Solving non-linear optimization problems by a trajectory approach.

This article focuses on solving non-linear optimization problems using a trajectory method that introduces a parameter to connect an easily solvable problem to the original problem via a continuous solution path. The method is applied to two location problems: the single facility Weber problem and a competitive facility location model based on the gravity (Huff) model with exponential distance decay. For the Weber problem, the trajectory starts at the center of gravity (parameter value zero) and moves to the Weber solution (parameter value one-half), with an improved approximate starting point proposed to enhance iterative methods. In the competitive location model, the trajectory begins at a trivial solution for zero decay parameter and progresses to the desired solution, with computational experiments in Orange County, California, demonstrating that the trajectory method reliably finds optimal or near-optimal facility locations, often outperforming the Nelder-Mead optimization method. The paper concludes that the trajectory approach is a versatile technique suitable for various unconstrained optimization problems, particularly location problems.