MATLAB Implementation Example of DFP Algorithm with Code Description

The DFP algorithm is a classical quasi-Newton optimization method, fully named the Davidon-Fletcher-Powell algorithm, which serves as a representative variable metric method. It approximates the inverse Hessian matrix to avoid direct computation of second-order derivatives, demonstrating excellent performance in unconstrained optimization problems. The core algorithmic approach involves several key steps: First, calculating the gradient of the objective function forms the foundation of all quasi-Newton methods. The distinctive feature of DFP lies in its unique correction formula, which updates the approximation of the inverse Hessian matrix using the displacement of the current iteration point and the gradient difference. During each iteration, the algorithm performs a one-dimensional search along the quasi-Newton direction, which can be implemented using either exact line search or inexact search strategies like Armijo condition. When implementing in MATLAB, several technical aspects require attention: First is gradient computation handling, which can be achieved through analytical differentiation or numerical difference methods. The inverse Hessian matrix is typically initialized as an identity matrix. The iteration process requires setting reasonable convergence criteria, such as gradient norm thresholds or function value change tolerances. Regarding memory management, the DFP algorithm needs to store an n×n matrix, which may become a bottleneck when handling high-dimensional problems. Common implementation challenges include: addressing stationary point issues in non-convex functions, ensuring step size selection stability, and managing accumulated rounding errors in numerical computations. Compared to the BFGS algorithm, DFP is more sensitive to the initial choice of the inverse Hessian matrix, requiring special attention during implementation. A typical MATLAB implementation would contain a main loop structure, gradient computation subroutine, line search module, and matrix update module. A well-structured implementation should also include detailed iteration information output for debugging and algorithm behavior analysis. In practical applications, the DFP algorithm is commonly used for medium-scale optimization problems, particularly when function evaluation costs are high, where its superlinear convergence rate provides significant advantages. Key implementation details include: - Gradient computation using either symbolic differentiation with MATLAB's Symbolic Math Toolbox or numerical approximation with finite differences - Line search implementation using fminbnd for exact search or while-loops with Wolfe conditions - Matrix update operations using efficient MATLAB matrix manipulation functions - Convergence checking with norm(gradient) < tolerance conditions