corner point theorem

Let $P$ be a primal linear programming problem with $f$ the objective function and polyhedron $R$ as its feasible region. A corner point, or extreme point of $P$ is a of $R$.

Corner Point Theorem. If $P$ has an optimal solution , then there is a corner point $p$ of $P$ such that $f(p)=a$. If another corner point $q$ also satisfies $f(x)=a$, then $f(\overline{pq})=\{a\}$. If $r$ is a third corner point such that $f(r)=a$, then $f(\mathrm{△}pqr)=\{a\}$.

Note that the line segment and triangle mentioned above are necessarily subsets of $R$.

A cousin of the above theorem is the following:

Theorem. If $P$ has an optimal solution occurring at an interior point of an edge $E$ (or a face $F$) on the boundary of the feasible region $R$, then $f(E)=\{a\}$ (or $f(F)=\{a\}$). In fact, if the solution occurs at an interior point of $R$, then the solution is satisfied by all points of $R$: $f(R)=\{a\}$. In other words, the objective function $f$ is constant on $R$.

On way to visualize the above theorems is to simplify them into the case when the objective function is a line $f(x)=mx+b$ on the “$x-y$ plane” and the feasible region is a line segment $I=[x_0,x_1]$ on the $x$-axis. It is easy to see now that the maximum (or minimum) of $f$ occurs at either $x_0$ or $x_1$. If the optimal value occurs either at both end points, or at an interior point $x_2\in (x_0,x_1)$, then $f$ is a horizontal line segment on $I$.

An application of the above theorems can be demonstrated by the following example: If the feasible region $R$ is a unit square and if corner points $(0,0),(1,1)$ satisfy the optimal solution of $P$, then all points on $\{(x,y)\mid x=y\}\cap R$ satisfy the solution. In particular, $(\frac{1}{2},\frac{1}{2})$, an interior point of $R$, satisfies the solution. As a result, all points of $R$ satisfy the solution.