An application of homotopy to solving linear programs

Consider a linear program inm inequality constraints andn nonnegative variables. An application of homotopy to the problem gives an algorithm similar to Dantzig's self-dual method. Howeve, the homotopy approach allows one to recognize several previously undescribed and potentially interesting properties. For example, the algorithm can be initiated in such a way as to produce a path which is primal-dual feasible. Moreover, one can theoretically identify an orthant with the property that if one initiates the algorithm at any point in that orthant then, after a ‘phase I’ requiring at most min{m, n} pivots, convergence is obtained in one step.