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

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Abstract

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.

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Abbreviations

X :

Vector of random variables

x :

Mean value of X

x L, x U :

Lower and upper bounds of x

U :

Vector of random variables in standard normal space

u :

Mean value of U

m :

Number of variables

p :

Number of constraint functions

ε :

Difference vector between X and x

R :

Vector of constraint reliabilities

J(·):

Objective function

g(·):

Vector of constraint functions

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

Value of surrogate models

(·)i :

The ith component of a vector

E(·):

Expectation of a random variable

P{·}:

Probability of a random variable

CDF:

Cumulative distribution function

F ε(·):

Vectorized CDF for ε

F U(·):

Vectorized CDF for U

β i :

The ith reliability index of the constraint functions

ϕ(·):

CDF of the standard normal distribution

RBO:

Reliability-based optimization

SSRBO:

Sequential surrogate reliability-based optimization

MCS:

Monte Carlo simulation

LSF:

Limit state function

MPP:

Most probable point

RBF:

Radial basis function

AMA:

Approximate moment approach

RIA:

Reliability index approach

PMA:

Performance measure approach

SORA:

Sequential optimization and reliability assessment

SLSV:

Single loop single variable

ASORA:

Advanced sequential optimization and reliability assessment

SLA:

Single-loop approach

AHA:

Adaptive hybrid approach

AH_SLM:

Adaptive hybrid single-loop method

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Acknowledgments

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

Funding

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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Authors and Affiliations

  1. School of Astronautics, Northwestern Polytechnical University, Xi’an, People’s Republic of China

    Xu Li, Chunlin Gong, Liangxian Gu, Hai Fang & Ruichao Gao

  2. Department of Mechanics, Huazhong University of Science and Technology, Wuhan, People’s Republic of China

    Zhao Jing

Authors
  1. Xu Li
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  2. Chunlin Gong
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  3. Liangxian Gu
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  4. Zhao Jing
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  5. Hai Fang
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  6. Ruichao Gao
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Correspondence to Chunlin Gong.

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Li, X., Gong, C., Gu, L. et al. A reliability-based optimization method using sequential surrogate model and Monte Carlo simulation. Struct Multidisc Optim 59, 439–460 (2019). https://doi.org/10.1007/s00158-018-2075-3

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  • DOI: https://doi.org/10.1007/s00158-018-2075-3

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