We will see how general matroids are algorithmically nice by being essentially the same as certain special systems of linear inequalities. We will see how oriented matroids (special as matroids) are an abstraction of general systems of linear inequalities which helps us to discover and derive algorithmically useful linear inequality theory.
To a large extent, "linear inequality theory" means "Farkas Lemma", which we study in some depth. From an especially simple oriented-matroid proof of Farkas lemma, we obtain a new linear programming pivot method, which remains to be tested.
[This is part one of a talk in two parts. Part two is scheduled for November 1, 1995]