A novel multi-objective evolutionary algorithm based on LLE manifold learning

In recent years, the LLE manifold learning algorithm is innovatively introduced to the multi-objective evolutionary algorithm to solve continuous multi-objective problems. It is according to the regularity that the Pareto set of a continuous multi-objective problem is a piece-wise continuous (m - 1)-dimensional manifold, and m is the objective number. The goal of the LLE manifold learning algorithm is to reduce the dimension of data. However, the existing LLE-based algorithm directly applies LLE manifold learning algorithm to the MOP modeling for individual reproduction. It has the following weaknesses: (1) the (d - 1)-dimensional manifold, which is constructed by refactoring coefficient, is not necessarily the manifold of the optimal solution space; (2) when resampling, the original information of samples is basically lost, especially in the case of d = 2, only keeping the scope of a linear interval. The distribution of the original samples information is completely lost and the cost of repeated calculation is higher; (3) the neighborhood relationship of resampling does not mean the neighborhood relationship of samples in PS space. So, we propose a new LLE modeling algorithm which is called O2O-LLE approach. The new O2O-LLE approach is inspired by LLE manifold learning idea and makes full use of the mapping function known in the MOP that the decision space is considered as the high-dimensional space and the object space is regarded as a low-dimensional space. Thus, the new modeling algorithm is no longer to build the overall low-dimensional space of the sample and then reflected back to the high-dimensional space, but it replaces directly constructing new samples in high-dimensional space. Thereby, the above four weaknesses are effectively avoided. Its steps are as follows: (1) mapping the samples from PS space to PF space; (2) searching K neighbors in PF space; (3) calculating LLE refactoring coefficient according to the K neighbors in PF space; (4) producing offspring sample according to the refactoring coefficient in PS space. Also, different from the early algorithm framework HMOEDA_LLE, the new algorithm framework O2O-LLE-RM does not include the genetic operation, so its efficiency is improved. To verify the performance of O2O-LLE-RM, several widely used test problems are employed to conduct the comparison experiments with three state-of-the-art multi-objective evolutionary algorithms: NSGA-II, RM-MEDA and Firefly. The simulated results show that the proposed algorithm has better optimization performance.