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EspañolCONSTRAINT (MATHEMATICS)
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.
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 is , which is the point with the lowest value of that satisfies the two constraints.
In standard form, constraints are written with a 'constraint function' on one side of the equation or inequality and 0 on the other side. In the example above, the constraints can be rewritten in standard form as
:
and
:
Equivalently, inequality constraints can be written in standard form with the opposite signs. Thus, the first constraint above can be written as
:
★ Linear programming
★ Nonlinear programming
★ Karush-Kuhn-Tucker conditions
★ Level set
★ Nonlinear programming FAQ
★ Mathematical Programming Glossary
| Contents |
| Example |
| See also |
| External links |
Example
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 is , which is the point with the lowest value of that satisfies the two constraints.
In standard form, constraints are written with a 'constraint function' on one side of the equation or inequality and 0 on the other side. In the example above, the constraints can be rewritten in standard form as
:
and
:
Equivalently, inequality constraints can be written in standard form with the opposite signs. Thus, the first constraint above can be written as
:
See also
★ Linear programming
★ Nonlinear programming
★ Karush-Kuhn-Tucker conditions
★ Level set
External links
★ Nonlinear programming FAQ
★ Mathematical Programming Glossary
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psst.. try this: add to faves
