Discrete Fenchel duality for a pair of integrally convex and separable convex functions

Abstract

Discrete Fenchel duality is one of the central issues in discrete convex analysis. The Fenchel-type min–max theorem for a pair of integer-valued M$$^{\natural }$$ ♮ -convex functions generalizes the min–max formulas for polymatroid intersection and valuated matroid intersection. In this paper we establish a Fenchel-type min–max formula for a pair of integer-valued integrally convex and separable convex functions. Integrally convex functions constitute a fundamental function class in discrete convex analysis, including both M$$^{\natural }$$ ♮ -convex functions and L$$^{\natural }$$ ♮ -convex functions, whereas separable convex functions are characterized as those functions which are both M$$^{\natural }$$ ♮ -convex and L$$^{\natural }$$ ♮ -convex. The theorem is proved by revealing a kind of box integrality of subgradients of an integer-valued integrally convex function. The proof is based on the Fourier–Motzkin elimination.

Structured Replacement Policies for Offshore Wind Turbines

“…Thus, the resulting downtime within a period is longer. Before proceeding to Lemma 3.3, we first review the notion of discrete concavity, which is more fully elucidated by Murota et al [45,46]. Definition 3.2.…”

Inclusion and Intersection Relations Between Fundamental Classes of Discrete Convex Functions

“…The reader is referred to [5,10,12,[26][27][28], [16,Section 3.4], and [20, Section 13] for more about integral convexity, A.3. L-convexity L-and L -convex functions form major classes of discrete convex functions [16,Chapter 7].…”