Duality in linear inequalities and linear programming

The subject of this paper is the theory of linear inequalities and linear programming. It aims, through use of theory of the duals, to set some formal connections between linear inequalities and linear programming problems, as well as to lay theoretical foundations for certain practical algorithms having been used to solve a general linear program.

In particular, this paper will attempt to present, at the end, the simplex algorithm invented by Dantzig, and the primal-dual algorithm developed later by Dantzig, Ford, and Fulkerson in terms of the general theory, and the duality theory of linear programs. Once the problems of foundation are uncovered, certain features of both algorithms will be presented, from which we could evaluate which one would be more advantegeous than the other.