FR Conjugate Gradient Algorithm for Minimum Value Optimization

This article explores the application of the Fletcher-Reeves (FR) conjugate gradient algorithm to find the minimum value of a function. A minimum value refers to a point where the function's output is lower than all surrounding points. Given a target function y with variable x, the algorithm initiates the search from a starting point x0 to locate the minimizing point xm, returning the corresponding minimum value fm.

The FR conjugate gradient algorithm is an efficient optimization method that significantly reduces computational overhead during minimization. In implementation, the algorithm calculates gradient vectors to determine search directions, with each iteration updating the direction using the Fletcher-Reeves formula: β = ||∇f(x_{k+1})||² / ||∇f(x_k)||². This approach ensures conjugate directions and accelerates convergence toward the minimum. Key implementation steps include gradient computation, direction updating, and line search optimization.

By employing the FR conjugate gradient algorithm, we can accurately identify function minima, thereby gaining deeper insights into function behavior and characteristics. This method is particularly valuable for large-scale optimization problems where Hessian matrix calculation is computationally expensive. We hope this discussion provides useful guidance for your optimization tasks.