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When constructing an investment portfolio, it is important to maximize returns while minimizing risks. This portfolio optimization can be considered as a multi-objective optimization problem that is solved by means of multi-objective evolutionary algorithms. The use of multi-objective evolutionary algorithms (MOEAs) provides an effective approach for dealing with the complex data involved in multi-objective optimization problems. However, current MOEAs often rely on a single strategy to obtain optimal solutions, leading to premature convergence and an insufficient population diversity. In this paper, a new MOEA called the Synergistic MOEA with Diffusion Population Generation (DPG-SMOEA) is proposed to address these limitations by integrating MOEAs with diffusion models. To train the diffusion model, a mixed memory pool strategy is optimized, which collects improved solutions from the MOEA/D-AEE, an optimized MOEA, as training samples. The trained model is then used to generate offspring. Considering the cold-start mechanism of the diffusion model, particularly during the training phase where it is not suitable for generating initial offspring, this paper adjusts and optimizes the collaborative strategy to enhance the synergy between the diffusion model and MOEA/D-AEE. Experimental validation of the DPG-SMOEA demonstrates the advantages of using diffusion models in low-dimensional and relatively continuous data analysis. The results show that the DPG-SMOEA performs well on the low-dimensional Hang Seng Index test dataset, while achieving average performance on other high-dimensional datasets, consistent with theoretical predictions. Overall, the DPG-SMOEA achieves better results compared to MOEA/D-AEE and other multi-objective optimization algorithms. [ABSTRACT FROM AUTHOR]

Multi-objective evolutionary algorithms (MOEA) have attracted much attention because of their good global exploration ability; however, their local search ability near the optimal value is weak, and for large-scale decision-variable optimization problems the number of populations and iterations required by MOEA are very large, so the optimization efficiency is low. Gradient optimization algorithms can overcome these difficulties well, but gradient search methods are difficult to apply to multi-objective optimization problems (MOPs). To this end, this paper introduces a stochastic weighting function based on the weighted average gradient and proposes two multi-objective stochastic gradient operators. Further, two efficient evolutionary algorithms, MOGBA and HMOEA, are developed. Their local search capability has been greatly enhanced while retaining the good global exploration capability by using different offspring update strategies for different subpopulations. Numerical experiments show that HMOEA has excellent capture ability for various Pareto formations, and it can easily solve multi-objective optimization problems with many objectives, which improves the efficiency by a factor of 5–10 compared with typical multi-objective evolutionary algorithms. HMOEA is further applied to the multi-objective aerodynamic optimization design of the RAE2822 airfoil and the ideal Pareto front is obtained, which indicates that HMOEA is an efficient optimization algorithm with potential applications in aerodynamic optimization design. [ABSTRACT FROM AUTHOR]