A Convergence indicator for Multi-Objective Optimisation Algorithms

The algorithms of multi-objective optimisation had a relative growth in the last years. Thereby, it's requires some way of comparing the results of these. In this sense, performance measures play a key role. In general, it's considered some properties of these algorithms such as capacity, convergence, diversity or convergence-diversity. There are some known measures such as  generational distance (GD), inverted generational distance (IGD), hypervolume (HV), Spread($\Delta$), Averaged Hausdorff distance ($\Delta_p$), R2-indicator, among others. In this paper, we focuses on proposing a new indicator to measure convergence based on the traditional formula for Shannon entropy. The main features about this measure are: 1) It does not require tho know  the true Pareto set and 2) Medium computational cost when compared with Hypervolume.

[1]  R. Gray.  Entropy and Information Theory , 1990, Springer New York.

[2]  Hisao Ishibuchi,et al.  Indicator-based evolutionary algorithm with hypervolume approximation by achievement scalarizing functions , 2010, GECCO '10.

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

[4]  Ricardo H. C. Takahashi,et al.  On the Performance Degradation of Dominance-Based Evolutionary Algorithms in Many-Objective Optimization , 2018, IEEE Transactions on Evolutionary Computation.

[5]  Piotr Czyzżak,et al.  Pareto simulated annealing—a metaheuristic technique for multiple‐objective combinatorial optimization , 1998 .

[6]  Dimitri P. Bertsekas,et al.  Nonlinear Programming , 1997 .

[7]  S. Schäffler,et al.  Stochastic Method for the Solution of Unconstrained Vector Optimization Problems , 2002 .

[8]  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.

[9]  Carlos A. Coello Coello,et al.  On the Influence of the Number of Objectives on the Hardness of a Multiobjective Optimization Problem , 2011, IEEE Transactions on Evolutionary Computation.

[10]  Hisao Ishibuchi,et al.  Difficulties in specifying reference points to calculate the inverted generational distance for many-objective optimization problems , 2014, 2014 IEEE Symposium on Computational Intelligence in Multi-Criteria Decision-Making (MCDM).

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

[12]  Ye Tian,et al.  PlatEMO: A MATLAB Platform for Evolutionary Multi-Objective Optimization [Educational Forum] , 2017, IEEE Computational Intelligence Magazine.

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

[14]  Eckart Zitzler,et al.  HypE: An Algorithm for Fast Hypervolume-Based Many-Objective Optimization , 2011, Evolutionary Computation.

[15]  C.A. Coello Coello,et al.  MOPSO: a proposal for multiple objective particle swarm optimization , 2002, Proceedings of the 2002 Congress on Evolutionary Computation. CEC'02 (Cat. No.02TH8600).

[16]  R. Lyndon While,et al.  A faster algorithm for calculating hypervolume , 2006, IEEE Transactions on Evolutionary Computation.

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

[18]  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.

[19]  Stefan Roth,et al.  Covariance Matrix Adaptation for Multi-objective Optimization , 2007, Evolutionary Computation.

[20]  Nicola Beume,et al.  Pareto-, Aggregation-, and Indicator-Based Methods in Many-Objective Optimization , 2007, EMO.

[21]  K. Schittkowski,et al.  NONLINEAR PROGRAMMING , 2022 .

[22]  Prospero C. Naval,et al.  An effective use of crowding distance in multiobjective particle swarm optimization , 2005, GECCO '05.

[23]  Qingfu Zhang,et al.  Multiobjective evolutionary algorithms: A survey of the state of the art , 2011, Swarm Evol. Comput..