High-performance differential evolution algorithm guided by information from individuals with potential
Introduction
The differential evolution (DE) algorithm proposed by Storn and Price [1] in 1995 is a swarm intelligence heuristic optimization algorithm in continuous space. Similar to other evolutionary algorithms (EAs), three operators, namely, mutation, crossover, and selection, are used to generate new solutions. The mutation operator is performed on the basis of the differential information from individuals in the current population. A probabilistic mechanism is used to perform the crossover operator. The algorithm updates the population through a greedy selection mechanism. The DE algorithm has good global optimization, simple structure, easy implementation, few control parameters, and strong search capability. DE has been proven to be an efficient global heuristic EA and has attracted the attention of many EA researchers. However, mutation strategy and control parameters will immensely influence the DE performance. Numerous DE variants with strategy and parameter adaptation have been proposed.
In DE, mutation operation is implemented on the basis of the difference among individuals in the current generation, thereby supporting the good optimization performance of DE. Many researchers have conducted studies on fitness values. However, the difference between the individuals of two successive generations that contains much information about evolution has not been fully utilized. In this study, the difference between the individuals of two successive generations is used in adjusting control parameters and selecting mutation strategies. The improvement of fitness values is utilized in classifying individuals. We have proposed the definitions of individuals with potential and individuals without potential in our previous work [2], which is determined by its difference in fitness values from those of previous individuals. Different individuals have different characteristics. Thus, strategy selection and parameter control are implemented on individuals with and without potential separately. In the strategy selection mechanism, different mutation strategies are used for individuals with and without potential separately, and the selection probability of each strategy is based on the average improvement in the fitness values with the mutation strategy. In the parameter control mechanism, the difference between the individuals of two successive generations is used to calculate the weight of the used parameters of each individual to tune the control parameters.
Good individuals contain information on good parameters, and the use of good parameters helps the next generation produce good individuals. With the experience learned from the previous generation of individuals with potential, the next evolution is likely to generate new individuals with potential. The separate learning from the previous individuals with and without potential can guide the current individuals with and without potential separately in a convincing manner. In this way, different individuals with different characteristics can evolve through different means that fit them.
The remainder of this paper is organized as follows: Section 2 briefly introduces the process of the basic DE. Section 3 reviews the relevant literature on strategy and parameter control in DE. Section 4 provides the details of the proposed high-performance (PDE). Section 5 discusses the comparison between the PDE and five advanced DE variants on CEC2005 and seven up-to-date EAs on CEC2014. Section 6 presents the application of PDE in the parameter estimation of a p-xylene (PX) oxidation kinetic model. Section 7 summarizes this paper.
Section snippets
Basic DE algorithm
This section introduces the process of the basic DE algorithm. For a population that contains N individuals , each individual that comprises D variables can be viewed as a solution in the search space of an optimization problem. G denotes the generation count, N is the population size, and D is the dimensionality of the optimization problem. In this study, DE is used to solve the following minimization optimization problem:
Literature review
The main parameters of DE include the population size N, the scaling factor F in the mutation strategy, and the crossover probability factor CR in the crossover strategy.
DE has been widely used to solve many optimization problems because of its simplicity, robustness, and effectiveness. However, its performance relies on the parameter control and strategy selection to some extent. Unsuitable strategies and parameters will immensely influence the algorithm’s performance and result in premature
Proposed method
In this section, the principle and process of the proposed PDE algorithm are introduced. PDE is presented in four subsections. Section 4.1 explains the general idea of PDE. Section 4.2 presents the mutation strategy adaptation mechanism. Section 4.3 introduces the control parameter adaptation mechanism. Section 4.4 provides the complete PDE algorithm.
Experimental results
Extensive experiments are performed to assess the optimization performance of PDE. Two benchmark suits, namely, CEC2005 and CEC2014, are used. Detailed information about the CEC2005 and CEC2014 benchmark suits can be found in [33] and [34], respectively.
Section 5.1 introduces the experimental setup and statistical analysis method, and the experimental results are presented in two subsections. In Section 5.2, a comparison between PDE and five state-of-the-art DEs is performed on CEC2005 with 30D
Application
In this section, PDE is applied to estimate the parameters of a kinetic model of the PX oxidation reaction.
Conclusion
In DE, numerous self-adaptive methods for adjusting parameters and selecting mutation strategy have been proposed. Several of them utilize fitness values, but few focus on the improvement of fitness values. The improvement of fitness values contains much information about evolution, such as the characteristic of an individual itself and the effectiveness of the used parameter and strategy. In this paper, an adaptive DE guided by information from individuals with potential is introduced. All
CRediT authorship contribution statement
Li Tian: Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Visualization, Data curation, Writing - original draft. Zhichao Li: Software, Validation, Visualization. Xuefeng Yan: Conceptualization,Project administration, Resources, Funding acquisition, Supervision, Writing - review & editing.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
The authors are grateful for the support of National Natural Science Foundation of China (21878081) and Fundamental Research Funds for the Central Universities under Grant of China (222201917006).
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