Solving Optimization Problems Using Quasi-Newton Methods

Quasi-Newton methods are highly efficient algorithms widely used in numerical optimization that approximate the Hessian matrix from Newton's method to avoid direct computation of second-order derivatives. These algorithms are particularly suitable for solving large-scale nonlinear optimization problems, with the BFGS algorithm being one of the most renowned quasi-Newton methods. When implementing quasi-Newton algorithms in MATLAB, several key programming steps are typically required: First, initialize the objective function and starting point using function handles like `@(x) objective(x)`. Then construct an initial approximation of the Hessian matrix or its inverse, often starting with an identity matrix. The algorithm iteratively updates this approximation matrix using gradient information and parameter updates through specific update formulas. In MATLAB, this can be implemented using `fminunc` with the 'quasi-newton' option or by custom coding the BFGS update formula: `H_new = H + (s*s')/(s'*y) - (H*y*y'*H)/(y'*H*y)` where `s` is the step vector and `y` is the gradient difference. Each iteration involves performing a line search along the current search direction to determine an appropriate step size, which can be automated using MATLAB's `fminbnd` or Wolfe condition implementations. Compared to traditional Newton's method, quasi-Newton algorithms significantly reduce computational complexity for high-dimensional problems by avoiding explicit Hessian matrix computation and storage. The BFGS algorithm ensures algorithm stability through its special matrix update formula that maintains positive definiteness of the approximation matrix. MATLAB provides an ideal environment for implementing these numerical optimization algorithms due to its efficient matrix operations and built-in visualization tools. Users can easily monitor convergence using `optimplot` functions, debug parameter settings, and visually compare performance across different optimization algorithms through convergence plots and objective function value tracking.