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.

Example

The following is a simple optimization problem:

$\min f(\bold x) = x_1^2+x_2^4$

subject to

$x_1 \ge 1$

and

$x_2 = 1, \,$

where $\bold x$ denotes the vector (x₁, x₂).

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 $(0,0)\,$ where $f(\bold x)$ has the lowest value. But this solution does not satisfy the constraints. The solution of the constrained optimization problem stated above is $\bold x = (1,1)$ , which is the point with the lowest value of $f(\bold x)$ 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

$c_1(\bold x)=1-x_1 \le 0$