Optimization Computational Methods: Quasi-Newton Method with Code Implementation

In optimization computational methods, the Quasi-Newton method is a widely-used optimization algorithm. This algorithm approximates the Hessian matrix of the objective function through a series of matrix updates, enabling the iterative solution for optimal values. The following code example demonstrates the BFGS (Broyden-Fletcher-Goldfarb-Shanno) implementation: The BFGS method implementation includes several key components: - Initialization of the approximation matrix as identity matrix - Gradient computation using automatic differentiation - Line search for optimal step size determination - Matrix update formula maintaining positive definiteness def bfgs_method(f, x0, eps=1e-5, max_iter=100): x = x0 B = np.eye(x0.shape[0]) # Initialize Hessian approximation as identity matrix g = grad(f)(x) # Compute initial gradient using automatic differentiation for i in range(max_iter): d = -np.dot(B, g) # Compute search direction using current Hessian approximation alpha = line_search(f, grad(f), x, d)[0] # Perform line search for optimal step size x_new = x + alpha * d # Update position with calculated step g_new = grad(f)(x_new) # Compute new gradient at updated position s = x_new - x # Position difference vector y = g_new - g # Gradient difference vector rho = 1 / np.dot(y, s) # Scaling factor for matrix update # BFGS matrix update formula maintaining positive definiteness B_new = (np.eye(x.shape[0]) - rho * np.outer(s, y)) @ B @ (np.eye(x.shape[0]) - rho * np.outer(y, s)) + rho * np.outer(s, s) if np.linalg.norm(g_new) < eps: # Convergence check based on gradient magnitude break x, g, B = x_new, g_new, B_new # Update variables for next iteration return x # Return optimized solution The algorithm iteratively updates both the solution approximation and the Hessian matrix approximation, providing superlinear convergence while avoiding direct Hessian computation.