Fibonacci indicator algorithm: A novel tool for complex optimization problems

Abstract

In this paper a new meta-heuristic algorithm is introduced. This optimization algorithm is inspired by the very popular tool among the technical traders in the stock market called the Fibonacci Indicator. The Fibonacci Indicator uses to predict possible local maximum and minimum prices, and periods in which the price of a stock will experience a significant amount of movement. The proposed Fibonacci Indicator algorithm is validated on several Benchmark functions up to 100 dimensions to have a comparison to algorithms such as DE extensions, PSO extensions, ABC, ABC-PS, CS, MCS and GSA in the ability of convergence and finding the global optimum in different research areas. Finally two engineering design problems are used to show the performance of the algorithm. Application of the proposed Fibonacci Indicator Algorithm in a wide set of benchmark functions has asserted its capability to deal with difficult optimization problems.

Introduction

In the last few decades, the use of meta-heuristic algorithms has been much improved to approach the optimized solution of nonlinear functions. A heuristic algorithm is a method to find the solution to an optimization problem by “trial-and-error”. How ever, these algorithms may not find the global best solution to the problem and might get trapped in the local optimum points. On the other hand, the metaheuristic algorithms finds the optimum solution by higher-level strategies employing trial-and-error,exploration, and exploitations. Particle Swarm Optimization (PSO) (Eberhart and Kennedy, 1995), Evolutionary Algorithms (EA) including Genetic Algorithm (GA) (Holland, 1975), Ant Colony Optimization (ACO) Bilchev and Parmee (1995), Dorigo and Blum (2005), and the Bee Algorithm (BA) (Pham et al., 2006) are among the most popular metaheuristic algorithms. Evolutionary algorithms and swarm intelligence-based algorithms are two main categories of population-based optimization (Karaboga and Akay, 2009). Genetic algorithms, Differential Evolution (DE) (Storm and Price, 1995), Meng and Pan (2016), Meng et al. (2018) and evolutionary strategy (ES) Rechenberg (1965), Schwefel (1965) have been the most popular techniques in evolutionary computation. Particle swarm optimization and Bees Algorithm are the most popular examples of swarm intelligence optimization. Two advantages of the different categories, i.e. evolutionary algorithms and swarm intelligence-based algorithms are presented below:

• Particularly useful in multi-modal and multi-objective optimization problems;

• Hybridize algorithms to each other;

A number of optimization algorithms can combine with each other and produce the hybrid algorithm with the synergy of both algorithm’s advantages and elimination of their disadvantages.

Global optimization can be applied to various branches of science, economics and engineering Bomze et al. (1997), Gergel (1997), Horst and Tuy (1996), Li et al. (2015), Rizk-Allah et al. (2016), Rizk-Allah et al. (2018). Generally, solving nonlinear optimization problems can be classified into deterministic and stochastic methods Li et al. (2015), Arora et al. (1995), Pardalos et al. (2000), Younis and Dong (2010). In deterministic methods, optimization problems are solved by creating deterministic progression of convergence at the global optimal solution. This method requires unfailing mathematical specification and responsively depends on the initial conditions. On the contrary, in the stochastic methods including heuristic and meta-heuristic methods, new points are randomly generated (Younis and Dong, 2010). The efficiency of the optimization algorithms is usually determined by their ability in finding the global best solution by the minimum cost usually corresponding to the number of function evaluations. Exploration and exploitation are two main strategies to find the global best solution. Poor exploring and very fast convergence of algorithms increase the chance of getting trapped in local minima. Furthermore, very slow converging and increasing function evaluations is not economical. The balance between exploration and exploitation is crucial to improve the efficiency of optimization algorithms (Li et al., 2015). Over the past decade, meta-heuristic algorithms such as GA Price et al. (2005), Yang et al. (2007), Chelouah and Siarry (2000) ACO, PSO (Jiang et al., 2007) and the artificial bee colony (ABC) have shown considerable successes in optimization algorithms (Ghanbari and Rhati, 2017). Previous researches show that ABC and GA have better exploration and slower convergence. However, ACO and PSO converge faster with more possibility of getting trapped in local optima Alshamlan et al. (2015), Premalatha and Natarajan (2009), Fidanova et al. (2014), Meng and Pan (2016). A nonlinear optimization problem can be formulated as a D-dimensional problem of the following type: Gergel (1997), Nguyen et al. (2014).

$$f\left(x\right)=\left\{\begin{array}{cc}min\hfill & f\left(x\right)\hfill \\ s.t.\hfill & 1\le x\le u\hfill \end{array}\right.$$

The objective function is defined by $f\left(x\right)$ and the $D$ dimensional vector of variables is $x=(x1,x2,\dots ,xD)$; lower and upper limits of variables are defined by $l=(l1,l2,\dots ,lD)$ and $u=(u1,u2,\dots ,uD)$.