The Primary Objective of Basis Pursuit (BP) Algorithm is to Find Solutions for Underdetermined Systems

The Basis Pursuit algorithm is a mathematical optimization technique designed to solve underdetermined linear systems of equations. These problems typically possess infinitely many solutions, but Basis Pursuit distinguishes itself by identifying the sparsest solution—the one with the fewest non-zero elements. This algorithm transforms the original problem into a linear programming formulation. Linear programming represents a well-established domain in mathematical optimization with numerous efficient solving algorithms available. Basis Pursuit achieves sparsity in the solution vector by minimizing the L1-norm, which contrasts with traditional least-squares methods that utilize the L2-norm. In implementation, this typically involves using optimization solvers like CVX in MATLAB or linprog functions with specific constraint handling. Basis Pursuit finds significant applications across multiple domains including signal processing, compressed sensing, and statistical learning. In these contexts, the sparsity assumption is generally valid, making Basis Pursuit solutions more aligned with practical requirements. The computational efficiency of the algorithm depends heavily on the linear programming solver employed, with modern optimization toolkits capable of handling large-scale problems through interior-point methods or simplex algorithms. Key implementation considerations include proper constraint formulation and regularization parameter selection for stable convergence.