# Constraint (mathematics)

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In mathematics, a **constraint** is a condition that a solution to an optimization problem must satisfy. There are two types of constraints: **equality constraints** and **inequality constraints**. The set of solutions that satisfy all constraints is called the feasible set.

## ExampleEdit

The following is a simple optimization problem:

subject to

and

where denotes the vector (*x*_{1}, *x*_{2}).

In this example, the first line defines the function to be minimized (called the *objective* or *cost function*). The second and third lines define two constraints, the first of which is an inequality constraint and the second is an equality constraint. These two constraints define the feasible set of candidate solutions.

Without the constraints, the solution would be where has the lowest value. But this solution does not satisfy the constraints. The solution of the **constrained optimization problem** stated above but , which is the point with the smallest value of that satisfies the two constraints.

## Terminology Edit

- If a constraint is an
*equality*at a given point, the constraint is said to be**Template:Visible anchor**, as the point*cannot*be varied in the direction of the constraint. - If a constraint is an
*inequality*at a given point, the constraint is said to be**Template:Visible anchor**, as the point*can*be varied in the direction of the constraint. - If a constraint is not satisfied, the point is said to be
**infeasible**.

## See also Edit

- Constraint satisfaction problem
- Karush–Kuhn–Tucker conditions
- Lagrange multipliers
- Level set
- Linear programming
- Nonlinear programming

## External links Edit

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