Study On The Fibonacci Numbers And The Golden Ratio For Function Optimization
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Date
2025
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Department of Mathematics, NITS
Abstract
The Fibonacci sequence and its associated Golden Ratio appear prominently in nature and various art forms. Numerous natural phenomena—such as the branching patterns of trees, the arrangement of leaves (phyllotaxis), the growth patterns of pineapples, the flower arrangements of artichokes, cross-sections of trees, and the configuration of pine cone bracts—display characteristics that correspond with Fibonacci numbers. This exploration includes various theorems and puzzles surrounding Fibonacci num bers, such as Binet’s formula, the Fibonacci Q-Matrix, Cassini’s identity, Fibonacci bamboozlement, the Golden rectangle, spiraling squares, the Golden spiral, and the Fibonacci spiral. Furthermore, Fibonacci numbers and the Golden ratio are critical in contemporary computational optimization, especially in developing uni-modal and multi-modal functions that implement the Golden Search (GS) method. Genetic Al gorithms (GA) are a popular method for function optimization, leveraging selection strategies like Tournament Selection (TS) or Roulette-Wheel Selection (RS) and em ploying crossover techniques such as Single-Point Crossover (SC) or Uniform Crossover (UC), alongside bit-wise mutation operators. This study presents an optimization anal ysis using seven benchmark functions—Sphere, Ackley, Rosenbrock, Rastrigin, Schaffer N.1, Cosine Mixture, and Schwefel—by applying both GA and GS hydridized GA. This hybrid method aims to enhance convergence accuracy and improve overall optimiza tion performance by skillfully marrying the global search abilities of GA with the local refinement offered by GS.