Discrete Midpoint Convexity

Abstract

For a function defined on the integer lattice, we consider discrete versions of midpoint convexity, which offer a unifying framework for discrete convexity of functions, including integral convexity, L -convexity, and submodularity. By considering discrete midpoint convexity for all pairs at ∞ -distance equal to two or not smaller than two, we identify new classes of discrete convex functions, called locally and globally discrete midpoint convex functions. These functions enjoy nice structural properties. They are stable under scaling and addition, and satisfy a family of inequalities named parallelogram inequalities. Furthermore, they admit a proximity theorem with the same small proximity bound as that for L -convex functions. These structural properties allow us to develop an algorithm for the minimization of locally and globally discrete midpoint convex functions based on the proximity-scaling approach and on a novel 2-neighborhood steepest descent algorithm.

Directed discrete midpoint convexity

“…Theorem 8.2 says that the sequence of points generated by the 1neighborhood steepest descent algorithm is bounded by the ℓ ∞ -distance between an initial point and the nearest minimizer. Similar facts are pointed out for L ♮ -convex function minimization [13,24,26] and globally/locally discrete midpoint convex function minimization [18].…”

“…In the same way as the arguments in [18], we can show the following proximity theorem for DDM-convex functions. We note that f α is also DDM-convex by Theorem 6.1 and x α corresponds to a minimizer 0 of f α (y) = f (x α + αy) by Corollary 3.3.…”

“…A function f : Z n → R ∪ {+∞} is said to be locally discrete midpoint convex if dom f is a discrete midpoint convex set and (1.2) holds for any pair (x, y) ∈ Z n × Z n with x − y ∞ = 2. It is shown in [18] that the following inclusion relations among function classes hold:…”

“…This paper, strongly motivated by [18], proposes a new type of discrete midpoint convexity between L ♮ -convexity and integral convexity, but it is independent of global/local discrete midpoint convexity with respect to inclusion relation. We name the new convexity directed discrete midpoint convexity (DDM-convexity) which forms the following classification…”

“…In Section 3, we discuss a relationship between DDM-convexity and known discrete convexities. For globally/locally discrete midpoint convex functions, Moriguchi et al [18] revealed a useful property, which is expressed by the so-called parallelgram inequality. We show that a similar parallelgram inequality holds for DDM-convex functions in Section 4.…”