An Algorithm Using Alternating Direction Method of Multipliers for Problem Solving

The Alternating Direction Method of Multipliers (ADMM) serves as an efficient algorithm for solving L1 regularization problems, basis pursuit (BP) problems, and LASSO problems. The ADMM algorithm demonstrates particular advantages when handling large-scale datasets, with its core methodology involving decomposing the original problem into several sub-problems and iteratively updating each sub-problem to gradually approach the optimal solution of the original problem. In solving L1 regularization problems, ADMM introduces Lagrange multipliers to handle constraints, effectively transforming the constrained optimization problem into an unconstrained formulation. From an implementation perspective, the algorithm typically involves alternating updates between the primal variables and dual variables, where the update steps often leverage proximal operators for efficient computation. For basis pursuit problems, ADMM incorporates penalty terms to achieve regularization, thereby enhancing the model's generalization capability. The algorithm structure typically includes an augmented Lagrangian formulation where the penalty parameter controls the trade-off between constraint satisfaction and objective minimization. ADMM also proves effective for solving LASSO problems by decomposing the objective function and solving components sequentially to reduce model complexity and improve solution sparsity. The algorithm's iterative nature allows for parallel computation of certain subproblems, making it suitable for distributed optimization scenarios. Key implementation aspects include proper parameter tuning, convergence criteria setting, and efficient linear algebra operations for variable updates. Therefore, ADMM represents a highly flexible and powerful optimization framework that finds applications across multiple scenarios, particularly benefiting from its convergence properties and scalability characteristics in large-scale optimization tasks.