ADMM Algorithm for Solving L1 Regularization, Basis Pursuit (BP), and Basis Pursuit Denoising (BPDN) Problems

In this paper, we present an algorithm based on the Alternating Direction Method of Multipliers (ADMM) to solve three types of optimization problems: L1 regularization problems, Basis Pursuit (BP) problems, and Basis Pursuit Denoising (BPDN) problems. ADMM is an effective optimization algorithm that decomposes complex problems into smaller subproblems through variable splitting and dual decomposition. The algorithm implementation typically involves alternating between solving L2-norm subproblems (using techniques like conjugate gradient methods) and L1-norm subproblems (solved via soft-thresholding operations), followed by dual variable updates. This approach provides better understanding of the mathematical properties and characteristics of L1 regularization, BP, and BPDN problems while enabling deeper investigation into their fundamental behaviors. In subsequent sections, we will detail the algorithm's advantages and limitations, along with practical implementation guidelines for real-world applications. We believe this algorithm will play a significant role in future research developments, particularly in compressed sensing, signal processing, and sparse optimization domains.