Frank-Wolfe Algorithm MATLAB Implementation

In this article, we introduce the Frank-Wolfe algorithm, an optimization method particularly effective for handling very large datasets while maintaining sparsity throughout the iterative process. This algorithm proves exceptionally valuable for practitioners working with substantial data volumes. The MATLAB implementation leverages key functions like quadprog for solving linear subproblems and includes efficient line search procedures. We will explore the algorithm's fundamental principles and provide detailed explanations of the corresponding MATLAB code structure in the following sections.

The core implementation involves two main computational steps per iteration: first solving a linear minimization subproblem to determine the descent direction, then performing an optimal step size calculation. The MATLAB code typically utilizes matrix operations for efficient constraint handling and incorporates convergence checks through gradient norms. Key algorithmic features include conditional gradient computations and adaptive step-size selection strategies that ensure monotonic objective function improvement.

For large-scale applications, the implementation employs sparse matrix operations to reduce memory usage and computational overhead. The code structure includes modular components for objective function evaluation, gradient computation, and termination condition verification, making it adaptable to various constraint types including polyhedral and simplex constraints commonly encountered in machine learning applications.