Steepest Descent Method: A Gradient-Based Optimization Algorithm for N-Dimensional Function Minimization

This documentation discusses the Steepest Descent Method, an approach for finding the minimum of N-dimensional objective functions by iteratively moving in the negative gradient direction. The implemented program uses this method to solve unconstrained optimization problems through iterative updates of the form x_{k+1} = x_k - α∇f(x_k), where α represents the step size determined by line search techniques.

It's worth noting that while the Steepest Descent Method serves as a fundamental optimization algorithm with straightforward implementation, it may exhibit limitations such as slow convergence rates (particularly for ill-conditioned problems) and susceptibility to local minima. The algorithm typically includes convergence checks based on gradient magnitude thresholds or maximum iteration limits. Therefore, selecting appropriate optimization algorithms based on specific problem characteristics—such as incorporating preconditioning or hybrid approaches—can lead to improved optimization performance.