Directed discrete midpoint convexity
Directed discrete midpoint convexity
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DOI：
10.1007/s13160-020-00416-0
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发表时间：
2020-01
影响因子：
0.9
通讯作者：
A. Tamura;Kazuya Tsurumi
中科院分区：
文献类型：
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作者：
A. Tamura;Kazuya Tsurumi
对于连续函数，中点凸性表征凸函数。通过考虑离散形式的中点凸性，研究了函数的几种离散凸性，包括积分凸性、L-凸性和整体/局部离散中点凸性。本文提出了一种新的离散中点凸性，它介于L-凸性和积分凸性之间，且与全局/局部离散中点凸性无关。这种新的凸性称为DDM-凸性，它具有L-凸性和全局/局部离散中点凸性所满足的良好性质。DDM-凸函数在标度下是稳定的，满足所谓的Einstein不等式和一个邻近定理，其小邻近界与L-凸函数的小邻近界相同。给出了DDM-凸性的几个特征，并给出了DDM-凸函数极小化的算法。我们还提出了连续变量的DDM-凸性，并给出了这些函数的邻近定理。
For continuous functions, midpoint convexity characterizes convex functions. By considering discrete versions of midpoint convexity, several types of discrete convexities of functions, including integral convexity, L-convexity and global/local discrete midpoint convexity, have been studied. We propose a new type of discrete midpoint convexity that lies between L-convexity and integral convexity and is independent of global/local discrete midpoint convexity. The new convexity, named DDM-convexity, has nice properties satisfied by L-convexity and global/local discrete midpoint convexity. DDM-convex functions are stable under scaling, satisfy the so-called parallelogram inequality and a proximity theorem with the same small proximity bound as that for L-convex functions. Several characterizations of DDM-convexity are given and algorithms for DDM-convex function minimization are developed. We also propose DDM-convexity in continuous variables and give proximity theorems on these functions.