Many practical integer programming problems involve variables with one or two-sided bounds. Dunkel and Schulz (A refined Gomory–Chvátal closure for polytopes in the unit cube, http://www.optimization-online.org/DB_FILE/2012/03/3404.pdf, 2012) considered a strengthened version of Chvátal–Gomory (CG) inequalities that use 0–1 bounds on variables, and showed that the set of points in a rational polytope that satisfy all these strengthened inequalities is a polytope. Recently, we generalized this result by considering strengthened CG inequalities that use all variable bounds. In this paper, we generalize further by considering not just variable bounds, but general linear constraints on variables. We show that all points in a rational polyhedron that satisfy such strengthened CG inequalities form a rational polyhedron. We also extend this polyhedrality result to mixed-integer sets defined by linear constraints.

许多实际的整数规划问题涉及具有一侧或两侧边界的变量。Dunkel 和 Schulz（单位立方体中多胞体的精炼 Gomory-Chvátal 闭包，http://www.optimization-online.org/DB_FILE/2012/03/3404.pdf, 2012）认为是 Chvátal-Gomory 的强化版本（ CG) 对变量使用 0-1 界限的不等式，并表明有理多面体中满足所有这些强化不等式的点集是多面体。最近，我们通过考虑使用所有变量边界的强化 CG 不等式来概括这个结果。在本文中，我们不仅考虑变量边界，还考虑变量的一般线性约束，从而进一步概括。我们表明，满足这种强化 CG 不等式的有理多面体中的所有点都形成有理多面体。