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First hitting time analysis of continuous evolutionary algorithms based on average gain

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Abstract

Runtime analysis of continuous evolutionary algorithms (EAs) is a hard topic in the theoretical research of evolutionary computation, relatively few results have been obtained compared to the discrete EAs. In this paper, we introduce the martingale and stopping time theory to establish a general average gain model to estimate the upper bound for the expected first hitting time. The proposed model is established on a non-negative stochastic process and does not need the assumption of Markov property, thus is more general. Afterwards, we demonstrate how the proposed model can be applied to runtime analysis of continuous EAs. In the end, as a case study, we analyze the runtime of (1, \(\lambda )\)ES with adaptive step-size on Sphere function using the proposed approach, and derive a closed-form expression of the time upper bound for 3-dimensional case. We also discuss the relationship between the step size and the offspring size \(\lambda \) to ensure convergence. The experimental results show that the proposed approach helps to derive tight upper bound for the expected first hitting time of continuous EAs.

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Acknowledgments

This work is supported by Humanity and Social Science Youth Foundation of Ministry of Education of China (14YJCZH216), National Natural Science Foundation of China (61370102, 61370177), Guangdong Natural Science Founds for Distinguished Young Scholar (2014A030306050), Guangdong Natural Science Foundation (2015A030310304), the Fundamental Research Funds for the Central Universities, SCUT (2015PT022), Guangdong High-level personnel of special support program (2014TQ01X664).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Guangdong University of Finance and Economics, Guangzhou, 510320, China

    Zhang Yushan & Hu Guiwu

  2. School of Software Engineering, South China University of Technology, Guangzhou, 510006, China

    Huang Han

  3. School of Mathematics and Big Data, Foshan University, Foshan, 528000, China

    Hao Zhifeng

Authors
  1. Zhang Yushan
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  2. Huang Han
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  3. Hao Zhifeng
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  4. Hu Guiwu
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Correspondence to Huang Han.

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Yushan, Z., Han, H., Zhifeng, H. et al. First hitting time analysis of continuous evolutionary algorithms based on average gain. Cluster Comput 19, 1323–1332 (2016). https://doi.org/10.1007/s10586-016-0587-4

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  • DOI: https://doi.org/10.1007/s10586-016-0587-4

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