ML-MOEA/SOM: A Manifold-Learning-Based Multiobjective Evolutionary Algorithm Via Self-Organizing Maps

Under mild conditions, it can be induced from the Karush–Kuhn–Tucker condition that the Pareto set, in the decision space, of a continuous Multiobjective Optimization Problems(MOPs) is a piecewise continuous ( 1) m D   manifold(where m is the number of objectives). One hand, the traditional Multiobjective Optimization Algorithms(EMOAs) cannot utilize this regularity property; on the other hand, the Regular Model-Based Multiobjective Estimation of Distribution Algorithm(RM-MEDA) only able to build the linear model of decision space using linear modelling algorithm, such as: the local principal component analysis algorithm(Local PCA).Aim at the shortcomings of EMOAs and RM-MEDA, the Manifold-Learning-Based Multiobjective Evolutionary Algorithm Via Self-Organizing Maps(ML-MOEA/SOM) is proposed for continuous multiobjective optimization problems. At each generation, first, via Self-Organizing Maps, the proposed algorithm learns such a nonlinear manifold in the decision space; then, new trial solutions is built through expanding the neurons of SOM with random noise; at the end, a nondominated sorting-based selection is used for choosing solutions for the next generation. Systematic experiments have shown that, overall, ML-MOEA/SOM outperforms NSGA-II, and is competitive with RM-MEDA in terms of convergence and diversity, on a set of test instances with variable linkages. We have demonstrated that, compared with NSGA-II and RM-MEDA, via self-Organizing maps, ML-MOEA/SOM can dig nonlinear manifold hidden in the decision space of multiobjective optimization problems.