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.

翻译：杰弗里翁定理是数学规划中的一个基本结果，用于评估拉格朗日松弛的质量——这是获取整数规划界限的标准技术。一个常被隐含的条件是可行解集为有限集或由有理线性约束描述。然而，我们通过具体实例表明，当此条件被舍弃时，杰弗里翁定理的结论未必成立。随后，我们提出了充分条件，以确保即使可行集非有限且无法用有限个线性约束描述时，该结果仍然有效。