## 3 Method
### Overall Framework of ICSO
In the context of “Improved HC”, \(X_{i}\) and \(X_{j}\) denote individuals that are formulated using Gaussian noise, whereas \(G_{best}\) and \(P_{best}\) represent the global and local optimal solutions, respectively, emulating principles from PSO. Within the framework of “VC”, \(D_{1}\), \(D_{2}\), and \(D_{3}\) indicate the dimensions of individuals across various dimensions. Concerning the “Selection based on _fitness_”, the fitness is calculated through two components: the gradient penalty and normalization. ### Problem Definition

The objective function of the GAN can be expressed as follows:

\[\min_{G}\max_{D}V(D,G)=\mathbb{E}_{x\sim p_{data}(x)}[\log D(x)]+\mathbb{E}_{z \sim p_{z}(z)}[\log(1-D(G(z)))] \tag{1}\]

where \(p_{data}(x)\) represents the distribution of real data; \(p_{z}(z)\) denotes the prior distribution of the latent variable \(z\). The Discriminator \(D\) is responsible for distinguishing between real data and data generated by the generator. The Generator \(G\) is designed to create data that closely mimic the distribution of real data. The latent variable \(z\) is randomly generated during the training process of GAN, which may lead to issues such as poor quality of generated samples and mode collapse. The main objective of optimizing the latent space is to find the optimal latent variable \(z\), thereby improving the quality and diversity of the samples generated by the generator. When considering the use of evolutionary algorithms to optimize the latent variable \(z\), we can define the fitness to guide the optimization of the randomly generated latent variable \(z\), which can be expressed as follows:

\[z^{*}=\operatorname*{arg\,min}_{z\in\mathbb{R}^{d}}F(F_{q},F_{d},GP) \tag{2}\]

where \(F_{q}\) and \(F_{d}\) are evaluation functions for assessing the quality of generation and the diversity of generation, respectively. \(GP\) is gradient-based regularization. ### Population Initialization

Population initialization is a crucial initial step in ICSO, where it generates \(n\) individuals to form the initial population \(\mathcal{P}_{0}\):

\[\mathcal{P}_{0}=\{X_{1},X_{2},\ldots,X_{n}\} \tag{3}\]

where each individual \(X_{i}\) is a vector with a fixed dimensionality. A high-quality initialization not only increases the probability of the algorithm finding the global optimal solution but also effectively reduces the risk of being trapped in local optima. The latent space of GAN is often generated using the Gaussian distribution. When employing evolutionary algorithms to optimize this latent space, the formula involved in initializing individuals is as follows:

\[X=\mu+\sigma\cdot\eta \tag{4}\]

where \(\mu\) represents the mean, which indicates the average value of a dataset, and its formula is as follows:

\[\mu=\int_{-\infty}^{\infty}xf(x)dx \tag{5}\]

where \(f(x)\) denotes the probability density function. In Equation (4), \(\sigma\) represents the standard deviation, which measures the dispersion of data points around the mean, and its formula is as follows:

\[\sigma=\sqrt{\int_{-\infty}^{\infty}(x-\mu)^{2}f(x)dx} \tag{6}\]

and \(\eta\) represents a random value drawn from the standard normal distribution \(\mathcal{N}(0,1)\), and its probability density function is as follows:

\[f(\eta)=\frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}\eta^{2}} \tag{7}\]Evaluation is a key operation in the evolutionary algorithm. By measuring the fitness of individuals, evaluation provides an effective evolutionary direction for a population. This paper approaches the problem from the perspective of multi-objective optimization, with a focus on two competing attributes of the generated samples: quality and diversity. On one hand, the generated samples need to exhibit high quality, meaning that they should be realistic enough to trick the discriminator. On the other hand, diversity must be ensured so that the distribution of generated samples is broad enough to prevent mode collapse.