Structural and Multidisciplinary Optimization

, Volume 59, Issue 2, pp 439–460 | Cite as

A reliability-based optimization method using sequential surrogate model and Monte Carlo simulation

  • Xu Li
  • Chunlin GongEmail author
  • Liangxian Gu
  • Zhao Jing
  • Hai Fang
  • Ruichao Gao
Research Paper


This paper presents a sequential surrogate model method for reliability-based optimization (SSRBO), which aims to reduce the number of the expensive black-box function calls in reliability-based optimization. The proposed method consists of three key steps. First, the initial samples are selected to construct radial basis function surrogate models for the objective and constraint functions, respectively. Second, by solving a series of special optimization problems in terms of the surrogate models, local samples are identified and added in the vicinity of the current optimal point to refine the surrogate models. Third, by solving the optimization problem with the shifted constraints, the current optimal point is obtained. Then, at the current optimal point, the Monte Carlo simulation based on the surrogate models is carried out to obtain the cumulative distribution functions (CDFs) of the constraints. The CDFs and target reliabilities are used to update the offsets of the constraints for the next iteration. Therefore, the original problem is decomposed to serial cheap surrogate-based deterministic problems and Monte Carlo simulations. Several examples are adopted to verify SSRBO. The results show that the number of the expensive black-box function calls is reduced exponentially without losing of precision compared to the alternative methods, which illustrates the efficiency and accuracy of the proposed method.


Expensive black box function Radial basis function Sequential sampling Reliability-based optimization Monte Carlo simulation 



Vector of random variables


Mean value of X

xL, xU

Lower and upper bounds of x


Vector of random variables in standard normal space


Mean value of U


Number of variables


Number of constraint functions


Difference vector between X and x


Vector of constraint reliabilities


Objective function


Vector of constraint functions

\( \widehat{\left(\cdotp \right)} \)

Value of surrogate models


The ith component of a vector


Expectation of a random variable


Probability of a random variable


Cumulative distribution function


Vectorized CDF for ε


Vectorized CDF for U


The ith reliability index of the constraint functions


CDF of the standard normal distribution


Reliability-based optimization


Sequential surrogate reliability-based optimization


Monte Carlo simulation


Limit state function


Most probable point


Radial basis function


Approximate moment approach


Reliability index approach


Performance measure approach


Sequential optimization and reliability assessment


Single loop single variable


Advanced sequential optimization and reliability assessment


Single-loop approach


Adaptive hybrid approach


Adaptive hybrid single-loop method



The authors also thank Dr. Xueyu Li for the helpful work to improve the study.

Funding information

The research is supported by the Fundamental Research Funds for the Central Universities (No. G2016KY0302) and the National Natural Science Foundation of China (No. 11572134).


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of AstronauticsNorthwestern Polytechnical UniversityXi’anPeople’s Republic of China
  2. 2.Department of MechanicsHuazhong University of Science and TechnologyWuhanPeople’s Republic of China

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