Numerical Algorithm for L1 Norm Minimization Using Homotopy Methods

This article presents a numerical algorithm based on homotopy methods for solving L1 norm minimization problems. Homotopy methods, originating from topological mathematics, transform complex optimization problems into simpler ones through continuous deformation paths. By implementing homotopy continuation techniques, our algorithm efficiently solves high-dimensional optimization problems with L1 regularization. The core implementation employs a predictor-corrector scheme that traces the solution path while maintaining sparsity through active set updates. L1 norm minimization has become fundamental in statistics and machine learning due to its ability to produce sparse solutions and enhance model interpretability. The provided MATLAB implementation includes key functions for homotopy parameterization, gradient computations, and optimality condition checks. Through this program, we aim to provide researchers with an effective tool for L1 minimization while promoting the application of homotopy methods in numerical computing, thereby offering new approaches for computational mathematics research.