Linear programming

In mathematics, linear programming (LP) problems are optimization problems in which the objective function and the constraints are all linear.

Theory

Geometrically, the linear constraints define a convex polyhedron, which is called the feasible region. Since the objective function is also linear, all local optima are automatically global optima. The linear objective function also implies that an optimal solution can only occur at a boundary point of the feasible region.

Related Topics:
Convex - Polyhedron

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There are two situations in which no optimal solution can be found. First, if the constraints contradict each other (for instance, x ≥ 2 and x ≤ 1) then the feasible region is empty and there can be no optimal solution, since there are no solutions at all. In this case, the LP is said to be infeasible.

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Alternatively, the polyhedron can be unbounded in the direction of the objective function (for example: maximize x1 + 3 x2 subject to x1 ≥ 0, x2 ≥ 0, x1 + x2 ≥ 10), in which case there is no optimal solution since solutions with arbitrarily high values of the objective function can be constructed.

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Barring these two pathological conditions (which are often ruled out by resource constraints integral to the problem being represented, as above), the optimum is always attained at a vertex of the polyhedron. However, the optimum is not necessarily unique: it is possible to have a set of optimal solutions covering an edge or face of the polyhedron, or even the entire polyhedron (This last situation would occur if the objective function were constant).

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