Editing Linear programming (section)

=== Existence of optimal solutions ===
Geometrically, the linear constraints define the [[feasible region]], which is a [[convex polytope]]. A [[linear functional|linear function]] is a [[convex function]], which implies that every [[local minimum]] is a [[global minimum]]; similarly, a linear function is a [[concave function]], which implies that every [[local maximum]] is a [[global maximum]].

An optimal solution need not exist, for two reasons. First, if the constraints are inconsistent, then no feasible solution exists: For instance, the constraints '''x'''&nbsp;≥&nbsp;2 and '''x'''&nbsp;≤&nbsp;1 cannot be satisfied jointly; in this case, we say that the LP is ''infeasible''. Second, when the [[polytope]] is unbounded in the direction of the gradient of the objective function (where the gradient of the objective function is the vector of the coefficients of the objective function), then no optimal value is attained because it is always possible to do better than any finite value of the objective function.