Revisiting Degeneracy, Strict Feasibility, Stability, in Linear Programming

Abstract

Currently, the simplex method and the interior point method are indisputably the most popular algorithms for solving linear programs. Unlike general conic programs, linear programs with a finite optimal value do not require strict feasibility in order to establish strong duality. Hence strict feasibility is often less emphasized. In this note we discuss that the lack of strict feasibility necessarily causes difficulties in both simplex and interior point methods. In particular, the lack of strict feasibility implies that every basic feasible solution is degenerate. We achieve this using facial reduction and simple means of linear algebra. Furthermore, we emphasize that facial reduction involves two steps where the first guarantees strict feasibility, and the second recovers full-row rankness of the constraint matrix.