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

Akihisa Tamura,Kazuo Murota

doi:10.1007/s13160-022-00499-x

Abstract

AbstractDiscrete 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.

Highlights

Discrete Fenchel duality is one of the central issues in discrete convex analysis [7, 16, 17, 19, 20]
Convex functions constitute a fundamental function class in discrete convex analysis, including both M ♮-convex functions and L ♮-convex functions, whereas separable convex functions are characterized as those functions which are both M ♮-convex and L ♮-convex
We summarize our present knowledge by compiling the results of this paper and the known facts in discrete convex analysis [17]. To this end we introduce notations for classes of functions f ∶ Zn → Z ∪ {+∞}: F = {f ∣ f is an integer-valued integrally convex function}, G = {f ∣ f is the integral conjugate of an integer-valued integrally convex function}, L = {f ∣ f is an integer-valued L♮-convex function}, M = {f ∣ f is an integer-valued M♮-convex function}, S = {f ∣ f is an integer-valued separable convex function}

Summary

Introduction

Discrete Fenchel duality is one of the central issues in discrete convex analysis [7, 16, 17, 19, 20]. The existence of such min–max formula guarantees the existence of a certificate of optimality for the problem of minimizing f (x) − g(x) over x ∈ Zn. The main result of this paper (Theorem 1.1 below) is the Fenchel-type min–max formula (1.3) where f is an integer-valued integrally convex function and g is an integer-valued separable concave function. Theorem 1.2 (Main technical result) Let f ∶ Zn → Z ∪ {+∞} be an integer-valued integrally convex function, x ∈ dom f , and B be an integral box.

Integrally convex functions

Main theorem and its implications

Fenchel duality for other function classes

G CntEx CntEx CntEx

Step 1: weak duality

Step 2: convex extension

Step 3: dual integrality

Step 4: finiteness assumption

Results