Abstract
For continuous functions, midpoint convexity characterizes convex functions. By considering discrete versions of midpoint convexity, several types of discrete convexities of functions, including integral convexity, L^natural-convexity and global/local discrete midpoint convexity, have been studied. We propose a new type of discrete midpoint convexity that lies between L^natural-convexity and integral convexity and is independent of global/local discrete midpoint convexity. The new convexity, named DDM-convexity, has nice properties satisfied by L^natural-convexity and global/local discrete midpoint convexity. DDM-convex functions are stable under scaling, satisfy the so-called parallelogram inequality and a proximity theorem with the same small proximity bound as that for L^{natural }-convex functions. Several characterizations of DDM-convexity are given and algorithms for DDM-convex function minimization are developed. We also propose DDM-convexity in continuous variables and give proximity theorems on these functions.
Highlights
For a continuous function f defined on a convex set S ⊆ Rn, it was proved by Jensen [12] that midpoint convexity defined by f (x) + f (y) ≥ 2f x + y 2 (∀x, y ∈ S)
The scaled function f defined by f (x) = f ( x) (x ∈ Zn) belongs to the same class, that is, global/local discrete midpoint convexity is closed with respect to scaling operations
Neither L ♮-convexity nor global nor local discrete midpoint convexity has this property, while integral convexity is closed with respect to individual sign inversion of variables
Summary
For a continuous function f defined on a convex set S ⊆ Rn , it was proved by Jensen [12] that midpoint convexity defined by f (x) + f (y) ≥ 2f x + y 2. A function f ∶ Zn → R ∪ {+∞} is said to be locally discrete midpoint convex if dom f is a discrete midpoint convex set and (1.2) holds for any pair (x, y) ∈ Zn × Zn with ‖x − y‖∞ = 2. The scaled function f defined by f (x) = f ( x) (x ∈ Zn) belongs to the same class, that is, global/local discrete midpoint convexity is closed with respect to scaling operations,. The merits of DDM-convexity relative to global/local discrete midpoint convexity are the following properties:. Neither L ♮-convexity nor global nor local discrete midpoint convexity has this property, while integral convexity is closed with respect to individual sign inversion of variables.
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