# A Deep Learning-Driven Black-Box Benchmark Generation Method via Exploratory Landscape Analysis
## 2 Related Works  
Landscape analysis employs mathematical and statistical techniques to quantify the characteristics of optimization problems, thereby effectively describing features and spaces. [PERSON] et al. [11] first proposed and developed a tunable landscape generator. [PERSON] et al. [12] subsumed a number of techniques employed to obtain knowledge about the properties of an unknown optimization by exploratory landscape analysis. [PERSON] et al. [13] showed fitness landscape analysis to better understand problems before deciding on approaches to solving the problem.  
Traditional benchmark suites, such as CEC and BBOB, have been widely used to assess algorithm performance across diverse landscape types. [PERSON] et al. [14] described that the CEC 2005 suite and the BBOB benchmark suite have similar distributions for some exploratory fitness landscape analysis measures. [PERSON] et al. [15] showed that the CEC benchmark suite provides poor coverage of the problem space. [PERSON] et al. [16] demonstrated that some benchmark functions show high performance correlation, which limits their utility in algorithm selection and generalization.  
To address this issue, researchers have proposed generative methods for creating benchmark functions. [PERSON] and [PERSON] [10] introduced a genetic programming-based framework for evolving symbolic functions that match specified ELA feature vectors. Although effective in increasing feature diversity, their method incurs high computational costs and offers limited extrapolation control.  
More recently, deep learning-based approaches have been proposed. [PERSON] et al. [6] developed a method that optimizes objective values using CMA-ES to approximate target ELA features, followed by training neural networks as surrogate functions. Their approach demonstrates the ability to interpolate and extrapolate within the ELA feature space, creating new functions with controllable and diverse landscape properties. [PERSON] et al. [17] proposed using affine recombination of existing BBOB functions to create new benchmark problems. [PERSON] et al. [18] showed that the promising performance of the trajectory-based per-run algorithm selection with warm-starting. [PERSON] et al. [6] showed that machine learn methods create functions that offer combinations of landscape properties not found in the current BBOB benchmark suite. While computationally efficient and interpretable, this method is restricted to convex combinations and does not support extrapolation beyond the span of existing feature vectors.  
Compared to these approaches, our method leverages CMA-ES optimization of topological map points and trains surrogate models using feed-forward neural networks. By integrating Inverse Distance Weighting Regression (IDWR), our method enhances extrapolation accuracy and reduces over-smoothing in rugged landscapes. Unlike previous methods, our framework enables controllable generation of benchmark functions that span both existing and novel regions of the ELA space, supporting both imitation and innovation in benchmark construction.