Simplex method, Linear programming, Duality.

We note that the well known simplex algorithm can be formulated in a very general setting of a possibly infinite-dimensional space over a linearly ordered field (e.g. the field of real numbers) and another linearly ordered vector space (e.g. the N-dimensional real vector space with the lexicographic ordering) over the same field. We recall the concept of basic solution and the concept of its primal and dual feasibility in the above described setting. We show (contrary to the generally accepted opinion) that a primal feasible basic solution yields a dual feasible solution and that a dual feasible basic solution yields a primal feasible solution. Then we present the primal and dual simplex method and point out differences between both algorithms.