A scalar optimization approach for averaged Hausdorff approximations of the Pareto front

This article presents a novel method to compute averaged Hausdorff () approximations of the Pareto fronts of multi-objective optimization problems. The underlying idea is to utilize directly the scalar optimization problem that is induced by the performance indicator. This method can be viewed as a certain set based scalarization approach and can be addressed both by mathematical programming techniques and evolutionary algorithms (EAs). In this work, the focus is on the latter where a first single objective EA for such approximations is proposed. Finally, the strength of the novel approach is demonstrated on some bi-objective benchmark problems with different shapes of the Pareto front.

[1]  D K Smith,et al.  Numerical Optimization , 2001, J. Oper. Res. Soc..

[2]  W. Kruskal,et al.  Use of Ranks in One-Criterion Variance Analysis , 1952 .

[3]  Carlos A. Coello Coello,et al.  Using the Averaged Hausdorff Distance as a Performance Measure in Evolutionary Multiobjective Optimization , 2012, IEEE Transactions on Evolutionary Computation.

[4]  Heike Trautmann,et al.  R2-EMOA: Focused Multiobjective Search Using R2-Indicator-Based Selection , 2013, LION.

[5]  G. Rudolph,et al.  Finding evenly spaced fronts for multiobjective control via averaging Hausdorff-measure , 2011, 2011 8th International Conference on Electrical Engineering, Computing Science and Automatic Control.

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

[7]  Carlos A. Coello Coello,et al.  A new multi-objective evolutionary algorithm based on a performance assessment indicator , 2012, GECCO.

[8]  Martin Brown,et al.  Directed Multi-Objective Optimization , 2005, Int. J. Comput. Syst. Signals.

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

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

[11]  Marco Laumanns,et al.  Computing Gap Free Pareto Front Approximations with Stochastic Search Algorithms , 2010, Evolutionary Computation.

[12]  Olivier Teytaud,et al.  Linear Convergence of Evolution Strategies with Derandomized Sampling Beyond Quasi-Convex Functions , 2013, Artificial Evolution.

[13]  Günter Rudolph,et al.  Evenly spaced Pareto fronts of quad-objective problems using PSA partitioning technique , 2013, 2013 IEEE Congress on Evolutionary Computation.

[14]  Nicola Beume,et al.  Multi-objective optimisation using S-metric selection: application to three-dimensional solution spaces , 2005, 2005 IEEE Congress on Evolutionary Computation.

[15]  R. D'Agostino,et al.  A Suggestion for Using Powerful and Informative Tests of Normality , 1990 .

[16]  Oliver Schütze,et al.  Hypervolume Maximization via Set Based Newton’s Method , 2014 .

[17]  S. Panda.  Multi-objective PID controller tuning for a FACTS-based damping stabilizer using Non-dominated Sorting Genetic Algorithm-II , 2011 .

[18]  Joan Fisher Box,et al.  Guinness, Gosset, Fisher, and Small Samples , 1987 .

[19]  Nicola Beume,et al.  Design and Management of Complex Technical Processes and Systems by means of Computational Intelligence Methods Gradient-based / Evolutionary Relay Hybrid for Computing Pareto Front Approximations Maximizing the S-Metric , 2007 .

[20]  M. J. D. Powell,et al.  On search directions for minimization algorithms , 1973, Math. Program..

[21]  Nicola Beume,et al.  An EMO Algorithm Using the Hypervolume Measure as Selection Criterion , 2005, EMO.

[22]  S. Daley,et al.  Optimal-tuning PID controller design in the frequency domain with application to a rotary hydraulic system , 1999 .

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

[24]  Günter Rudolph,et al.  Homogene Approximation der Paretofront bei mehrkriteriellen Kontrollproblemen , 2012, Autom..

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

[26]  Gary B. Lamont,et al.  Multiobjective evolutionary algorithms: classifications, analyses, and new innovations , 1999 .

[27]  Welch Bl.  THE GENERALIZATION OF ‘STUDENT'S’ PROBLEM WHEN SEVERAL DIFFERENT POPULATION VARLANCES ARE INVOLVED , 1947 .

[28]  Eckart Zitzler,et al.  Evolutionary algorithms for multiobjective optimization: methods and applications , 1999 .

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

[30]  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.

[31]  Yousef Naranjani,et al.  Simple cell mapping method for multi-objective optimal feedback control design , 2013, International Journal of Dynamics and Control.

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

[33]  Günter Rudolph,et al.  Finding Evenly Spaced Pareto Fronts for Three-Objective Optimization Problems , 2012, EVOLVE.

[34]  Carlos A. Coello Coello,et al.  Solving Multiobjective Optimization Problems Using an Artificial Immune System , 2005, Genetic Programming and Evolvable Machines.

[35]  Marco Laumanns,et al.  Scalable multi-objective optimization test problems , 2002, Proceedings of the 2002 Congress on Evolutionary Computation. CEC'02 (Cat. No.02TH8600).

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

[37]  Honggang Wang,et al.  Zigzag Search for Continuous Multiobjective Optimization , 2013, INFORMS J. Comput..

[38]  Heike Trautmann,et al.  On the properties of the R2 indicator , 2012, GECCO '12.

[39]  Thomas Ertl,et al.  Computer Graphics - Principles and Practice, 3rd Edition , 2014 .

[40]  Tobias Friedrich,et al.  Convergence of Hypervolume-Based Archiving Algorithms , 2014, IEEE Transactions on Evolutionary Computation.

[41]  M. Bartlett.  Properties of Sufficiency and Statistical Tests , 1992 .

[42]  Michael T. M. Emmerich,et al.  Time Complexity and Zeros of the Hypervolume Indicator Gradient Field , 2012, EVOLVE.