[Paper Review] Surrogate-assisted reliability-based design optimization: a survey and a new general framework

Maliki Moustapha, Bruno Sudret|arXiv (Cornell University)|Jan 10, 2019

Probabilistic and Robust Engineering DesignDecision Sciences85 references3 citations

TL;DR

This paper proposes a modular, non-intrusive framework for reliability-based design optimization (RBDO) that integrates adaptive surrogate modeling, reliability analysis, and optimization as independent blocks. By combining Kriging and support vector machines with Monte Carlo and subset simulation, and using CMA-ES or SQP for optimization, the framework achieves robust solutions with only 154 finite element evaluations for a 16-dimensional problem, demonstrating high efficiency and accuracy compared to traditional methods.

ABSTRACT

Reliability-based design optimization (RBDO) is an active field of research with an ever increasing number of contributions. Numerous methods have been proposed for the solution of RBDO, a complex problem that combines optimization and reliability analysis. Classical approaches are based on approximation methods and have been classified in review papers. In this paper, we first review classical approaches based on approximation methods such as FORM, and also more recent methods that rely upon surrogate modelling and Monte Carlo simulation. We then propose a general framework for the solution of RBDO problems that includes three independent blocks, namely adaptive surrogate modelling, reliability analysis and optimization. These blocks are non-intrusive with respect to each other and can be plugged independently in the framework. After a discussion on numerical considerations that require attention for the framework to yield robust solutions to various types of problems, the latter is applied to three examples (using two analytical functions and a finite element model). Kriging and support vector machines together with their own active learning schemes are considered in the surrogate model block. In terms of reliability analysis, the proposed framework is illustrated using both crude Monte Carlo and subset simulation. Finally, the covariance-matrix adaptation - evolution scheme (CMA-ES), a global search algorithm, or sequential quadratic programming (SQP), a local gradient-based method, are used in the optimization block. The comparison of the results to benchmark studies show the effectiveness and efficiency of the proposed framework.

Motivation & Objective

• To address the computational burden of traditional RBDO methods that rely on approximation techniques like FORM.
• To overcome the limitations of existing surrogate-assisted RBDO approaches by proposing a unified, modular framework.
• To enable flexible integration of diverse surrogate models, reliability analysis techniques, and optimization algorithms without interdependence.
• To validate the framework's robustness and efficiency across problems with varying complexity and probabilistic input types.
• To demonstrate the framework's effectiveness using analytical functions and a finite element model with minimal performance function evaluations.

Proposed method

• The framework decomposes RBDO into three non-intrusive, independent blocks: adaptive surrogate modeling, reliability analysis, and optimization.
• Adaptive surrogate modeling employs Kriging and support vector machines with active learning to iteratively enrich the model with strategically selected points.
• Reliability analysis is performed using either crude Monte Carlo or subset simulation to estimate failure probabilities accurately.
• Optimization is carried out using either global (CMA-ES) or local (SQP) algorithms, selected independently from the other blocks.
• The framework uses an augmented input space to handle mixed deterministic and random variables, enabling consistent treatment of all probabilistic input types.
• Convergence of the surrogate enrichment is monitored via a criterion that ensures sufficient accuracy near the limit-state surface, validated through bootstrap-resampled Monte Carlo testing.

Experimental results

Research questions

• RQ1 How can a general, modular framework be designed to decouple surrogate modeling, reliability analysis, and optimization in RBDO?
• RQ2 What is the impact of combining different surrogate models (Kriging, SVM), reliability methods (crude Monte Carlo, subset simulation), and optimizers (CMA-ES, SQP) on solution accuracy and efficiency?
• RQ3 Can the proposed framework achieve high accuracy with minimal performance function evaluations, especially for high-dimensional and nonlinear problems?
• RQ4 How can surrogate model accuracy be validated near the limit-state surface when failure probabilities are critical to the design?
• RQ5 To what extent does the framework outperform classical approximation-based RBDO methods like FORM in terms of robustness and reliability?

Key findings

• The framework solved a 16-dimensional RBDO problem using only 154 finite element model evaluations, demonstrating high computational efficiency.
• The final Kriging surrogate model accurately predicted the 95th percentile of deflection (10 cm) with a bootstrap-estimated standard deviation, confirming model reliability.
• The optimal design assigned maximum cross-sectional areas to bars in radial direction (group 2), while non-radial, unconnected bars were minimized, aligning with structural intuition.
• The framework’s modular design allowed seamless integration of diverse methods—Kriging/SVM, Monte Carlo/subset simulation, CMA-ES/SQP—without modification.
• Validation via 1,000-point Monte Carlo sampling with 500 bootstrap replications confirmed that the surrogate model slightly underestimated the true quantile but remained within a tight confidence interval.
• The framework was successfully implemented in the UQLab platform, ensuring broad dissemination and extensibility for future research and engineering applications.