The development and application of the sparse search direction in thresholding-type sparse optimization algorithms

Sparse optimization problems are widely applied in fields such as wireless communication and image processing, as these scenarios often involve sparse signals or vectors as well as noise. Compressed sensing refers to a typical type of sparse optimization problem that aims to find the optimal solution that minimizes the objective function subject to certain sparsity constraints. The purpose of this project is to study a class of classical sparse optimization problems in compressed sensing, in order to provide a novel sparse search direction for threshold algorithms, thus reducing the influence of thresholding operators on the objective functions. It aims to design a new search direction, make it sparse, and improve the existing threshold class algorithm to improve the performance of sparse optimization algorithms. Three phases are involved in the technical development of the project. The first step in improving a threshold algorithm is to evaluate the existing algorithm and identify a direction of improvement. The second step is to explore a new search direction, analyze the relationship between the current point sparsity, the search direction sparsity, and the objective function decline, and then design a new thresholding-type algorithm. Finally, the algorithm parameters are modeled, the idea of machine learning is applied to find the optimal parameters, and the new algorithm is verified and analyzed based on the experimental results. This project is expected to contribute significantly to solving a variety of complex problems through practical application.