TITLE: Generating random linear programming problems
QUESTION [7 upvotes]: I've just finished writing a a linear programming problem solver which uses the simplex method.  Now I would like to start optimizing my solver but before I can do this, I need a way of reliably testing it's performance.
What is a good algorithm for generating random linear programming problems of arbitrary size?  If possible I would also like to be able to control whether a solution exists or not and I would like to ensure that the origin is a vertex on the simplex.

REPLY [5 votes]: If you want a given vector $v \in {\mathbb R}^n$ as a solution:

take a random $m \times n$ matrix $A$
choose a vector $b \in{\mathbb R}^m$ such that $b \ge A v$.
You can generate b by taking $b=Av+\delta, \delta_i\ge0$ 

Then the constraints can be $A x \le b$.
Generically, if $m \ge n$ and $b - Av$ has at least $n$ zero entries, $v$ will be a basic solution.
If you want a problem that is unbounded 

pick vectors $u$ and $v$ with $v \ne 0$.
Take a random matrix $A$.
For each $j$ such that $(Av)_j > 0$, multiply row $j$ of $A$ by
$-1$, so you get a matrix $A$ with $A v \le 0$.
Choose $b$ such that $b \ge A u$.  Then $u + t v$ satisfies the constraints $Ax \le b$ for all $t \ge 0$.
Take for the objective any vector $c \in
{\mathbb R}^n$ such that $c^T v > 0$.

If you want a problem that is infeasible, take the dual of an unbounded problem.