Generalized Fibonacci search for optimization of unconstrained one-and two-dimensional unimodal functions - Chong, Chin Yoon * (2023)
Undeniably, optimization problems of smooth functions have been extensively investigated by many researchers with the use of calculus throughout the years. On the contrary, most practical problems involve the optimization of non-smooth or discontinuous functions, which may require non-calculus methods. One approach is the Fibonacci Search. This study investigates the minimization of the unconstrained one- and two-dimensional unimodal functions by generalizing the Fibonacci Search. In the first part of this research, we begin with the investigation of line search methods using Fibonacci numbers on one-dimensional unimodal functions, f(x). Starting from a fixed point, we use the Initial Interval Search to create an initial interval to bound the global optimum of f(x). Next, we consider the “ratio length of 1” from the conventional Fibonacci Search to implement a different approach to the Fibonacci Search by taking the lower successive Fibonacci numbers as the initial ratios with specifying the number of iterations to be used beforehand, where the number of iterations to achieve the optimum nth sub-Fibonacci number, Sn(a,b,p,q), where a,b ∈Z+∪{0} and p,q ∈Z+ such that S0 = a, S1=b and Sn+2=pSn+1 + qSn, for n≥0, a+b>0, and q=2p2−1. Furthermore, we generate possible combinations of a, b, p and q to guarantee the achievement the optimum of f(x). Due to the efficiency, the optimistic nth sub-Fibonacci number, Sn(a=1,b=3,p=2,q=7) is chosen, and we name this number as Fibo1327. Next, we execute the Bracketing Search to narrow the interval of uncertainty by developing a macro program in Microsoft Excel to optimize several one-dimensional benchmark functions. Using the same width of the initial interval and based on the necessary and sufficient conditions, the Fibo1327 Search requires less iterations to achieve the approximate optimum and out-performs the Golden Section Search and other Fibonacci-like Search methods such as the Fibonacci, Lucas and Pell approaches. In the second part of this research, we introduce a new search method, namely the Swing Descent method by extending the Generalized Fibonacci Search to solve a two-dimensional unimodal function, f(x,y). The Swing Descent method also involves the use of Initial Interval and Bracketing searches. Without knowing an initial domain that bounds the global optimum point, the Swing Descent method only needs the initial point on f(x,y) and the initial step sizes along the x- and y-axes. We develop a macro program in Microsoft Excel to execute the algorithm of the Swing Descent method to optimize a number of two-dimensional benchmark functions. From the initial point, together with the Fibo1327 Search and the subsequent step sizes produced recursively from the initial step sizes, the Swing Descent method is repeated iteratively on the cross-sectional functions along the x- and y-axes, until the desired approximate optimum of f(x,y) is reached within the boundary restricted by the tolerance. It performed effectively in locating the approximate optimum point. We conclude that the optimization problem of one- and two-dimensional unimodal functions, whether smooth or non-smooth, continuous or discontinuous, can be solved approximately by the Generalized Fibonacci Search.