A new unidimensional search method for optimization: The 5/9 method

This paper presents a new unidimensional search method for nonlinear and unconstrained optimization based on considering the relationship between the three search points of the initial interval of uncertainty. In the algorithm forming the initial interval, each point is generated in such a way that every one of them will be two times as far as the previous one. Hence, when the algorithm is terminated, it is concluded with certainty that the last three points will satisfy the condition (c1 − a1) = 1/3(b1 − a1) where a1, b1, and c1 are the search points and [a1, b1] is to be the initial interval of uncertainty. The new method described in this contribution is essentially based on the above mentioned relationship. In our method, the condition (ck − ak) = 1/3(bk − ak) is always held in each iteration. We attempt to derive an iteration formula reducing the interval, and also satisfying the above mentioned equality for the new iteration. The performance of the new method, named the 5/9 method, is analyzed in terms of the most popular and widely used criteria; the number of iterations, the number of function evaluations, and the computer (CPU) time in comparison with the most effectual non-derivative methods, the Quadratic Interpolation, Golden Section, RMS, and AM methods, using 10 test functions.