# Constraints

Constraints need to be satisfied for an optimization to be acceptable. Constraints may also be associated with a DOE. While not used in the evaluation of the DOE, constraints can be useful while visualizing DOE results. Limits on displacement or stress are common examples.

## Constraint Categories

All constraints in an optimization problem can be placed into the following distinct categories:
Inequality Constraint
One sided condition that must be satisfied.
$\begin{array}{cc}{g}_{j}\left(x\right)\le 0& j=1,...,m\end{array}$
Equality Constraint
Precise condition that must be satisfied.
$\begin{array}{cc}{h}_{k}\left(x\right)=0& k=1,...,{m}_{h}\end{array}$
Side Constraint
Bounds on the input variables that limit the region of search for the optimum.
${x}_{i}^{L}\le {x}_{i}\le {x}_{i}^{U}$

## Constraint Types

Constraints can be defined as type Deterministic or Random (probabilistic) when setting up an Optimization in HyperStudy, depending on the design requirements.
Deterministic
Deterministic constraints enable you to manually define a Bound Type, Bound Value, and evaluation source for the output response(s).
Random
Random problem formulations take into account the variability in the design and study the corresponding variability in the performances. This aspect is studied under reliability and robustness.
Random constraints require you to modify the CFD Limit if the reliability requirement is different than the default value of 99.00%. The CFD Limit is the reliability requirement on the constraint; that is the probability of (Output Response >= 0) > 99.00%.

## Standard Constraint Enforcement

Constraints violations can be treated in the following ways:
Standard Enforcement
Constraints are considered feasible when they are within a small percentage of difference between their bounds. This type of enforcement is conventional.
Strict Enforcement
Constraints must be perfectly satisfied with no margin. This type of enforcement may require additional iterations from an optimizer for convergence.
Percent of Constraint Bound
Constraints must be violated by more than this value in the converged design. Strict enforcement only uses this tolerance for equality constraints.
When the Constraint Bound = 0.0
In general, constraint values are normalized to their bound value. One exception is if the absolute bound value is less than this parameter.
Tip: The recommended range is 1.0e-6 ~ 1.0.