quality:corruption, quality:correction, structural:summarization
Bounded and unbounded feasible sets

Feasible sets may be bounded or unbounded. For example, the feasible set defined by the constraint set { x ≥ 0, y ≥ 0} is unbounded because in some directions there is no limit on how far one can go and still be in the feasible region. In contrast, the feasible set formed by the constraint set { x ≥ 0, y ≥ 0, x + 2 y ≤ 4} is bounded because the extent of movement in any direction is limited by the constraints.

In linear programming problems with n variables, a necessary but insufficient condition for the feasible set to be bounded is that the number of constraints be at least n + 1 (as illustrated by the above example).

If the feasible set is unbounded, there may or may not be an optimum, depending on the specifics of the objective function. For example, if the feasible region is defined by the constraint set { x ≥ 0, y ≥ 0}, then the problem of maximizing x + y has no optimum since any candidate solution can be improved upon by increasing x or y; yet if the problem is to minimize x + y, then there is an optimum (specifically at (x, y) = (0, 0)).