An Empirical Investigation of the Optimality and Monotonicity Properties of Multiobjective Archiving Methods

Most evolutionary multiobjective optimisation (EMO) algorithms explicitly or implicitly maintain an archive for an approximation of the Pareto front. A question arising is whether existing archiving methods are reliable with respect to their convergence and approximation ability. Despite theoretical results available, it remains unknown how these archivers actually perform in practice. In particular, what percentage of solutions in their final archive are Pareto optimal? How frequently do they experience deterioration during the archiving process? Deterioration means archiving a new solution which is dominated by some solution discarded previously. This paper answers the above questions through a systematic investigation of eight representative archivers on 37 test instances with two to five objectives. We have found that (1) deterioration happens to all the archivers; (2) the deterioration degree can vary dramatically on different problems; (3) some archivers clearly perform better than others; and (4) several popular archivers sometime return a population with most solutions being the non-optimal. All of these suggest the need of improvement of current archiving methods.

[1]  Joshua D. Knowles,et al.  Bounded archiving using the lebesgue measure , 2003, The 2003 Congress on Evolutionary Computation, 2003. CEC '03..

[2]  Saúl Zapotecas Martínez,et al.  Approaches for Many-Objective Optimization: Analysis and Comparison on MNK-Landscapes , 2015, Artificial Evolution.

[3]  David W. Corne,et al.  Properties of an adaptive archiving algorithm for storing nondominated vectors , 2003, IEEE Trans. Evol. Comput..

[4]  Shengxiang Yang,et al.  Shift-Based Density Estimation for Pareto-Based Algorithms in Many-Objective Optimization , 2014, IEEE Transactions on Evolutionary Computation.

[5]  Jonathan E. Fieldsend,et al.  Using unconstrained elite archives for multiobjective optimization , 2003, IEEE Trans. Evol. Comput..

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

[7]  Marco Laumanns,et al.  A Tutorial on Evolutionary Multiobjective Optimization , 2004, Metaheuristics for Multiobjective Optimisation.

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

[9]  Shengxiang Yang,et al.  A test problem for visual investigation of high-dimensional multi-objective search , 2014, 2014 IEEE Congress on Evolutionary Computation (CEC).

[10]  Thomas Stützle,et al.  A Large-Scale Experimental Evaluation of High-Performing Multi- and Many-Objective Evolutionary Algorithms , 2018, Evolutionary Computation.

[11]  Jonathan E. Fieldsend,et al.  University staff teaching allocation: formulating and optimising a many-objective problem , 2017, GECCO.

[12]  David Corne,et al.  Bounded Pareto Archiving: Theory and Practice , 2004, Metaheuristics for Multiobjective Optimisation.

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

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

[15]  B. Naujoks,et al.  Non‐monotonicity of Observed Hypervolume in 1‐Greedy S‐Metric Selection , 2013 .

[16]  Hisao Ishibuchi,et al.  Many-Objective Test Problems to Visually Examine the Behavior of Multiobjective Evolution in a Decision Space , 2010, PPSN.

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

[18]  Marco Laumanns,et al.  Convergence of stochastic search algorithms to finite size pareto set approximations , 2008, J. Glob. Optim..

[19]  Kiyoshi Tanaka,et al.  Working principles, behavior, and performance of MOEAs on MNK-landscapes , 2007, Eur. J. Oper. Res..

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

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

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

[23]  Joshua D. Knowles,et al.  Some multiobjective optimizers are better than others , 2003, The 2003 Congress on Evolutionary Computation, 2003. CEC '03..

[24]  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.

[25]  Xin Yao,et al.  Multiline Distance Minimization: A Visualized Many-Objective Test Problem Suite , 2018, IEEE Transactions on Evolutionary Computation.

[26]  Günter Rudolph,et al.  Convergence properties of some multi-objective evolutionary algorithms , 2000, Proceedings of the 2000 Congress on Evolutionary Computation. CEC00 (Cat. No.00TH8512).

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

[28]  Shengxiang Yang,et al.  Pareto or Non-Pareto: Bi-Criterion Evolution in Multiobjective Optimization , 2016, IEEE Transactions on Evolutionary Computation.

[29]  Marco Laumanns,et al.  On Sequential Online Archiving of Objective Vectors , 2011, EMO.

[30]  Christian Fonteix,et al.  Multicriteria optimization using a genetic algorithm for determining a Pareto set , 1996, Int. J. Syst. Sci..

[31]  Nicola Beume,et al.  SMS-EMOA: Multiobjective selection based on dominated hypervolume , 2007, Eur. J. Oper. Res..

[32]  Carlos M. Fonseca,et al.  An Improved Dimension-Sweep Algorithm for the Hypervolume Indicator , 2006, 2006 IEEE International Conference on Evolutionary Computation.

[33]  Mario Köppen,et al.  Substitute Distance Assignments in NSGA-II for Handling Many-objective Optimization Problems , 2007, EMO.

[34]  Kalyanmoy Deb,et al.  An Evolutionary Many-Objective Optimization Algorithm Using Reference-Point-Based Nondominated Sorting Approach, Part I: Solving Problems With Box Constraints , 2014, IEEE Transactions on Evolutionary Computation.

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

[36]  Joseph R. Kasprzyk,et al.  Evolutionary multiobjective optimization in water resources: The past, present, and future , 2012 .

[37]  Marco Laumanns,et al.  Stochastic convergence of random search methods to fixed size Pareto front approximations , 2011, Eur. J. Oper. Res..

[38]  Huidong Jin,et al.  Adaptive, convergent, and diversified archiving strategy for multiobjective evolutionary algorithms , 2010, Expert Syst. Appl..

[39]  Thomas Stützle,et al.  Automatic Component-Wise Design of Multiobjective Evolutionary Algorithms , 2016, IEEE Transactions on Evolutionary Computation.

[40]  Thomas Hanne,et al.  On the convergence of multiobjective evolutionary algorithms , 1999, Eur. J. Oper. Res..