Constraint (mathematics)
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In
Example
The following is a simple optimization problem:
subject to
and
where denotes the vector (x1, x2).
In this example, the first line defines the function to be minimized (called the
Without the constraints, the solution would be (0,0), 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 smallest value of 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 cannot 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. Under certain conditions, as for example in convex optimization, 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
Global constraints
Global constraints[2] are constraints representing a specific relation on a number of variables, taken altogether. Some of them, such as the alldifferent
constraint, can be rewritten as a conjunction of atomic constraints in a simpler language: the alldifferent
constraint holds on n variables , and is satisfied if the variables take values which are pairwise different. It is semantically equivalent to the conjunction of inequalities . Other global constraints extend the expressivity of the constraint framework. In this case, they usually capture a typical structure of combinatorial problems. For instance, the regular
constraint expresses that a sequence of variables is accepted by a deterministic finite automaton.
Global constraints are used
See also
References
- ISBN 0-521-31498-4.
- OCLC 162587579.
- OCLC 771185146.
Further reading
- Beveridge, Gordon S. G.; Schechter, Robert S. (1970). "Essential Features in Optimization". Optimization: Theory and Practice. New York: McGraw-Hill. pp. 5–8. ISBN 0-07-005128-3.
External links
- Nonlinear programming FAQ Archived 2019-10-30 at the Wayback Machine
- Mathematical Programming Glossary Archived 2010-03-28 at the Wayback Machine