Constraint ( mathematics ) - wikipedia Constraint ( mathematics ) For constraints in Hamiltonian mechanics, see constraint ( classical mechanics ), first class constraint, primary constraint, and holonomic constraint. This article includes a list of references, related reading or external links, but its sources remain unclear because it lacks inline citations. Please help to improve this article by introducing more precise citations. ( September 2016 ) 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. Contents 1 Example 2 Terminology 3 Hard and soft constraints 4 See also 5 External links Example The following is a simple optimization problem : min f ( x ) = x 1 2 + x 2 4 ( displaystyle min f ( ( mathbf ( x ) ) ) = x_ ( 1 ) ^ ( 2 ) + x_ ( 2 ) ^ ( 4 ) ) subject to x 1 ≥ 1 ( displaystyle x_ ( 1 ) geq 1 ) and x 2 = 1, ( displaystyle x_ ( 2 ) = 1, ) where x ( displaystyle ( mathbf ( x ) ) ) denotes the vector ( x, x ). 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 f ( x ) ( 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 x = ( 1, 1 ) ( displaystyle ( mathbf ( x ) ) = ( 1, 1 ) ), which is the point with the smallest value of f ( x ) ( displaystyle f ( ( mathbf ( x ) ) ) ) that satisfies the two constraints. Terminology If an inequality constraint holds with equality at the optimal point, the constraint is said to be binding, as the point can not be varied in the direction of the constraint even though doing so would improve the value of the objective function. If an inequality constraint holds as a strict inequality at the optimal point ( that is, does not hold with equality ), the constraint is said to be non-binding, as the point could be varied in the direction of the constraint, although it would not be optimal to do so. If a constraint is non-binding, the optimization problem would have the same solution even in the absence of that constraint. If a constraint is not satisfied at a given point, the point is said to be infeasible. Hard and soft 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. See also Constraint algebra Constraint satisfaction problem Karush -- Kuhn -- Tucker conditions Lagrange multipliers Level set Linear programming Nonlinear programming External links Nonlinear programming FAQ Mathematical Programming Glossary Retrieved from `` https://en.wikipedia.org/w/index.php?title=Constraint_(mathematics)&oldid=846036219 '' Categories : Mathematical optimization Constraint programming Hidden categories : Articles lacking in - text citations from September 2016 All articles lacking in - text citations Talk Contents About Wikipedia Български Català Čeština Deutsch Español Esperanto Euskara Français Ido Nederlands Română Slovenščina Українська 中文 6 more Edit links This page was last edited on 15 June 2018, at 21 : 18 ( UTC ). Text is available under the Creative Commons Attribution - ShareAlike License ; additional terms may apply. By using this site, you agree to the Terms of Use and Privacy Policy. Wikipedia ® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization. About Wikipedia

Question: what does it mean for a constraint to be binding
Answer: If an inequality constraint holds with equality at the optimal point