Convergence speed in multi‐objective metaheuristics: Efficiency criteria and empirical study

Solving optimization problems using a reduced number of objective function evaluations is an open issue in the design of multi-objective optimization metaheuristics. The usual approach to analyze the behavior of such techniques is to choose a benchmark of known problems, to perform a predetermined number of function evaluations, and then, apply a set of performance indicators in order to assess the quality of the solutions obtained. However, this sort of methodology does not provide any insights of the efficiency of each algorithm. Here, efficiency is defined as the effort required by a multi-objective metaheuristic to obtain a set of non-dominated solutions that is satisfactory to the user, according to some pre-defined criterion. Indeed, the type of solutions of interest to the user may vary depending on the specific characteristics of the problem being solved. In this paper, the convergence speed of seven state-of-the-art multi-objective metaheuristics is analyzed, according to three pre-defined efficiency criteria. Our empirical study shows that SMPSO (based on a particle swarm optimizer) is found to be the best overall algorithm on the test problems adopted when considering the three efficiency criteria. Copyright © 2010 John Wiley & Sons, Ltd.

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

[2]  Qingfu Zhang,et al.  Multiobjective optimization Test Instances for the CEC 2009 Special Session and Competition , 2009 .

[3]  Michel Gendreau,et al.  Metaheuristics in Combinatorial Optimization , 2022 .

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

[5]  F. Glover,et al.  Handbook of Metaheuristics , 2019, International Series in Operations Research & Management Science.

[6]  David H. Wolpert,et al.  No free lunch theorems for optimization , 1997, IEEE Trans. Evol. Comput..

[7]  Carlos A. Coello Coello,et al.  A New Proposal for Multiobjecive Optimization Using Particle Swarm Optimization and Rough Sets Theory , 2006, PPSN.

[8]  Carlos A. Coello Coello,et al.  A new proposal for multi-objective optimization using differential evolution and rough sets theory , 2006, GECCO '06.

[9]  Lothar Thiele,et al.  A Tutorial on the Performance Assessment of Stochastic Multiobjective Optimizers , 2006 .

[10]  Christian Blum,et al.  Metaheuristics in combinatorial optimization: Overview and conceptual comparison , 2003, CSUR.

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

[12]  Joshua D. Knowles,et al.  ParEGO: a hybrid algorithm with on-line landscape approximation for expensive multiobjective optimization problems , 2006, IEEE Transactions on Evolutionary Computation.

[13]  Enrique Alba,et al.  SMPSO: A new PSO-based metaheuristic for multi-objective optimization , 2009, 2009 IEEE Symposium on Computational Intelligence in Multi-Criteria Decision-Making(MCDM).

[14]  Carlos A. Coello Coello,et al.  EMOPSO: A Multi-Objective Particle Swarm Optimizer with Emphasis on Efficiency , 2007, EMO.

[15]  Francisco Luna,et al.  jMetal: a Java Framework for Developing Multi-Objective Optimization Metaheuristics , 2006 .

[16]  Enrique Alba,et al.  Parallel Metaheuristics: A New Class of Algorithms , 2005 .

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

[18]  Enrique Alba,et al.  Design Issues in a Multiobjective Cellular Genetic Algorithm , 2007, EMO.

[19]  Carlos A. Coello Coello,et al.  The Micro Genetic Algorithm 2: Towards Online Adaptation in Evolutionary Multiobjective Optimization , 2003, EMO.

[20]  Jouni Lampinen,et al.  GDE3: the third evolution step of generalized differential evolution , 2005, 2005 IEEE Congress on Evolutionary Computation.

[21]  Tadahiko Murata,et al.  Cellular Genetic Algorithm for Multi-Objective Optimization , 2001 .

[22]  Jouni Lampinen,et al.  DE’s Selection Rule for Multiobjective Optimization , 2001 .

[23]  Kirsten Schmieder,et al.  Registration of CT and Intraoperative 3-D Ultrasound Images of the Spine Using Evolutionary and Gradient-Based Methods , 2008, IEEE Transactions on Evolutionary Computation.

[24]  Marco Laumanns,et al.  Performance assessment of multiobjective optimizers , 2002 .

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

[26]  Raphael T. Haftka,et al.  Response surface approximation of Pareto optimal front in multi-objective optimization , 2007 .

[27]  Jonathan E. Fieldsend,et al.  Full Elite Sets for Multi-Objective Optimisation , 2002 .

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

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

[30]  Qingfu Zhang,et al.  Multiobjective Optimization Problems With Complicated Pareto Sets, MOEA/D and NSGA-II , 2009, IEEE Transactions on Evolutionary Computation.

[31]  Janez Demsar,et al.  Statistical Comparisons of Classifiers over Multiple Data Sets , 2006, J. Mach. Learn. Res..

[32]  Meng-Sing Liou,et al.  Multiobjective Optimization Using Coupled Response Surface Model and Evolutionary Algorithm. , 2005 .

[33]  Enrique Alba,et al.  AbYSS: Adapting Scatter Search to Multiobjective Optimization , 2008, IEEE Transactions on Evolutionary Computation.

[34]  Hamidreza Eskandari,et al.  FastPGA: A Dynamic Population Sizing Approach for Solving Expensive Multiobjective Optimization Problems , 2006, EMO.

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

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

[37]  Marco Laumanns,et al.  SPEA2: Improving the strength pareto evolutionary algorithm , 2001 .

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

[39]  Enrique Alba,et al.  A Study of Convergence Speed in Multi-objective Metaheuristics , 2008, PPSN.

[40]  David Corne,et al.  The Pareto archived evolution strategy: a new baseline algorithm for Pareto multiobjective optimisation , 1999, Proceedings of the 1999 Congress on Evolutionary Computation-CEC99 (Cat. No. 99TH8406).