Uniformity And Diversity Strategies On MOEA

Evolutionary algorithms seek out the Pareto optimal solutions from population to population. The group search feature for multi-objective optimization problems has obvious advantages and attracts wide attentions to scholars. A number of multi-objective evolutionary algorithms have been developed in the past decade. In many real-life applications of multi-objective optimization, the decision maker may not be interested in having an unduly large number of Pareto optimal solutions. Therefore, it is a non-trivial task to find a manageable number of Pareto optimal solutions which are evenly distributed along the Pareto front. In this paper, we study the uniformity and diversity of Pareto solutions and propose three ameliorate strategies.A monotonic increasing function is utilized to transform each individual objective function into the one so that the curve shape of the non-dominant solutions of the transformed multi-objective problem is close to the hyper-plane whose intercept of coordinate axes is equal to one in the original objective function space. We define the notion of weak distance preserving transformation, and argue for impact on distribution of the Pareto solutions. Consequently, we can approximate the Pareto optimal solutions that are uniformly distributed over the Pareto front using the advanced decomposition technique of MOEA/D. Numerical results show that the proposed algorithm has a good performance.A dynamic weight design method is proposed in this paper. In most of the multi-objective evolutionary algorithms based on aggregating objectives, the weight vectors are user-supplied or generated randomly, and they are static in the algorithms. If the Pareto front shape is friendly, the algorithms can work well, otherwise, it might fail. In this paper, we propose a dynamic weight design method which is based on the projection of the current non-dominant solutions and equidistant interpolation. Even if the PF is complex, we can find evenly distributed solutions by this method. Some test instances are constructed to compare the performance of the MOEA/D used dynamic weight design method with that of MOEA/D. The results indicate that the weight design method can dramatically improve the performance of the algorithms.We propose a multi-objective evolutionary algorithm based on competition and cooperation between multi-groups. It aims to divide the function space into several small regions according to the influence factor of each sub-group. Each sub-group includes an internal set and an external set. The internal set is composed of some best individuals in this sub-region, while the external set is composed of those individuals that are found before in this sub-region. Multi-group strategy makes the individuals only compare with the ones in the same sub-group, which largely reduces its complexity and the selection pressure for the dominant solutions in some sub-group. It is of great significance to maintain its population diversity. The competition and cooperation mechanism between the multi-groups can improve the computational efficiency of the algorithm. The external set can maintain diversity in population of the sub-region. There are a few of individuals in an internal set. The individual which has better objective values will dominate the population in the sub-region at once. The diversity of the population might be lost. Because there is a external set in each sub-region, the individual, which is replaced by the internal set or new generated one, will not be eliminated immediately, even though it is an dominated individual. But it will keep several generations for external set to get an opportunity to participate in crossover and mutation. However, it slightly increases the complexity of the algorithm, since the algorithm only randomly replace the external individuals. Numerical results show that the proposed algorithm has a good performance.