Corner Point¶

Definition¶

In an LP with $n$ decision variables, a corner point is a point determined by the intersection of constraint boundaries with $n$ independent boundary equations. At least $n$ constraints are therefore active there, counting nonnegativity boundaries. A corner point is feasible only if it satisfies all constraints.

Interpretation¶

Corner points are the points where boundary lines of the constraints cross. In an LP with two variables, a corner is found by turning two constraints into equations and solving them together.

The feasible pentagon with its five corners labelled, and the point (50, 0), where two boundary lines meet outside the region, marked as infeasible.

The orange points are the five feasible corners of the Giapetto region, from $(0,0)$ round to $(0,80)$ . The point $(50,0)$ , where the finishing line $2x_s+x_t=100$ meets the axis $x_t=0$ $0$ , is also a corner point in the sense of the Definition, but it lies outside the region because $50>40$ violates the demand limit $x_s\le40$ $40$ .

Every intersection of two boundaries is a candidate corner. Filled points are feasible corners; hollow ones lie outside the region.

With $n$ variables, a corner needs $n$ boundary equations that pin down exactly one point. In two variables these are two different lines that are not parallel: the lines $x_s=0$ and $x_s=40$ never meet, and writing the same line twice fixes no point. The five boundary lines of the figure, three constraints and two axes, form $10$ pairs; one pair is parallel, and of the other nine intersection points only five are feasible.

Corner points matter because an LP that has an optimum attains it at a feasible corner (corner point optimality). The graphical method uses this directly, and the simplex method later finds corners by algebra. A common slip is to treat every corner point as feasible: a corner satisfies its two defining constraints with equality, but the other constraints must still be checked.

Key Results¶

In two dimensions, two nonparallel active boundaries determine a corner.
Extra active constraints can occur at a degenerate corner.
Not every intersection of boundaries is feasible.

Examples¶

The illustrated feasible corner $(20,60)$ solves $2x_s+x_t=100$ and $x_s+x_t=80$ $80$ . The boundary intersection $(50,0)$ is infeasible because it violates the soldier-demand limit; it is not an extreme point of the feasible set. Feasible corners correspond to basic feasible solutions of the augmented model.

The lines $x_2=5$ and $2x_1+x_2=9$ intersect at $(2,5)$ .
The nonnegativity boundaries meet at the origin $(0,0)$ .

Prerequisites¶

Sources¶

OR Modelling · Week 1

Aliases: extreme point, corner-point solution Keywords: vertex of the feasible region

Last updated: 20 October 2026 03:02