并且证明了拉格朗日对偶界与通过凸松弛得到的下界是相等的；
We prove that the Lagrangian dual bound is identical to the lower bound obtained by convex relaxation.
应用已有的极点理论，优化相伴凸组合的界并改进整数规划问题的目标函数及变量的界。
Assuming knowledge of extreme points, we develop bounds for associated convex combinations and improve bounds on the integer programming problems objective function and variables.
应用推荐