## 1 Introduction  
Hybrid evolutionary algorithms, by integrating the advantages of multiple search strategies, have proven to be robust and efficient in solving complex optimization problems. When applied to a generative adversarial network (GAN), they exhibit significant potential, achieving notable success in fields such as advanced prototyping [1], and enhancing robotic functionalities [2], optimizing logistics operations [3], and making improvements in predictive maintenance [4]. In recent years, with increasing attention on sustainable optimization studies, researchers have begun to explore how to incorporate sustainability principles into the integration of GANs and evolutionary algorithms, aiming to reduce computational resource consumption, improve model efficiency, and balance environmental responsibility and social impact while advancing technological progress. Despite their promising prospects, practical applications of GANs based on evolutionary algorithms face challenges, including deformation and distortion during generation, and a lack of diversity in outcomes, which can appear monotonous. To overcome these issues, researchers have developed improved models like DCGAN [5], Wasserstein GAN [6], and StyleGAN [7], significantly enhancing the quality and diversity of outputs and thus increasing the realism and innovativeness of results. GAN is one of the most important technologies which has made significant progress in the industry in recent years. At present, many researchers have studied GAN from different aspects: (1) objective function-based ones, (2) model structure-based ones, and (3) latent space-based ones. Objective function-based ones can improve the stability of adversarial training, reduce the problems of mode collapse and gradient vanishing during training, enhance the quality of generated images, and promote the convergence of models, e.g., the recurrent stacked GAN (RSGAN), the spectral normalized GAN (SNGAN), and the least squares GAN (LSGAN). However, the design of objective functions is often problem dependent, requiring different optimization strategies for different tasks and datasets. For model structure-based ones, since the model structure plays a decisive role in the performance of GAN, the expression ability of the generator and discriminator can be significantly enhanced by using complex model structure. Because the generator learns the real data distribution more effectively, it thus improves the quality of generated images. In addition, an excellent model structure also helps to alleviate problems such as gradient vanishing and mode collapse during training. However, the complex model structure causes the training process to become more computationally expensive. The optimization of the latent space through interpolation, transformation, and traversal can improve the robustness of model input and the controllability of generated sample. This helps to build more expressive latent space, which improves the quality and diversity of generated images. Compared with the objective function-based ones and the model structure-based ones, the optimization of the latent space has the following advantages: By adjusting the latent vector with different dimensions or directions, a finer control over the generated samples can be achieved. At the same time, the latent space of GAN is completely independent of other components, such as objective function and model structure. The latent space of GAN have emerged as a pivotal research frontier, and optimizing this space through evolutionary algorithms presents notable advantages. By simulating the process of biological evolution, evolutionary algorithms systematically explore the latent space, transcending the constraints of random initialization. In contrast to gradient descent, which can often converge to local optima, yielding samples that may not be globally optimal, evolutionary algorithms delve deeply into the search space, identifying latent vectors that are closer to global optimality.