Consider the relaxation of an integer programming (IP)  problem in which the feasible region is replaced by the intersection of  the linear programming (LP) feasible region and the corner  polyhedron for a particular LP basis. Recently a primal-dual ascent algorithm has been given for solving this relaxation. Given an optimal  solution of this relaxation, we state criteria for selecting a new LP  basis for which the associated relaxation is stronger. These criteria  may be successively applied to obtain either an optimal IP solution or a lower bound on the cost of such a solution. Conditions are given  for equality of the convex hull of feasible IP solutions and the  intersection of all corner polyhedra.