Projection and convolution operations for integrally convex functions

Abstract

This paper considers projection and convolution operations for integrally convex functions, which constitute a fundamental function class in discrete convex analysis. It is shown that the class of integrally convex functions is stable under projection, and this is also the case with the subclasses of integrally convex functions satisfying local or global discrete midpoint convexity. As is known in the literature, the convolution of two integrally convex functions may possibly fail to be integrally convex. We show that the convolution of an integrally convex function with a separable convex function remains integrally convex. We also point out in terms of examples that the similar statement is false for integrally convex functions with local or global discrete midpoint convexity.

Directed discrete midpoint convexity

“…Example 6.1. We borrow the example in [16,Example 4.2], which shows that L ♮convexity and global/local discrete midpoint convexity may not be closed under the convolutions. Let S 1 = {(0, 0, 0), (1, 1, 0)} and S 2 = {(0, 0, 0), (0, 1, 1)} which are DDM-convex.…”

“…, n). By the same arguments of [16,Proposition 4.7], this can be extended to DDM-convexity. Proposition 6.9.…”

A survey of fundamental operations on discrete convex functions of various kinds

“…The reader is referred to [10], [31,Section 3.4], and [36,Section 13] for more about integral convexity, and [24] and [26] for recent developments.…”

“…The results for L -convex, Lconvex, and M -convex functions are given in [31,Theorem 7.11], [31,Theorem 7.10], and [31,Theorem 6.15], respectively. The statements for integrally convex functions, globally discrete midpoint convex functions, locally discrete midpoint convex functions, and jump M -convex functions are obtained recently in [24,Theorem 3.3], [24,Theorem 3.5], [24,Theorem 3.6], and [37], respectively.…”

“…With arithmetic or numerical operations on function values we can define nonnegative multiplication of function values f (x) → a f (x) with a ≥ 0, addition of a linear function f (x) → f (x) + n i=1 c i x i with c ∈ R n , projection (partial minimization) f (x) → inf z f (y, z), sum f 1 + f 2 of two functions f 1 and f 2 , convolution ( f 1 f 2 )(x) = inf{ f 1 (y) + f 2 (z) | x = y + z, y, z ∈ Z n } of two functions f 1 and f 2 , etc. Stability of discrete convexity under these operations has been investigated for many function classes in discrete convex analysis [19,24,26,27,31,39]. By collecting these results scattered in the literature and by adding new observations and examples, we will present a fairly comprehensive survey on the operations on discrete convex sets and functions.…”