Geoffrion's theorem beyond finiteness and rationality

Abstract

Geoffrion's theorem is a fundamental result from mathematical programming assessing the quality of Lagrangian relaxation, a standard technique to get bounds for integer programs. An often implicit condition is that the set of feasible solutions is finite or described by rational linear constraints. However, we show through concrete examples that the conclusion of Geoffrion's theorem does not necessarily hold when this condition is dropped. We then provide sufficient conditions ensuring the validity of the result even when the feasible set is not finite and cannot be described using finitely-many linear constraints.

Figures (2)

• Figure 1: The region defined by the inequality $-\sqrt{2}x+y\geqslant0$ (resp. $-\sqrt{2}x+y\leqslant0$) is shown in blue (resp. red). The ray defined by the intersection of the red and blue regions is shown in black. The set $X$ contains the integral points in the red region (including the points in the $x$-axis). Then the only integer point in $X$ that satisfies the inequality $-\sqrt{2}x+y\geqslant0$ is (0,0).
• Figure 2: The set of points considered in Example \ref{['ex:ex2']}.

Theorems & Definitions (16)

• Theorem : Geoffrion's theorem geoffrion1974lagrangian
• Example 1.1
• Example 1.2
• Proposition 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Proposition 2.1
• proof
• proof : Proof of Proposition \ref{['prop:slater']}