Monte Carlo integration with adaptive variance selection for improved stochastic efficient global optimization

  • Felipe CarraroEmail author
  • Rafael Holdorf Lopez
  • Leandro Fleck Fadel Miguel
  • André Jacomel Torii
Research Paper


In this paper, the minimization of computational cost on evaluating multidimensional integrals is explored. More specifically, a method based on an adaptive scheme for error variance selection in Monte Carlo integration (MCI) is presented. It uses a stochastic efficient global optimization (sEGO) framework to guide the optimization search. The MCI is employed to approximate the integrals, because it provides the variance of the error in the integration. In the proposed approach, the variance of the integration error is included into a stochastic kriging framework by setting a target variance in the MCI. We show that the variance of the error of the MCI may be controlled by the designer and that its value strongly influences the computational cost and the exploration ability of the optimization process. Hence, we propose an adaptive scheme for automatic selection of the target variance during the sEGO search. The robustness and efficiency of the proposed adaptive approach were evaluated on global optimization stochastic benchmark functions as well as on a tuned mass damper design problem. The results showed that the proposed adaptive approach consistently outperformed the constant approach and a multi-start optimization method. Moreover, the use of MCI enabled the method application in problems with high number of stochastic dimensions. On the other hand, the main limitation of the method is inherited from sEGO coupled with the kriging metamodel: the efficiency of the approach is reduced when the number of design variables increases.


Stochastic kriging Efficient global optimization Integral minimization Adaptive target variance Robust optimization 


Funding information

This study received financial support from the Brazilian research funding agencies CNPq and CAPES.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Felipe Carraro
    • 1
    Email author
  • Rafael Holdorf Lopez
    • 1
  • Leandro Fleck Fadel Miguel
    • 1
  • André Jacomel Torii
    • 2
  1. 1.Center for Optimization and Reliability in Engineering (CORE), Civil Engineering DepartmentFederal University of Santa CatarinaFlorianópolisBrazil
  2. 2.Center for Optimization and Reliability in Engineering (CORE), Civil Engineering DepartmentFederal University for Latin American IntegrationFoz do IguaçuBrazil

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