There exist many multi-objective optimization problems (MOPs) containing several inequality and equality constraints in practical applications, which are known as CMOPs. CMOPs pose great challenges for existing multi-objective evolutionary algorithms (MOEAs) since the difficulty in balancing the objective minimization and constraint satisfaction. Without loss of generality, the distribution of the Pareto set for a continuous m-objective CMOP can be regarded as a piecewise continuous manifold of dimension (m − 1). According to this property, a self-organizing map (SOM) approach for constrained multi-objective optimization problems is proposed in this article. In the proposed approach, we adopt the strategy of two population evolution, in which one population is evolved by considering all the constraints and the other population is used to assist in exploring the areas. In the evolutionary stage, each population is assigned a self-organizing map for discovering the population distribution structure in the decision space. After the topological mapping, we utilize the extracted neighborhood relationship information to generate promising offspring solutions. Afterwards, the neuron weight vectors of SOM are updated by the objective vectors of the surviving offsprings. Through the proposed approach, we can make the population efficiently converge to the feasible region with suitable levels of diversity. In the experiments, we compare the proposed method with several state-of-the-art approaches by using 48 benchmark problems. The evaluation results indicate that the overwhelmingly superior performance of the proposed method over the other peer algorithms on most of the tested problems. The source code is available at https://github.com/hccccc92918/CMOSMA.

Although multiobjective particle swarm optimizers (MOPSOs) have performed well on multiobjective optimization problems (MOPs) in recent years, there are still several noticeable challenges. For example, the traditional particle swarm optimizers are incapable of correctly discriminating between the personal and global best particles in MOPs, possibly leading to the MOPSOs lacking sufficient selection pressure toward the true Pareto front (PF). In addition, some particles will be far from the PF after updating, this may lead to invalid search and weaken the convergence efficiency. To address the abovementioned issues, we propose a competitive swarm optimizer with probabilistic criteria for many-objective optimization problems (MaOPs). First, we exploit a probability estimation method to select the leaders via the probability space, which ensures the search direction to be correct. Second, we design a novel competition mechanism that uses winner pool instead of the global and personal best particles to guide the entire population toward the true PF. Third, we construct an environment selection scheme with the mixed probability criterion to maintain population diversity. Finally, we present a swarm update strategy to ensure that the next generation particles are valid and the invalid search is avoided. We employ various benchmark problems with 3–15 objectives to conduct a comprehensive comparison between the presented method and several state-of-the-art approaches. The comparison results demonstrate that the proposed method performs well in terms of searching efficiency and population diversity, and especially shows promising potential for large-scale multiobjective optimization problems.

Because of the high cost of experimental data acquisition, the limited size of the sample set available when conducting tissue structure ultrasound evaluation can cause the evaluation model to have low accuracy. To address such a small-sample problem, the sample set size can be expanded by using virtual samples. In this study, an ultrasound evaluation method for the primary α phase grain size based on the generation of virtual samples by a generative adversarial network (GAN) was developed. TC25 titanium alloy forgings were treated as the research object. Virtual samples were generated by the GAN with a fully connected network of different sizes used as the generator and discriminator. A virtual sample screening mechanism was constructed to obtain the virtual sample set, taking the optimization rate as the validity criterion. Moreover, an ultrasound evaluation optimization problem was constructed with accuracy as the target. It was solved by using support vector machine regression to obtain the final ultrasound evaluation model. A benchmark function was adopted to verify the effectiveness of the method, and a series of experiments and comparison experiments were performed on the ultrasound evaluation model using test samples. The results show that the learning accuracy of the original small samples can be increased by effective virtual samples. The ultrasound evaluation model built based on the proposed method has a higher accuracy and better stability than other models.

The popular trend of today’s music can be obtained by deep excavation, analysis, and prediction of the audience’s preferences. Using huge music library resources and user behavior to form music big data and truly realizing the aggregation of audience preferences determine the popular development trend of music. Therefore, this paper will apply data mining (DM) technology, introduce neural network (NNS) theory, establish a prediction model of music fashion trend, predict and evaluate the music fashion trend according to the selected evaluation index, find the change of music fashion trend in time, and provide decision-making basis for music fashion trend. In this paper, the prediction of music popularity trend based on NNS and DM technology is studied. In the prediction of the number of songs played by 10 artists, the NNS algorithm proposed in this paper reduces the prediction effect from the original 0.074 and 0.045 to 0.044 and 0.032, respectively, and the error rates are reduced by 35.7% and 29.4%, respectively, compared with the learning algorithm and the decision tree algorithm. Among the three methods, the NNS algorithm in this paper has the highest accuracy. Therefore, it can be proved that the model proposed in this paper is more suitable for predicting the trend of music popularity. In the end, it can accurately control the trend of pop music and also realize the aggregation of user preferences to determine the trend of pop music.