la rams crop top Abstract In this paper, for every relation R on a vector space V, we consider the R-vector space ( V , R ) $(V,R)$ and define the notions of R-convexity, R-convex hull, and R-extreme point in this space.Some examples are provided to compare them with the reference cases.The effects of some operations on R-convex sets are investigated.In particular, it is shown that the R-interior of an R-convex set is also an R-convex set under some restrictions on R.
Also, we give some equivalent conditions for R-extremeness.Moreover, the notions of R-convex and R-affine maps on R-vector spaces are defined, and some results that assert the relation between an R-convex map f and its R-epigraph under some limitations on R are considered.Several propositions, such as R-continuous maps preserve R-compact sets and R-affine maps preserve R-convex sets, are presented, and some results on click here the composition of R-convex and R-affine maps are considered.Finally, some applications of R-convexity are investigated in optimization.
More precisely, we show that the extrema values of R-affine R-continuous maps are reached on R-extreme points.Moreover, local and global minimum points of an R-convex map f on R-convex set K are considered.