L-convex Functions and M-convex Functions

In the field of nonlinear programming (in continuous variables) convex analysis [22, 23] plays a pivotal role both in theory and in practice. An analogous theory for discrete optimization (nonlinear integer programming), called " discrete convex analysis " [18, 17], is developed for L-convex and M-convex functions by adapting the ideas in convex analysis and generalizing the results in matroid theory. The Land M-convex functions are introduced in [18] and [13, 14], respectively. Definitions of Land M-convexity. Let V be a nonempty finite set and Z be the set of integers. For any function g : Z V → Z ∪ {+∞} define dom g = {p ∈ Z V | g(p) < +∞}, called the effective domain of g. g(p) + g(q) ≥ g(p ∨ q) + g(p ∧ q) (p, q ∈ Z V), ∃r ∈ Z : g(p + 1) = g(p) + r (p ∈ Z V), where p ∨ q = (max(p(v), q(v)) | v ∈ V) ∈ Z V , p ∧ q = (min(p(v), q(v)) | v ∈ V) ∈ Z V , and 1 is the vector in Z V with all components being equal to 1. A set D ⊆ Z V is said to be an L-convex set if its indicator function δ D (defined by: δ D (p) = 0 if p ∈ D, and = +∞ otherwise) is an L-convex function, i.e., if (i) D ̸ = ∅, (ii) p, q ∈ D ⇒ p ∨ q, p ∧ q ∈ D, and (iii) p ∈ D ⇒ p ± 1 ∈ D. A function f : Z V → Z∪{+∞} with dom f ̸ = ∅ is called M-convex if it satisfies (M-EXC) For x, y ∈ dom f and u ∈ supp + (x − y), there exists v ∈ supp − (x − y) such that f (x) + f (y) ≥ f (x − χ u + χ v) + f (y + χ u − χ v) where, for any u ∈ V , χ u is the characteristic vector of u (defined by: χ u (v) = 1 if v = u, and = 0 otherwise), and supp + (z) = {v ∈ V | z(v) > 0} (z ∈ Z V), supp − (z) = {v ∈ V | z(v) < 0} (z ∈ Z V). A set B ⊆ Z …