Balinski-Tucker simplex tableaus: Dimensions, degeneracy degrees, and interior points of optimal faces

This paper shows the relationship between degeneracy degrees and multiplicities in linear programming models. The usual definition of degeneracy is restricted to vertices of a polyhedron. We introduce degeneracy for nonempty subsets of polyhedra and show that for linear programming models for which the feasible region contains at least one vertex holds that the dimension of the optimal face is equal to the degeneracy degree of the optimal face of the corresponding dual model. This result is obtained by means of so-called Balinski-Tucker Simplex Tableaus. Furthermore, we give a strong polynomial algorithm for constructing such a Balinski-Tucker Simplex Tableau when an optimal interior point solution is known.