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A Level-Value Estimation Method and Stochastic Implementation for Global Optimization

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Abstract

In this paper, we propose a new method, namely the level-value estimation method, for finding global minimizer of continuous optimization problem. For this purpose, we define the variance function and the mean deviation function, both depend on a level value of the objective function to be minimized. These functions have some good properties when Newton’s method is used to solve a variance equation resulting by setting the variance function to zero. We prove that the largest root of the variance equation equals the global minimal value of the corresponding optimization problem. We also propose an implementable algorithm of the level-value estimation method where importance sampling is used to calculate integrals of the variance function and the mean deviation function. The main idea of the cross-entropy method is used to update the parameters of sample distribution at each iteration. The implementable level-value estimation method has been verified to satisfy the convergent conditions of the inexact Newton method for solving a single variable nonlinear equation. Thus, convergence is guaranteed. The numerical results indicate that the proposed method is applicable and efficient in solving global optimization problems.

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Notes

  1. The three parameters H,V and B are used to give the upper bounds of CΔ, \(\operatorname{Var}(Y_{1})\) in (48) and the absolute value of lower bound of the objective function f over ℝn, respectively.

  2. The scopes of parameters α,β and q are suggested in [23]. In the numerical tests of this paper, we set α=0.8,β=0.80 and q=6.

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Acknowledgements

The authors would like to thank Professor A. Hoffmann and Professor C.-S. Lee for their help in improving the quality of the initial manuscript. They are very grateful to Professor Q.-C. Zhao and the anonymous reviewers for their constructive comments which play a crucial role on the final version of this paper. This work was supported by Natural Science Foundation of China (61170308), the Natural Science Foundation of FuJian Province (2011J01008) and FuJian Province Eduction Department (JA11033).

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Authors and Affiliations

  1. College of Mathematics and Computer Science, Fuzhou University, Fuzhou, 350108, China

    Zheng Peng

  2. Department of Mathematics, Shanghai University, Shanghai, 200444, China

    Donghua Wu & Quan Zheng

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  1. Zheng Peng
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  2. Donghua Wu
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  3. Quan Zheng
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Correspondence to Zheng Peng.

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Communicated by Qianchuan Zhao.

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Peng, Z., Wu, D. & Zheng, Q. A Level-Value Estimation Method and Stochastic Implementation for Global Optimization. J Optim Theory Appl 156, 493–523 (2013). https://doi.org/10.1007/s10957-012-0151-1

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