L-convex Functions and M- Convex Functions Encyclopedia of Optimization (kluwer)

In the eld of nonlinear programming (in continuous variables) convex analysis [21, 22] 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. Denitions of Land M-convexity. Let V be a nonempty nite set and Z be the set of integers. For any function g : Z V ! Z [ f+1g dene dom g = fp 2 Z V j g(p) < +1g, called the eective domain of g.