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Summary: Solving Inequalities and Proving Farkas's Lemma Made Easy
David Avis and Bohdan Kaluzny
March 20, 2003
1 Introduction
Every college student has learned how to solve a system of linear equations, but how many
would know how to solve Ax b for x 0 or show that there is no solution? Solving a
system of linear inequalities has traditionally been taught only in higher level courses and is
given an incomplete treatment in introductory linear algebra courses. For example, the text
of Strang [4] presents linear programming and states Farkas's lemma. It does not, however,
include any proof of the niteness of the simplex method or a proof of the lemma. Recent
developments have changed the situation dramatically. Re nements of the simplex method by
Bland [1] in the 1970s lead to simpler proofs of its niteness, and Bland's original proof was
simpli ed further by several authors. In this paper we will use a variant of Bland's pivot rule
to solve a system of inequalities directly, without any need for introducing linear programming.
We give a simple proof of the niteness of the method, based on ideas contained in the paper
of Fukuda and Terlaky [3] on the related criss-cross method. Finally, if the system is infeasible,
we show how the termination condition of the algorithm gives a certi cate of infeasibility, thus
proving the Farkas Lemma. Terminology and notation used here follows that of Chvatal's linear
programming book [2].
We consider the following problem. Given a matrix A = [a ij ] 2 < mn , and a column vector
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