Success-Failure Method for Minimum Value Search Algorithm

The Success-Failure Method is a numerical optimization technique designed to locate function minimum values. This algorithm progressively approaches the minimum point by alternating between successful and failed steps through iterative step size adjustments.

The core algorithm operates as follows: Starting from an initial point, the method performs exploratory moves with a given step size. When a trial step succeeds (i.e., the function value decreases), the step size is increased to accelerate progress. Conversely, when a trial step fails (function value doesn't decrease), the step size is reduced and the search direction is reversed. Through repeated step size and direction adjustments, the algorithm gradually converges toward the neighborhood of the minimum value. In code implementation, this typically involves maintaining three key variables: current position, step size, and function evaluation count.

This method offers simple intuition and works effectively for low-dimensional optimization problems. However, step size selection requires careful consideration - overly large steps may cause oscillations around the minimum, while excessively small steps lead to slow convergence. For complex high-dimensional functions, combining this method with other optimization techniques may improve efficiency. The algorithm can be implemented with a while loop containing conditional statements that check function value changes after each step.

As a direct search method, the Success-Failure Algorithm doesn't require derivative information, making it suitable for cases where derivatives are difficult to compute. However, due to its relatively slow convergence rate, it's typically applied in scenarios with moderate accuracy requirements or smaller-scale problems. The method's termination condition often combines step size tolerance and maximum iteration limits to prevent infinite loops.