An Overview of Weighted and Unconstrained Scalarizing Functions

Scalarizing functions play a crucial role in multi-objective evolutionary algorithms MOEAs based on decomposition and the R2 indicator, since they guide the population towards nearly optimal solutions, assigning a fitness value to an individual according to a predefined target direction in objective space. This paper presents a general review of weighted scalarizing functions without constraints, which have been proposed not only within evolutionary multi-objective optimization but also in the mathematical programming literature. We also investigate their scalability upi¾źto 10 objectives, using the test problems of Lame Superspheres on the MOEA/D and MOMBI-II frameworks. For this purpose, the best suited scalarizing functions and their model parameters are determined through the evolutionary calibrator EVOCA. Our experimental results reveal that some of these scalarizing functions are quite robust and suitable for handling many-objective optimization problems.

[1]  Rubén Saborido,et al.  On the use of the $$L_{p}$$Lp distance in reference point-based approaches for multiobjective optimization , 2015, Ann. Oper. Res..

[2]  Margaret M. Wiecek,et al.  An improved algorithm for solving biobjective integer programs , 2006, Ann. Oper. Res..

[3]  Evangelos Triantaphyllou,et al.  Multi-criteria Decision Making Methods: A Comparative Study , 2000 .

[4]  Hisao Ishibuchi,et al.  Simultaneous use of different scalarizing functions in MOEA/D , 2010, GECCO '10.

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

[6]  Tobias Friedrich,et al.  Don't be greedy when calculating hypervolume contributions , 2009, FOGA '09.

[7]  Hisao Ishibuchi,et al.  A Study on the Specification of a Scalarizing Function in MOEA/D for Many-Objective Knapsack Problems , 2013, LION.

[8]  V. Bowman.  On the Relationship of the Tchebycheff Norm and the Efficient Frontier of Multiple-Criteria Objectives , 1976 .

[9]  Thomas Stützle,et al.  A Two-Phase Local Search for the Biobjective Traveling Salesman Problem , 2003, EMO.

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

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

[12]  Hiroyuki Sato,et al.  MOEA/D using constant-distance based neighbors designed for many-objective optimization , 2015, 2015 IEEE Congress on Evolutionary Computation (CEC).

[13]  Guido Carpinelli,et al.  Exponential weighted method and a compromise programming method for multi-objective operation of plug-in vehicle aggregators in microgrids , 2014 .

[14]  Bernabé Dorronsoro,et al.  A Survey of Decomposition Methods for Multi-objective Optimization , 2014, Recent Advances on Hybrid Approaches for Designing Intelligent Systems.

[15]  Ozden Ustun,et al.  Proposal of a nonlinear multi-objective genetic algorithm using conic scalarization to the design of cellular manufacturing systems , 2015 .

[16]  Hiroyuki Sato,et al.  Inverted PBI in MOEA/D and its impact on the search performance on multi and many-objective optimization , 2014, GECCO.

[17]  Peter J. Fleming,et al.  Methods for multi-objective optimization: An analysis , 2015, Inf. Sci..

[18]  K. Lewis,et al.  Pareto analysis in multiobjective optimization using the collinearity theorem and scaling method , 2001 .

[19]  E. Hughes.  Multiple single objective Pareto sampling , 2003, The 2003 Congress on Evolutionary Computation, 2003. CEC '03..

[20]  Jian-Bo Yang,et al.  Quantitative parametric connections between methods for generating noninferior solutions in multiobjective optimization , 1999, Eur. J. Oper. Res..

[21]  Mark Fleischer,et al.  The measure of pareto optima: Applications to multi-objective metaheuristics , 2003 .

[22]  Carlos A. Coello Coello,et al.  Improved Metaheuristic Based on the R2 Indicator for Many-Objective Optimization , 2015, GECCO.

[23]  Lotfi A. Zadeh,et al.  Optimality and non-scalar-valued performance criteria , 1963 .

[24]  Bilel Derbel,et al.  Force-Based Cooperative Search Directions in Evolutionary Multi-objective Optimization , 2013, EMO.

[25]  Tao Zhang,et al.  Localized Weighted Sum Method for Many-Objective Optimization , 2018, IEEE Transactions on Evolutionary Computation.

[26]  Peter J. Fleming,et al.  Diversity Management in Evolutionary Many-Objective Optimization , 2011, IEEE Transactions on Evolutionary Computation.

[27]  Bo Zhang,et al.  Balancing Convergence and Diversity in Decomposition-Based Many-Objective Optimizers , 2016, IEEE Transactions on Evolutionary Computation.

[28]  Bernhard Sendhoff,et al.  Adapting Weighted Aggregation for Multiobjective Evolution Strategies , 2001, EMO.

[29]  Shengxiang Yang,et al.  Improving the multiobjective evolutionary algorithm based on decomposition with new penalty schemes , 2017, Soft Comput..

[30]  P. Papalambros,et al.  A NOTE ON WEIGHTED CRITERIA METHODS FOR COMPROMISE SOLUTIONS IN MULTI-OBJECTIVE OPTIMIZATION , 1996 .

[31]  Tadahiko Murata,et al.  Many-Objective Optimization for Knapsack Problems Using Correlation-Based Weighted Sum Approach , 2009, EMO.

[32]  Hisao Ishibuchi,et al.  Behavior of Multiobjective Evolutionary Algorithms on Many-Objective Knapsack Problems , 2015, IEEE Transactions on Evolutionary Computation.

[33]  Michael T. M. Emmerich,et al.  Test Problems Based on Lamé Superspheres , 2007, EMO.

[34]  Ralph E. Steuer,et al.  An interactive weighted Tchebycheff procedure for multiple objective programming , 1983, Math. Program..

[35]  Kalyanmoy Deb,et al.  A Multimodal Approach for Evolutionary Multi-objective Optimization (MEMO): Proof-of-Principle Results , 2015, EMO.

[36]  Jasbir S. Arora,et al.  Survey of multi-objective optimization methods for engineering , 2004 .

[37]  Refail Kasimbeyli,et al.  A Nonlinear Cone Separation Theorem and Scalarization in Nonconvex Vector Optimization , 2009, SIAM J. Optim..

[38]  Carlo Meloni,et al.  Dynamic objectives aggregation methods for evolutionary portfolio optimisation. A computational study , 2012, Int. J. Bio Inspired Comput..

[39]  Kaisa Miettinen,et al.  On scalarizing functions in multiobjective optimization , 2002, OR Spectr..

[40]  Tao Zhang,et al.  Pareto Adaptive Scalarising Functions for Decomposition Based Algorithms , 2015, EMO.

[41]  Xin Yao,et al.  Many-Objective Evolutionary Algorithms , 2015, ACM Comput. Surv..

[42]  Emil Björnson,et al.  Multiobjective Signal Processing Optimization: The way to balance conflicting metrics in 5G systems , 2014, IEEE Signal Processing Magazine.

[43]  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.

[44]  Duan Li,et al.  Convexification of a noninferior frontier , 1996 .

[45]  Carlos A. Coello Coello,et al.  MOMBI: A new metaheuristic for many-objective optimization based on the R2 indicator , 2013, 2013 IEEE Congress on Evolutionary Computation.

[46]  Gabriele Eichfelder,et al.  An Adaptive Scalarization Method in Multiobjective Optimization , 2008, SIAM J. Optim..

[47]  María Cristina Riff,et al.  A new algorithm for reducing metaheuristic design effort , 2013, 2013 IEEE Congress on Evolutionary Computation.

[48]  John E. Dennis,et al.  Normal-Boundary Intersection: A New Method for Generating the Pareto Surface in Nonlinear Multicriteria Optimization Problems , 1998, SIAM J. Optim..

[49]  Evan J. Hughes,et al.  MSOPS-II: A general-purpose Many-Objective optimiser , 2007, 2007 IEEE Congress on Evolutionary Computation.

[50]  Hisao Ishibuchi,et al.  Optimization of Scalarizing Functions Through Evolutionary Multiobjective Optimization , 2007, EMO.

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

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

[53]  Rubén Saborido,et al.  Global WASF-GA: An Evolutionary Algorithm in Multiobjective Optimization to Approximate the Whole Pareto Optimal Front , 2017, Evolutionary Computation.

[54]  Loo Hay Lee,et al.  A study on multi-objective particle swarm optimization with weighted scalarizing functions , 2014, Proceedings of the Winter Simulation Conference 2014.

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

[56]  Ignacy Kaliszewski,et al.  A modified weighted tchebycheff metric for multiple objective programming , 1987, Comput. Oper. Res..

[57]  Rubén Saborido,et al.  A preference-based evolutionary algorithm for multiobjective optimization: the weighting achievement scalarizing function genetic algorithm , 2015, J. Glob. Optim..

[58]  Hisao Ishibuchi,et al.  Adaptation of Scalarizing Functions in MOEA/D: An Adaptive Scalarizing Function-Based Multiobjective Evolutionary Algorithm , 2009, EMO.