Abstract

: The present knowledge about L-convex and M-convex functions are surveyed on the basis of [27]. Keywords: Discrete convexity, Submodular function, Matroid Introduction In the field of nonlinear programming (in continuous variables) convex analysis [30, 31] plays a pivotal role both in theory and in practice. An analogous theory for discrete optimization (nonlinear integer programming), called "discrete convex analysis" [24, 25], is developed for L-convex and M-convex functions by adapting the ideas in convex analysis and generalizing the results in matroid theory. The L- and M-convex functions are introduced in [24] and [22, 23], respectively. A polyhedral theory on L- and M-convex functions is developed recently in [29]. Definitions of L-convexity and M-convexity Let V be a nonempty finite set and Z be the set of integers. For any function g : Z V ! Z[ f+1g define dom g = fp 2 Z V j g(p) ! +1g, called the effective domain of g. A function g : Z V ! Z[f+1g with dom g 6= ;...

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