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♮-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♮-convex functions and L♮-convex functions, whereas separable convex functions are characterized as those functions which are both M♮-convex and L♮-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.
|Japan Journal of Industrial and Applied Mathematics
|Published - 2022 5月
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