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arXiv · 2509.15972

In ratio section method and algorithms for minimizing unimodal functions

Vladimir Kodnyanko
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Abstract

This paper proposes a new method for section an interval in a given ratio intended for minimizing unimodal functions. The ratio section search is capable of quickly recognizing monotone functions and functions with a flat bottom, which contributes to increasing its performance, as measured by the number of minimized function evaluations. The method is implemented as passive and active algorithms. A comparison of the performance of the developed method with that of the classical methods of bisection search and the golden section search was performed on the basis of the data used to minimize twenty unimodal functions of various types. For all types of functions, the passive algorithm is 2.26 times faster than the bisection search and 1.72 times faster than the golden section method. Thus, the proposed method turned out to be the fastest of the known methods of cutting off segments intended for minimizing unimodal functions. The active algorithm is faster: for all types of functions, these indicators are 3.31 and 2.52, respectively. The fastest combined Brent method was also modernized. After the golden section procedure is replaced with a procedure for dividing a segment in a given ratio, a numerical experiment is conducted. The modernized method is 1.69 times faster than its prototype. Moreover, the performance of the active algorithm for dividing a segment at a given ratio exceeds that of the Brent method by 1.48 times for all types of functions. The modernized Brent method is approximately 4 times faster than the bisection search and 3 times faster than the golden section method.

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Vladimir Kodnyanko. 2026-04-29. In ratio section method and algorithms for minimizing unimodal functions. https://arxiv.org/abs/2509.15972

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