Convex Optimization Toolkit for Solving L1-Norm Minimization Problems

L1-norm minimization represents a crucial class of convex optimization problems with extensive applications in signal processing, machine learning, and statistical modeling. Its core objective is to find sparse solutions through L1 regularization for feature selection or noise suppression. Modern convex optimization toolkits typically offer the following six prominent solution approaches for these problems:

Linear Programming Transformation Method Converts the original problem into linear programming form and solves it using simplex methods or interior-point algorithms. This classical approach is particularly suitable for small to medium-scale problems, where implementation involves constructing constraint matrices and objective vectors compatible with LP solvers like MATLAB's linprog or Python's scipy.optimize.linprog.

Proximal Gradient Method Applies proximal operators to handle the non-differentiable L1 term while using gradient descent for the smooth component. Ideal for objective functions decomposable into smooth and non-smooth parts, with code implementation typically involving gradient calculations for the smooth part and soft-thresholding operations for the L1 norm.

Alternating Direction Method of Multipliers (ADMM) Decomposes problems into multiple subproblems through variable splitting and augmented Lagrangian methods, solved iteratively. Excels in large-scale distributed computing scenarios, where implementation requires separate updates for different variable blocks and dual variable adjustments.

Iterative Shrinkage-Thresholding Algorithm (ISTA/FISTA) Fast algorithms based on soft-thresholding operations, with FISTA achieving O(1/k²) convergence speed through Nesterov acceleration. Particularly suitable for high-dimensional sparse recovery problems, implemented using iterative thresholding steps with momentum updates in FISTA.

Coordinate Descent Method Optimizes sequentially along coordinate directions, updating only single variables per iteration. Demonstrates significant efficiency for strongly sparse models, commonly used in LASSO regression with implementations featuring cyclic or random coordinate selection strategies.

Second-Order Cone Programming (SOCP) Reformulation Transforms problems into second-order cone programming through convex relaxation, solved via interior-point methods. Offers theoretical advantages in constrained L1 optimization scenarios like basis pursuit denoising, requiring cone constraint formulations in optimization solvers.

These methods possess distinct characteristics: linear programming and SOCP guarantee global optimality but involve higher computational costs; gradient-based algorithms better suit large-scale data; while ADMM excels in parallelization. Practical selection requires balancing problem scale, precision requirements, and computational resources, with considerations for algorithm-specific parameters and convergence criteria in implementation.