Convexity with respect to a set

Abstract

The necessity of characterising the set of efficient and non-efficient points of a vectorial programming problem in integers (see chapter 12) leads to the approach of the convexity with respect to a set. An analysis of the set SI of those solutions of a linear system of inequations having all the coordinates integer numbers shows that this set is not convex. But it has the property that, for any choice of a pair of its points, each convex combination of its elements having all the co-ordinates integer numbers belongs to SI. Starting from this remark, L. Lup§a (1980[150]) first defined the notion of convex set with respect to a given set and studied its properties. This notion corresponds to today’s notion of a strong 2-convex set with respect to a given set. The notion of strong convexity with respect to a given set was introduced in L. Lup§a (1981 [152]), showing by means of an example that the notions of convexity with respect to a set and strong convexity with respect to a set are distinct. After a deeper investigation of these sets, L. Lupsa (1982) introduced the notion of n-convexity with respect to a set, which is equivalent to the strong n-convexity from this chapter. The paper of L. Lup§a (1986[163]) is a synthesis of four types of convexity properties: strong n-convexity with respect to a set, slack n-convexity with respect to a set, strict convexity with respect to a set and slack convexity with respect to a set. The examples presented in that paper show that the notions are distinct from each other and also distinct both from the classical convexity and from the cone convexity. J.E. Martinez-Legaz and I. Singer (1990) proved, when they defined the segmential convexity, that V = Rn and M = Zn, the segmential convexity with respect to (Zn,τ’) is equivalent to the strong 2- convexity. In general these notions are different.KeywordsNatural NumberNonempty SubsetConvexity SpaceNonnegative Real NumberOuter PointThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.