SIAM Journal on Optimization, Volume 30, Issue 4, Page 2809-2840, January 2020.
In this paper, we compare the definitions of convex sets and convex functions in finite dimensional integer spaces introduced by Adivar and Fang, Borwein, and Giladi, respectively. We show that their definitions of convex sets and convex functions are equivalent. We also provide exact formulations for convex sets, convex cones, affine sets, and convex functions and we analyze the separation between convex sets in finite dimensional integer spaces. As an application, we consider an integer linear programming problem with linear inequality constraints and obtain some necessary or sufficient optimality conditions by employing the image space analysis. We finally provide some computational results based on the above-mentioned optimality conditions.

中文翻译：

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$ \ mathbb {Z} ^ n $中的凸分析及其在整数线性规划中的应用

SIAM优化杂志，第30卷，第4期，第2809-2840页，2020年1月。
在本文中，我们比较了Adivar和Fang，Borwein和Giladi分别引入的有限维整数空间中凸集和凸函数的定义。我们证明了它们对凸集和凸函数的定义是等效的。我们还提供了凸集，凸锥，仿射集和凸函数的精确公式，并分析了有限维整数空间中凸集之间的间隔。作为应用，我们考虑具有线性不等式约束的整数线性规划问题，并通过使用图像空间分析获得一些必要或充分的最优性条件。最后，我们根据上述最优条件提供了一些计算结果。