The authors discuss techniques of linear programming based on the theory of convex cones. A primal method is first given. A method based on a dual condition, which is used to derive the maximum, and on a primal method for the computation of the solution is described in detail.
To construct feasible algorithms that implement the dual method the theory of the extreme rays of the pointed polyhedral cone given by the intersection of a subspace with the nonnegative orthant is developed.
The results of major importance are: (1) in all stages the approach allows a direct and simple geometrical interpretation; (2) the developped theory of strictly tangent relaxation is independent of the particular method used to solve a linear programming problem and allows to reduce the dimension of the problem; (3) the possibility of parallel implementation of the algorithms; (4) for the dual method a restart capability in the sense that it is possible to quickly recompute the maximum after certain changes in the parameters.