Solving Basis Pursuit Problems Using ADMM Algorithm

The ADMM (Alternating Direction Method of Multipliers) algorithm serves as an efficient tool for solving Basis Pursuit problems, particularly suited for applications like sparse signal recovery. The core objective of Basis Pursuit is to find the sparsest solution vector given an observation matrix and measurement vector, essentially formulating a convex optimization problem with L1 regularization.

ADMM decomposes the original problem into multiple subproblems for alternating optimization. The implementation typically begins by introducing auxiliary variables to separate the L1-norm term from other components in the objective function, followed by constructing an augmented Lagrangian function. Each iteration consists of three key steps: updating the primal variable (often involving least-squares solutions using matrix inversion or conjugate gradient methods), updating the auxiliary variable (achieved through soft-thresholding operations that promote sparsity), and updating the Lagrangian multiplier (similar to a gradient ascent step).

This approach combines the coordination benefits of dual decomposition with the convergence efficiency of multiplier methods, making it particularly suitable for large-scale sparse optimization. The algorithm's strength lies in its ability to break down complex problems into simpler subproblems while maintaining relative robustness to parameter selection. It finds widespread applications in compressed sensing and signal processing domains. During iteration, balancing the primal and dual residuals serves as a critical metric for controlling convergence, where appropriate tolerance thresholds can be implemented in code through while-loops or convergence checks.