Constraint set

A constraint set is a subset of a domain $U\subseteq \mathbb{R}^n$ described by one or more constraints, commonly written as a level set

where $g:U\to \mathbb{R}^m$ is a function and $g^{-1}$ denotes preimage .

Constraint sets are central in constrained optimization (see Lagrange multipliers ). When $c$ is a regular value of $g$, the constraint set typically has good local structure and is a natural domain for an implicitly defined function .

Examples:

• The unit circle is the constraint set $\{(x,y)\in\mathbb{R}^2:\ x^2+y^2=1\}$.
• The affine plane in $\mathbb{R}^3$ given by $x+2y-z=0$ is the constraint set $\{(x,y,z): x+2y-z=0\}$.