Exploring Simultaneous Equation in Univariate Unconstrained Problems: Intersection Strategies

Abstract

The Steepest Descent Intersect (SDI) is an algorithm developed to solve unconstrained optimization problems, particularly those with multiple local minima or maxima. The algorithm uses simultaneous equation techniques to find the horizontal line-objective function intersection, generating a promising initial point that converges to a local minimum. This process continues until no intersection occurs, indicating the current solution has reached the lowest position, which represents the global solution. SDI can build a "bridge" within the valley for multimodal and wavy functions, allowing it to escape from the valley bottom and increase its chances of finding the global minimum. The steepest descent is sensitive to step size, with large steps causing large errors and small steps slowing down convergence rates. SDI’s characteristics help handle the sensitive issue of steepest descent regarding step size. The simulation results show the reliability of SDI in generating promising initial points and identifying the global minimum point, making it a good algorithm for solving univariate unconstrained optimization problems.