A boundary-point LP solution method and its application to dense linear programs

This paper presents a linear programming solution method that generates a sequence of boundary-points belonging to faces of the feasible polyhedron. The method is based on a steepest descent search by iteratively optimising over a two-dimensional cross section of the polyhedron. It differs from extreme point algorithms such as the simplex method in that optimality is detected by identifying an optimal face of the polyhedron which is not necessarily an extreme point. It also differs from the polynomial-time methods such as the ellipsoid algorithm or projective scaling method that avoids the boundary of the feasible polyhedron. Limited computational analysis with an experimental code of the method, EZLP, indicates that our method performs quite well in total solution time when the number of variables and the density of the constraint matrix increase.