Theoretical Investigation And Feature Analysis Of Fibonacci Tree Optimization For Solving Multimodal Functions

Typical intelligent optimization algorithms,such as Genetic Algorithm and Particle Swarm Optimization,and Differential Evolution Algorithm,are confronted with premature convergence and thus drop into local optima when they are applied to solve multi-model functions which have multiple global optima and local optima.The features of these algorithms were under continuous theoretical investigation,leading to fruitful studies.Some of these algorithms and their improved versions,thanks to excellent performances,are widely used in solving complex optimization problems in engineering.Premature convergence,as one of the major features of such algorithms,becomes an area which lots of new algorithm models have tried to research and improve in recent years.The basic structure and schema of typical intelligent algorithms are employed and designed for solving global optima of the target functions.In order to solve multiple solutions of a multi-model function,many new algorithms introduce new operators or propose new schema on the basis of typical intelligent algorithms,with the result that algorithms become more complex and new problems or issues appear.How to address these new problems or further improve the algorithms are hot issues for research in the field.For the purpose of addressing premature convergence of typical intelligent optimization algorithms when solving global optima of multimodal functions and tackling the limitation of those algorithms in solving multiple local optima,this dissertation develops a computational intelligent optimization algorithm—Fibonacci tree optimization(FTO)—to solve global optima and multiple optima of n-dimensional multimodal functions.FTO is built as a Fibonacci structure on the basis of the Fibonacci search method for solving unidimensional unimodal function and metaheuristic search of intelligent algorithms.FTO has the feature of global randomness and alternates between global and local searches in the search space of target functions so as to protect FTO from trapping into local optima while solving global optima.Meanwhile,FTO itself is capable of automatically solving multiple local optima of multimodal functions in a probabilitydriven manner.In this dissertation,the principle and features of FTO are described and analyzed,and the characteristic of FTO for solving global optima and multiple local optima is also investigated.The structure and highlights of the study are as follows:Firstly,the mechanism of FTO is analyzed.The features of global randomness and alternation between global and local searches,as displayed during the operation of FTO,are investigated.Two rules for building a Fibonacci tree are elaborated in details and the concepts of basic structure,Fibonacci tree and Fibonacci tree chain are also defined.Two core configuration parameters,namely Fibonacci tree nested depth and depth,are developed.FTO searches the results globally and locally in target search space alternatively,prevents the algorithm to trap into local optima when solving the global optima of a multimodal function,and thus demonstrates a superior validity.Secondly,Fibonacci tree-end self-adaptive radius is proposed to improve convergence schema of FTO.The convergence in ?-neighbourhood of local extremum of target function and the concomitant stagnation are analyzed and proven.The results show that Fibonacci tree-end self-adaptive radius significantly improves the precision of solution and enhances the convergence ability of FTO.Thirdly,FTO properties are analyzed and proven.While global randomness is investigated,the accessibility,progressivity and convergence properties of FTO are proven based on probability theory.Time complexity of FTO is also examined.Global randomness,as a key feature of FTO,protects FTO from trapping into local optima when solving the global optima of multimodal functions.The results show that FTO is able to access global optima with a probability of 1,that FTO displays progressivity towards target solution,and that FTO is able to converge to global optima with a probability of 1.Fourthly,the mechanism and characteristic of FTO in solving multimodal optimization problems is investigated.A Fibonacci tree chain 8)for automatically solving multimodal optimization problems in a probability-driven manner is developed.The probability property of 8)is analyzed and proven.8)is developed based on global randomness and Fibonacci tree-end self-adaptive radius without increasing the key configuration parameters of FTO.FTO automatically solves multimodal optimization problems in a probability-driven manner by generating 8)s to hit multiple local extremums of target function.Fifthly,the applicability and feasibility of using the FTO feature for solving practical optimization problems is investigated.The functions and specifications of FTO simulation platform are described,and a solving operation demo is performed.Fibonacci search method was proven as an optimal strategy of solving unimodal optimization problems using segmentation method.This dissertation proposes FTO on the basis of Fibonacci search method and metaheuristic search idea of intelligent optimization algorithms.The generated Fibonacci tree conforms to the requirements of Fibonacci number and golden section ratio,and the FTO structure is,therefore,exquisite ad compact.While Fibonacci number is the key configuration parameter,global randomness is the vital feature of FTO.FTO does global searching and local searching alternatively in search space of target function and demonstrates effective performance.The schema of FTO is capable of solving multiple optima of multimodal functions,and Fibonacci tree chain,which automatically solves multimodal optimization problems in a probability-driven manner,is an application of the features of FTO.The study of FTO provides new ideas and a novel method in the research and application of intelligent optimization algorithms.