论文标题
定向离散的中点凸度
Directed Discrete Midpoint Convexity
论文作者
论文摘要
对于连续函数，中点凸性表征了凸功能。通过考虑中点凸度的离散版本，已经研究了几种类型的函数凸的离散凸度，包括积分凸，l $^\ natural $ -convexity和全局/局部/局部/局部离散的中点凸度。我们提出了一种新型的离散中点凸度，该凸度位于l $^\天然$ -Convexity和积分凸度之间，并且独立于全球/本地离散中点凸度。新的凸度，名为DDM-Convexity，具有L $^\ Natural $ -Convexity和全球/局部/本地离散中点凸的良好属性。 DDM-CONVEX函数在缩放率下是稳定的，满足所谓的平行格不等式和与L $^{\ natural} $ - 凸功能相同的小接近度结合的接近定理。给出了DDM-Convexity的几种表征，并开发了DDM-ConVex功能最小化的算法。我们还提出了连续变量中的DDM-CONVEXITY，并在这些函数上给出接近定理。
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$^\natural$-convexity and global/local discrete midpoint convexity, have been studied. We propose a new type of discrete midpoint convexity that lies between L$^\natural$-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$^\natural$-convexity and global/local discrete midpoint convexity. DDM-convex functions are stable under scaling, satisfy the so-called parallelgram inequality and a proximity theorem with the same small proximity bound as that for L$^{\natural}$-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.
