An Algorithm Utilizing Alternating Direction Method of Multipliers for Problem Solving

The Alternating Direction Method of Multipliers (ADMM) is an algorithmic framework widely applied to optimization problems, particularly effective for solving convex optimization problems with separable structures. It demonstrates excellent performance in handling L1 regularization problems, basis pursuit (BP) problems, and LASSO regression, decomposing complex problems into more manageable subproblems through variable splitting techniques. The core concept of ADMM involves decomposing the original problem into multiple subproblems and approaching the global optimum through alternating optimization of these subproblems. Its advantage lies in leveraging problem structure to break down the optimization process into sequential steps, where each step focuses on partial variables, thereby simplifying computation. In implementation, this typically involves alternating between primal variable updates and dual variable updates using gradient-based methods. For L1 regularization problems, ADMM can decompose the objective function into smooth and non-smooth components for separate optimization. The smooth component is typically solved using gradient descent or Newton's method with backtracking line search, while the non-smooth component can be efficiently handled through soft-thresholding operations implemented via proximal operators. In basis pursuit (BP) problems, ADMM transforms the problem into a constrained optimization form by introducing auxiliary variables, then alternately optimizes primal and auxiliary variables. This method effectively handles non-smoothness in BP problems while maintaining fast convergence rates through iterative shrinkage-thresholding algorithms. For LASSO regression problems, ADMM similarly demonstrates its advantages by decomposing the LASSO objective function into a least-squares problem and an L1 regularization problem. It achieves efficient optimization by alternately solving these two subproblems, where each step admits closed-form solutions through coordinate descent implementations. The convergence of ADMM is generally guaranteed for convex optimization problems, but practical applications require appropriate parameter selection (such as penalty parameters) to balance convergence speed and accuracy. The algorithm's parallelism gives it additional advantages in large-scale problems, enabling accelerated solution processes through distributed computing frameworks that implement variable updates across multiple processors.