[论文解读] Discrete Midpoint Convexity

Satoko Moriguchi, Kazuo Murota+2 more|arXiv (Cornell University)|Aug 7, 2017

Optimization and Variational Analysis参考文献 34被引用 5

一句话总结

本文引入了局部与全局离散中点凸函数，作为离散凸性的一种统一框架，推广了L^{ atural}-凸性和整数凸性。通过在ℓ∞-距离≥2时施加离散中点凸性，作者建立了诸如平行四边形不等式、缩放与加法下的稳定性以及具有小常数界限的邻近性定理等结构性质——从而实现了一种新颖的邻近性-缩放算法，用于函数最小化。

ABSTRACT

For a function defined on a convex set in a Euclidean space, midpoint convexity is the property requiring that the value of the function at the midpoint of any line segment is not greater than the average of its values at the endpoints of the line segment. Midpoint convexity is a well-known characterization of ordinary convexity under very mild assumptions. For a function defined on the integer lattice, we consider the analogous notion of discrete midpoint convexity, a discrete version of midpoint convexity where the value of the function at the (possibly noninteger) midpoint is replaced by the average of the function values at the integer round-up and round-down of the midpoint. It is known that discrete midpoint convexity on all line segments with integer endpoints characterizes L$^{ atural}$-convexity, and that it characterizes submodularity if we restrict the endpoints of the line segments to be at $\ell_\infty$-distance one. By considering discrete midpoint convexity for all pairs at $\ell_\infty$-distance equal to two or not smaller than two, we identify new classes of discrete convex functions, called local and global discrete midpoint convex functions, which are strictly between the classes of L$^{ atural}$-convex and integrally convex functions, and are shown to be stable under scaling and addition. Furthermore, a proximity theorem, with the same small proximity bound as that for L$^{ atural}$-convex functions, is established for discrete midpoint convex functions. Relevant examples of classes of local and global discrete midpoint convex functions are provided.

研究动机与目标

• 将离散凸性概念（如L^{ atural}-凸性、整数凸性及子模性）统一于离散中点凸性的共同框架之下。
• 基于ℓ∞-距离约束，定义并分析新的离散凸函数类别：局部与全局离散中点凸函数。
• 为这些新函数类别建立结构性质（如平行四边形不等式、缩放与加法下的稳定性）。
• 证明邻近性定理，其邻近常数与L^{ atural}-凸函数相同，从而实现高效最小化。
• 基于邻近性定理与一种新颖的2-邻域最速下降法，设计用于最小化局部与全局离散中点凸函数的邻近性-缩放算法。

提出的方法

• 通过不等式 f(x) + f(y) ≥ f(⌈(x+y)/2⌉) + f(⌊(x+y)/2⌋) 对所有 x,y ∈ ℤⁿ 定义离散中点凸性。
• 引入两类新函数：局部与全局离散中点凸函数，分别要求该不等式在ℓ∞-距离恰好为2或≥2的点对上成立。
• 证明这些类在加法与缩放下保持稳定，并满足一族平行四边形不等式。
• 证明这些类在包含关系上严格介于L^{ atural}-凸函数与整数凸函数之间。
• 基于邻近性定理与一种新颖的2-邻域最速下降法，开发邻近性-缩放算法。
• 通过在整数邻域上进行最小-最大优化，利用凸包络表示定义弱离散中点凸性。

实验结果

研究问题

• RQ1 离散中点凸性能否作为ℤⁿ上已知离散凸函数类别的统一框架？
• RQ2 局部与全局离散中点凸函数具有哪些结构性质？它们与L^{ atural}-凸函数及整数凸函数相比如何？
• RQ3 这些新类别是否具有与L^{ atural}-凸函数类似的邻近性定理，且邻近常数较小？
• RQ4 能否基于邻近性-缩放与邻域下降法设计出适用于这些函数的最小化算法？
• RQ5 弱离散中点凸性（与凸包络的不等式关系）是否等价于整数凸性，即使不假设有效定义域为整数凸？

主要发现

• 局部与全局离散中点凸函数满足一族平行四边形不等式，推广了L^{ atural}-凸函数的已知不等式。
• 这些函数在加法与缩放下保持稳定，保留了L^{ atural}-凸函数的关键代数性质。
• 邻近性定理成立，且邻近常数与L^{ atural}-凸函数相同，从而支持高效最小化。
• 局部与全局离散中点凸函数类在包含关系层次上严格介于L^{ atural}-凸函数与整数凸函数之间。
• 基于邻近性定理与2-邻域最速下降法，开发了一种新的最小化算法，利用邻近性-缩放策略。
• 弱离散中点凸性（与凸包络的不等式关系）即使在不假设有效定义域为整数凸的情况下，也等价于整数凸性。