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Constraint is an equation that defines a restriction of solutions of an optimization problem to a so called feasible set.
We distinguish between equality and inequality constraints.
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Example
A simple optimization problem can be defined as
subject to
and - x2 = 1-.
where x = (x1, x2) . In this example, the first line defines the function to be minimized (sometimes called the objective or cost function). The second and third lines defines two constraints, the first of which is inequality constraint and the second is equality constraint. These two constraints define the feasible set of points in which we search for the solution.
Without the constraints, the solution would be (0,0) where f(x) has the lowest value. But this solution does not satisfy the constraints (it is not included in the feasible set). The solution of the constraint optimization problem defined above is x=(1,1), which is the point with the lowest value of f(x) among all feasible points.
In standard form we write constraints in such a way that there is 0 on one side of equation or inequality. What is on the other side is called a constraint function. In the above example, constraints can be rewritten as
and
Different authors use different agreement with regard to the sign in inequality constraints, i.e. some authors would define the first constraint function with the opposite sign such that [edit]
See also [edit] In mathematics, the term optimization refers to the study of problems that have the form Given: a function f : A R from some set A to the real numbers Sought: an element x0 in A such that f(x0) ≤ f(x) for all x in A (minimization) or such that...
In mathematics, nonlinear programming (NLP) is the process of solving a system of equalities and inequalities over a set of unknown real variables, along with an objective function to be maximized or minimized. ...
In mathematics, the Karush-Kuhn-Tucker conditions (also known as the Kuhn-Tucker or the KKT conditions) are necessary for a solution in nonlinear programming to be optimal. ...
External links - Nonlinear programming FAQ
- Mathematical Programming Glossary
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