MATLAB Implementation of Frank-Wolfe Algorithm (FW) with Code Optimization Details

The Frank-Wolfe algorithm (also known as the conditional gradient method) is a classical optimization algorithm particularly suitable for solving constrained convex optimization problems. The core principle involves solving linear approximation problems at each iteration to determine search directions, gradually converging toward the optimal solution. When implementing the Frank-Wolfe algorithm in MATLAB, the following key steps are essential: Initialization: Select a feasible starting point, which can be any point within the constraint set or a heuristic-based initial solution. In MATLAB code, this typically involves defining initial variables and constraint boundaries using arrays or matrices. Gradient Computation: At each iteration, calculate the gradient of the objective function using MATLAB's symbolic differentiation or numerical gradient functions like `gradient()` or custom analytical derivatives. This determines the descent direction at the current point. Linear Optimization Subproblem: Solve a linear optimization problem to identify the optimal search direction. MATLAB's `linprog()` function efficiently handles this subproblem, especially when constraints are linear and well-structured. This step is computationally simpler than solving the original nonlinear problem. Step Size Selection: Implement either fixed step sizes or dynamic line search methods. MATLAB implementations often use `fminbnd()` for exact line searches or armijo condition implementations for adaptive step sizing during solution updates. Convergence Checking: Monitor stopping criteria based on gradient norms (using `norm()` function), objective value changes, or maximum iteration limits. Typical convergence thresholds range from 1e-6 to 1e-8 for gradient norms. A significant advantage of the Frank-Wolfe algorithm is its ability to handle linear optimizations over constraint sets without projection operations, making it particularly efficient for high-dimensional problems with complex constraints. However, convergence rates are generally slower near optimal solutions compared to projection-based methods. In MATLAB implementations, developers can leverage built-in optimization tools like `linprog()` for linear subproblems while integrating gradient computation and iterative update logic. The algorithm finds applications in machine learning (sparse optimization), signal processing (basis pursuit), and engineering optimization, especially for large-scale sparse problems where constraint handling is computationally challenging.