Optimization Computation: Newton's Method + Conjugate Gradient Method Implementation

In this discussion, we further explore MATLAB program implementations combining Newton's Method and Conjugate Gradient Method for optimization computations. Newton's Method is an iterative approach for finding function minima or maxima, which constructs quadratic approximations of the original function at each iteration. The method typically involves calculating the Hessian matrix (second derivatives) and gradient vector, then solving the linear system HΔx = -∇f(x) to update the solution. Conjugate Gradient Method is another iterative technique for solving large-scale linear systems and optimization problems, utilizing gradient information from previous iterations to determine search directions for subsequent computations. This approach avoids storing large matrices and is particularly efficient for sparse systems. The combination of these methods enhances computational efficiency and precision by using Conjugate Gradient to solve the Newton equation when the Hessian is large or sparse. Key MATLAB implementations involve: - Function handles for objective function and gradient calculations - Hessian-vector products for memory-efficient operations - Line search procedures with Wolfe conditions - Convergence criteria based on gradient norms and function improvements This hybrid approach has found widespread applications in practical problems including machine learning, engineering design, and financial modeling. Further research can focus on developing more efficient programs with adaptive tolerance settings, preconditioning techniques, and parallel computing capabilities to solve various optimization challenges.