Sequential Optimization and Reliability Assessment (SORA)
A reliabilitybased design optimization method. Reliabilitybased design optimization (RBDO) methods take uncertainties in the design into account and search for designs that satisfy the design requirements with a required probability of success.
 Objective
 min f(x, r, p)
 Constraints
 P(g(x, r, p ≤ 0.0) > PS
 x
 Deterministic input variables
 r
 Random input variables (affect the design but are subject to uncertainties)
 p
 Pure random parameters (variables we have no control over but affect the design, such as humidity and temperature)
Usability Characteristics
 An extension of 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.
 Sequential Optimization and Reliability Assessment is the most accurate of the three RBDO methods available in HyperStudy. It is also the most expensive.
 Sequential Optimization and Reliability Assessment terminates if one of the conditions below are
met:
 One of the two convergence criterias are met.
 The absolute objective change is less than a convergence tolerance value (Termination Criteria) and there is no constraint violation (Constraint Violation Tol. (%)).
 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 maximum number of allowable iterations (Maximum Iterations) is reached.
 One of the two convergence criterias are met.
 The reliability analysis is carried out by searching for the most probable point (MPP). Issues such as nonuniqueness of the MPP and highly nonlinear output response functions can reduce the accuracy of the reliability calculation.
 The number of evaluations in each iteration is automatically set and varies due to the finite difference calculations used in the sensitivity calculation. The number of evaluations in each iteration is dependent of the number of variables. The evaluations required for the finite difference are executed in parallel. The evaluations required for the line search are executed sequentially.
Settings
Parameter  Default  Range  Description 

Maximum Iterations  25  > 0  Maximum number of iterations allowed. 
Robust Optimization  No  No or Yes 
Defines whether this is a
robust optimization or not.

Robust Min %  95.0 

Defines the percentile value of robust optimization for minimization objective. 
Robust Max %  5.0 

Defines the percentile value of robust optimization for maximization objective. 
On Failed Evaluation  Terminate optimization 


Parameter  Default  Range  Description 

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
$\overline{u}$
(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.

Termination Criteria  1.0e4  > 0.0  Termination tolerance. If the absolute or relative change
of the objective value is less than this value, and the
constraint violation is not larger than this value, then
Sequential Optimization and Reliability Assessment will be terminated. There
also must not be any constraint with an allowable violation
that has been exceeded in the last design.
Where,
$f$
is the objective;
$k$
is the current iteration number;
${c}_{\mathrm{max}}$
is the maximum constraint
violation;
${g}_{\mathrm{max}}$
is the allowable constraint
violation; Termination
Criteria is the value of the termination criteria.
$$\{\begin{array}{l}{c}_{\mathrm{max}}^{k}\le {g}_{\mathrm{max}}\\ if\text{}\left{f}^{k}{f}^{k1}\right\text{TerminationCriteria}\\ or\text{}\frac{\left{f}^{k}{f}^{k1}\right}{\left{f}^{k1}\right+{10}^{10}}\text{TerminationCriteria}\end{array}$$
An exception is
when the current objective is worse than the previous
objective and the constraint violation of the previous
design is within allowable violation, Sequential Optimization and Reliability Assessment will be terminated.
$$\{\begin{array}{l}{c}_{\mathrm{max}}^{k1}\le {g}_{\mathrm{max}}\\ \{\begin{array}{l}{f}^{k}>{f}^{k1},\text{minimization}\\ {f}^{k}{f}^{k1},\text{maximization}\end{array}\end{array}$$
