Approximating Extreme Points of Infinite Dimensional Convex Sets

The property that an optimal solution to the problem of minimizing a continuous concave function over a compact convex set in ℝⁿ is attained at an extreme point is generalized by the Bauer Minimum Principle to the infinite dimensional context. The problem of approximating and characterizing infinite dimensional extreme points thus becomes an important problem. Consider now an infinite dimensional compact convex set in the nonnegative orthant of the product space ℝ^∞. We show that the sets of extreme points E_N of its corresponding finite dimensional projections onto ℝ^N converge in the product topology to the closure of the set of extreme points E of the infinite dimensional set. As an application, we extend the concept of total unimodularity to infinite systems of linear equalities in nonnegative variables where we show when extreme points inherit integrality from approximating finite systems. An application to infinite horizon production planning is considered.