A geometric simplex method in infinite-dimensional spaces

Abstract

We expand the basic geometric elements of the simplex method to linear programs in locally convex topological vector spaces and provide conditions under which the method converges in value to optimality. This setting generalizes many previous investigations of the simplex method, which are restricted to Hilbert spaces or otherwise specially structured instances. Our generality is obtained by avoiding the ``algebraic'' machinery of pivoting via column operations, which has required stronger topological conditions in establishing a connection between basic feasible solutions and extreme point structure. We show that our definition of polytopes captures optimization over the Hilbert cube, a quintessential object in infinite-dimensional spaces known for its surprisingly complicated properties. Moreover, all polytopes (under our definition) have exposed extreme points connected by edge paths.

Figures (4)

• Figure 1: An illustration of the basic notions in the paper. Here $A = \{\alpha_1, \alpha_2, \dots \alpha_5\}$, $A(p) = \{\alpha_1, \alpha_5\}$, $\mathcal{H}(p) = H_{\alpha_1} \cap H_{\alpha_5} = \{p\}$, and $s_{\alpha_1}(p) = s_{\alpha_5}(p) = 0$ with $s_{\alpha_j}(p) > 0$ for $j = 2,3,4$.
• Figure 2: A counter-example to the converse of \ref{['prop:intersect-then-extreme']}.
• Figure 3: An illustration of extreme points, edges, and adjacent extreme points. Here $S_1 = \{ x \in \mathbb R^3 \mid -x_1 \le 0$, $S_2 = \{x \in \mathbb R^3 \mid -x_2 \le 0\}$, $S_3 = \{x \in \mathbb R^3 \mid -x_3 \le 0\}$, and $S_4 = \{ x \in \mathbb R^3 \mid x_1 + x_2 + x_3 \le 1\}$.
• Figure 4: Visualizing the construction of the Schauder basis at an extreme point.

Theorems & Definitions (44)

• Proposition 1
• proof
• Example 1
• Example 2: Hilbert cube
• Lemma 1
• Lemma 2
• Proposition 2
• proof
• Theorem 1
• Lemma 3: Non-degeneracy