Large scale set covering problems have often been approached by constructive greedy heuristics, and much research has been devoted to the design and evaluation of various greedy criteria for such heuristics. A criterion proposed by Caprara et al. (1999) is based on reduced costs with respect to the yet unfulfilled constraints, and the resulting greedy heuristic is reported to be superior to those based on original costs or ordinary reduced costs.

We give a theoretical justification of the greedy criterion proposed by Caprara et al. by deriving it from a global optimality condition for general non-convex optimisation problems. It is shown that this criterion is in fact greedy with respect to incremental contributions to a quantity which at termination coincides with the deviation between a Lagrangian dual bound and the objective value of the feasible solution found.

覆盖问题的大规模集合通常通过建设性贪婪启发式来处理，并且大量研究致力于设计和评估此类启发式的各种贪婪标准。Caprara 等人提出的标准。（1999）基于相对于尚未实现的约束的降低成本，据报道，由此产生的贪婪启发式优于基于原始成本或普通降低成本的那些。

我们给出了 Caprara 等人提出的贪心标准的理论依据。通过从一般非凸优化问题的全局最优条件推导出来。结果表明，对于在终止时与拉格朗日对偶界限与找到的可行解的目标值之间的偏差一致的量的增量贡献，该标准实际上是贪婪的。