Characterizing extreme points as basic feasible solutions in infinite linear programs

We consider linear programs with countably many equality constraints and variables, i.e., Countably Infinite Linear Programs (CILPs). CILPs subsume infinite network flow problems [4] and non-stationary infinite horizon deterministic and stochastic dynamic programs [3]. Recall that a vector x in a convex subset S of a vector space is called an extreme point of S if it cannot be expressed as a strict convex combination of two distinct vectors in S [1]. In standard form finite-dimensional linear programs (henceforth LPs), a feasible solution is an extreme point if and only if it can be obtained as the unique solution to the system of equations derived from the equality constraints by setting a subset of variables to zero (Theorem 2.3 in [2]). This subset consists of the so-called non-basic variables while the remaining variables are termed basic. This has proven critical in LPs as for example in designing pivot operations that exchange a variable from the non-basic set with a variable in the basic set, leading to the Simplex method. Unfortunately, although optimal solutions of CILPs often occur at extreme points, Example 1 illustrates that the above finite dimensional characterization of extreme points does not extend to CILPs heretofore limiting research in this area. Example 1 (modified from [4]) Consider the following CILP