ARSM-Based Sequential Optimization and Reliability Assessment (SORA_ARSM)

Reliability and robustness based optimization methods require many design evaluations, therefore improving their efficiency is one of the issues. ARSM-Based Sequential Optimization and Reliability Assessment attempts to address this issue by using Adaptive Response Surface Method (ARSM).

In this process, response surfaces are created and an optimization is carried out on the surfaces. During deterministic optimization and reliability analysis, response surfaces are adaptively updated for increased accuracy.

Usability Characteristics

ARSM-Based Sequential Optimization and Reliability Assessment is the most efficient of the three RBDO methods available in HyperStudy, but it is also the least accurate.
An extension of ARSM-Based Sequential Optimization and Reliability Assessment is implemented in HyperStudy to allow for robust design optimization. Robust design optimization attempts to minimize the objective variance in order to reduce its sensitivity to design variations and consequently increase the design's robustness. The implementation in HyperStudy is based on the use of percentiles for the objective function and is turned on via the Robust Optimization setting in the Specification step.
ARSM-Based Sequential Optimization and Reliability Assessment terminates if one of the conditions are met.
- One of the three convergence criteria are met.
  - The absolute objective change is less than a convergence tolerance value (Termination Criteria) and there is no constraint violation (Constraint Violation Tol. (%)) in the last design.
  - The relative objective change is less than a convergence tolerance value (Termination Criteria) and there is no constraint violation (Constraint Violation Tol. (%)) in the last design.
  - The absolute change and relative change of each input variable (Termination Criteria) is less than a convergence tolerance value. Also, there is no constraint violation (Constraint Violation Tol. (%)) in the last design.
- The maximum number of allowable iterations (Maximum Iterations) is reached.
An exception is when the absolute change and relative change of each input variable is less than this value in the last two consecutive designs or when we have found feasible designs and the best feasible design has not been improved during the last two consecutive iterations. When this occurs, ARSM-Based Sequential Optimization and Reliability Assessment will be terminated.
The reliability analysis is carried out by searching for the most probable point (MPP). Issues such as non-uniqueness of the MPP and highly non-linear output response functions can reduce the accuracy of the reliability calculation.
The algorithm begins with N+1 evaluations, where N is the number of design variables. Evaluations in subsequent iterations are generated sequentially.

Settings

In the Specifications step, change method settings from the Settings and More tabs.

Note: For most applications the default settings work optimally, and you may only need to change the Maximum Iterations and Robust Optimization.

Table 1. Settings Tab
Parameter	Default	Range	Description
Maximum Iterations	25	> 0	Maximum number of iterations allowed.
Angle Convergence Tol.	0.25	> 0.0	Angle convergence tolerance for inverse MPP search, in unit of degrees. If the angle between the vector of $\bar{u}$ (design point in standard normal distribution space) and the negative gradient falls within the tolerance, then inverse MPP search is regarded as converged. Tip: A smaller value favors a higher precision of reliability analysis, but more computational effort is needed.
Robust Optimization	No	No or Yes	Defines whether this is a robust optimization or not. No Do not use robust optimization. Yes Use robust optimization.
Robust Min %	95.0	> 50 < 100	Defines the percentile value of robust optimization for minimization objective.
Robust Max %	5.0	> 0 < 50	Defines the percentile value of robust optimization for maximization objective.
On Failed Evaluation	Terminate optimization	Terminate optimization Ignore failed evaluations	Terminate optimization Terminates with an error message when an analysis run fails. Ignore failed evaluations Ignores the failed analysis run.

Table 2. More Tab
Parameter	Default	Range	Description
Termination Criteria	1.0e-4	> 0.0	If the absolute or relative change of the objective value is less than this value, or the absolute or relative change of the input variables is less than this value, and the constraint violation is not larger than this value, then ARSM-Based Sequential Optimization and Reliability Assessment will be terminated. Also, there must not be any constraints with an allowable violation that has been exceeded in the last design. An exception is when the absolute change and relative change of each input variable is less than this value in the last two consecutive designs or when we have found feasible designs and the best feasible design has not been improved during the last two consecutive iterations, ARSM-Based Sequential Optimization and Reliability Assessment will be terminated. ${\begin{cases} c_{\max}^{k} \leq g_{\max} \\ [\| \frac{f^{k} - f^{k - 1}}{\| f^{k - 1} \| + 10^{- 10}} \| < ε] . o r . [\| f^{k} - f^{k - 1} \| < ε] . o r . [{\begin{cases} \| \frac{x_{j}^{k} - x_{j}^{k - 1}}{\| x_{j}^{k - 1} \| + 10^{- 10}} \| < ε, j = 1, ..., n \\ \| x_{j}^{k} - x_{j}^{k - 1} \| < ε \end{cases}] \end{cases}$ Where, $x$ is the input variable; $f$ is the objective; $n$ is the number of input variables; $k$ is the current iteration number; $c_{\max}$ is the maximum constraint violation; $g_{\max}$ is the allowable constraint violation; $ε$ is the value of the termination criteria. $\begin{array}{l} i f ({\begin{cases} \| \frac{x_{j}^{i} - x_{j}^{i - 1}}{\| x_{j}^{i - 1} \| + 10^{- 10}} \| < ε, j = 1, ..., n \\ \| x_{j}^{i} - x_{j}^{i - 1} \| < ε \end{cases}) \\ . o r . ({\begin{matrix} c_{\max}^{k - 2} \leq g_{\max} \\ (c_{\max}^{i} > g_{\max}) . o r . (\begin{array}{l} f^{i} \geq f^{k - 2}, \min i m i z a t i o n \\ f^{i} \leq f^{k - 2}, \max i m i z a t i o n \end{array}) \end{matrix}) \\ i = k, k - 1 \end{array}$
Move Limit Fraction	0.1	0.0 < Move Limit Fraction < 1.0	Move limit fraction. Smaller values allow more steady convergence (smaller fluctuation of the output response values), but more computational effort could be consumed. The value will be adaptively updated during the optimization process.
Initial Linear Move	By DV Initial	By DV Initial By DV Bounds	By DV Initial Initial move = Initial Input Perturbation * Move Limit Fraction * abs(INI) (Default when initial value of input variable is non-zero) An exception is that initial move will be set to minimum move if it is less than minimum move. Minimum move = Minimal Move Factor * (UB-LB) if (UB-LB) is less than 1. Minimum move = Minimal Move Factor if (UB-LB) is not less than 1 and absolute value of INI is less than 1. Minimum move = Minimal Move Factor * min((UB-LB),abs(INI)) if (UB-LB) is not less than 1 and absolute value of INI is not less than 1. By DV Bounds Initial move = Initial Input Perturbation * Move Limit Fraction * (UB-LB) (Default when initial value of input variable is zero) INI: Initial input variable value. LB, UB: Lower and upper bounds on input variable.
Minimal Move Factor	0.1	0.0 < Minimal Move Factor < Move Limit Fraction	Minimal move factor. See the usage of Minimal Move Factor in Initial Linear Move.
Initial Input Perturbation	1.1	≠ 0.0	Initial input variable perturbation value. Larger values result in a wider spread of the initial $N$ designs ( $N$ is the number of input variables; the $N$ designs together with the start design can determine a linear response surface). Adaptive Response Surface Method will search the design space more widely.
Use SVD	No	No or Yes	No Adaptive Response Surface Method is terminated in case of soft convergence (when the current design is the same or little change (1.0e-15) to one of the history designs). Yes Singular Value Decomposition is activated to build the response surfaces and the optimization process will be continued. $\begin{array}{l} \| x_{i}^{k} - x_{i}^{j} \| \leq 10^{- 15}, i = 1, ..., n \\ \forall j, j \in {1 \leq j < k} \end{array}$ Where, $k$ is the current iteration number; $n$ is the number of input variables; $xj$ is one of the history designs.
Revision	A	A or B	Used to help when there is a convergence difficulty. By default, "A" is selected meaning the legacy algorithm.