Constraint (mathematics)

In mathematics, a constraint is a condition of an optimization problem that the solution must satisfy. There are several types of constraints—primarily equality constraints, inequality constraints, and integer constraints. The set of candidate solutions that satisfy all constraints is called the feasible set.

The following is a simple optimization problem:

subject to

and

where ${\displaystyle \mathbf {x} }$ denotes the vector (x₁, x₂).

In this example, the first line defines the function to be minimized (called the objective function, loss function, or cost function). The second and third lines define two constraints, the first of which is an inequality constraint and the second of which is an equality constraint. These two constraints are hard constraints, meaning that it is required that they be satisfied; they define the feasible set of candidate solutions.

Without the constraints, the solution would be (0,0), where ${\displaystyle f(\mathbf {x} )}$ has the lowest value. But this solution does not satisfy the constraints. The solution of the constrained optimization problem stated above is ${\displaystyle \mathbf {x} =(1,1)}$, which is the point with the smallest value of ${\displaystyle f(\mathbf {x} )}$ that satisfies the two constraints.

If the problem mandates that the constraints be satisfied, as in the above discussion, the constraints are sometimes referred to as hard constraints. However, in some problems, called flexible constraint satisfaction problems, it is preferred but not required that certain constraints be satisfied; such non-mandatory constraints are known as soft constraints. Soft constraints arise in, for example, preference-based planning. In a MAX-CSP problem, a number of constraints are allowed to be violated, and the quality of a solution is measured by the number of satisfied constraints.