Scaling, proximity, and optimization of integrally convex functions

Abstract

In discrete convex analysis, the scaling and proximity properties for the class of Lconvex functions were established more than a decade ago and have been used to design efficient minimization algorithms. For the larger class of integrally convex functions of n variables, we show here that the scaling property only holds when n ≤ 2, while a proximity theorem can be established for any n, but only with a superexponential bound. This is, however, sufficient to extend the classical logarithmic complexity result for minimizing a discrete convex function of one variable to the case of integrally convex functions of any fixed number of variables.∞ -distance for f α . Proximity-scaling algorithmS0: Find an initial vector x with f (x) < +∞, and set α := 2 log 2 K ∞ .

Projection and convolution operations for integrally convex functions

“…Integral convexity can be characterized by a local condition under the assumption that the effective domain is an integrally convex set. 1,10]). Let f : Z n → R ∪ {+∞} be a function with an integrally convex effective domain.…”

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

“…Here is a supplement to Proposition 3.6 about scaling for two-dimensional sets. Part (1) for integrally convex sets is due to [26,Theorem 3.2]. Part (2) for M -convex sets follows from the statement for L -convex sets in Proposition 3.5 and Remark 2.2 in Section 2.6, while Part (3) for M-convex sets is almost a triviality.…”

“…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.…”

Integrality of subgradients and biconjugates of integrally convex functions

“…Integral convexity of a function can be characterized by a local condition under the assumption that the effective domain is an integrally convex set. 1,9]). Let f : Z n → R∪{+∞} be a function with an integrally convex effective domain.…”

“…The extension to functions with general integrally convex effective domains is straightforward, which is found in [11]. Theorem 2.1 is proved in [1, Proposition 3.3] when the effective domain is an integer interval and in [9] for the general case.…”