-Optimal Solutions Through Dimensionality Reduction for Certain Large-Dimensional Multi-Objective Optimization Problems

Many real-world applications of multi-objective optimization involve a large number (10 or more) of objectives. Existing evolutionary multi-objective optimization (EMO) methods are applied only to problems having smaller number of objectives (about five or so) for the task of finding a well-representative set of Pareto-optimal solutions, in a single simulation run. Having adequately shown this task, EMO researchers/practitioners must now investigate if these methodologies can really be used for a large number of objectives. The major impediments in handling large number of objectives relate to stagnation of search process, increased dimensionality of Pareto-optimal front, large computational cost, and difficulty in visualization of the objective space. These difficulties, do and would continue to persist, in M -objective optimization problems, having an M -dimensional Pareto-optimal front. In this paper, we propose an EMO procedure for solving those large-objective (M) problems, which degenerate to possess a lowerdimensional Pareto-optimal front (lower than M). Such problems may often be observed in practice, as ones with similarities in optimal solutions for different objectives – a matter which may not be obvious from the problem description. The proposed method is a principal component analysis (PCA) based EMO procedure, which progresses iteratively from the interior of the search space towards the Pareto-optimal region by adaptively finding the correct lower-dimensional interactions. The efficacy of the procedure is demonstrated by solving up to 30-objective optimization problems.

[1]  Tomas Gal,et al.  Redundant objective functions in linear vector maximum problems and their determination , 1977 .

[2]  E Sandgren,et al.  TOPOLOGICAL DESIGN OF STRUCTURAL COMPONENTS USING GENETIC OPTIMIZATION METHOD , 1990 .

[3]  C. Fonseca,et al.  GENETIC ALGORITHMS FOR MULTI-OBJECTIVE OPTIMIZATION: FORMULATION, DISCUSSION, AND GENERALIZATION , 1993 .

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

[5]  Peter J. Fleming,et al.  Multiobjective optimization and multiple constraint handling with evolutionary algorithms. II. Application example , 1998, IEEE Trans. Syst. Man Cybern. Part A.

[6]  Thomas Hanne,et al.  Consequences of dropping nonessential objectives for the application of MCDM methods , 1999, Eur. J. Oper. Res..

[7]  J. Tenenbaum,et al.  A global geometric framework for nonlinear dimensionality reduction. , 2000, Science.

[8]  J. Branke,et al.  Guidance in evolutionary multi-objective optimization , 2001 .

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

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

[11]  Mikkel T. Jensen,et al.  Guiding Single-Objective Optimization Using Multi-objective Methods , 2003, EvoWorkshops.

[12]  Kalyanmoy Deb,et al.  Multi-objective evolutionary algorithms: introducing bias among Pareto-optimal solutions , 2003 .

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

[14]  Kevin Tucker,et al.  Response surface approximation of pareto optimal front in multi-objective optimization , 2004 .

[15]  Marco Farina,et al.  A fuzzy definition of "optimality" for many-criteria optimization problems , 2004, IEEE Trans. Syst. Man Cybern. Part A.

[16]  Kalyanmoy Deb,et al.  Finding Knees in Multi-objective Optimization , 2004, PPSN.

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

[18]  Kalyanmoy Deb,et al.  Integrating User Preferences into Evolutionary Multi-Objective Optimization , 2005 .

[19]  Kalyanmoy Deb,et al.  Comparing Classical Generating Methods with an Evolutionary Multi-objective Optimization Method , 2005, EMO.

[20]  Kalyanmoy Deb,et al.  Searching for Robust Pareto-Optimal Solutions in Multi-objective Optimization , 2005, EMO.

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

[22]  Alexander V. Lotov,et al.  Interactive Decision Maps , 2013 .