Convex Analysis and Duality over Discrete Domains

The aim of this paper is to establish a fundamental theory of convex analysis for the sets and functions over a discrete domain.By introducing conjugate/biconjugate functions and a discrete duality notion for the cones over discrete domains,we study duals of optimization problems whose decision parameters are integers.In particular,we construct duality theory for integer linear programming,provide a discrete version of Slater’s condition that implies the strong duality and discuss the relationship between integrality and discrete convexity.