Convex envelopes for ray-concave functions

Explicit convex and concave envelopes through polyhedral subdivisions

5 In this paper, we derive explicit characterizations of convex and concave envelopes of several 6 nonlinear functions over various subsets of a hyper-rectangle. These envelopes are obtained 7 by identifying polyhedral subdivisions of the hyper-rectangle over which the envelopes can be 8 constructed easily. In particular, we use these techniques to derive, in closed-form, the concave 9 envelope...

Short Proofs of the Separation Theorems for L-convex/concave and M-convex/concave Functions

Recently K. Murota has introduced concepts of L-convex function and Mconvex function as generalizations of those of submodular function and base polyhedron, respectively, and has shown separation theorems for L-convex/concave functions and for M-convex/concave functions. The present note gives short alternative proofs of the separation theorems by relating them to the ordinary separation theore...

On convex-concave perception based functions

Perception based function (PBF) is given by the set of rules Ri:“If X is Ti then Y is Si”, where Ti is a linguistic term describing some fuzzy intervals Ai on the domain of real values of X and Si is a linguistic description of the shape of the function Y(X) on this interval. The methods of explicitation of such type of rules when Si are given by words QUICKLY INCREASING AND SLIGHTLY CONVEX, SL...

On Self-Concordant Convex-Concave Functions

In this paper, we introduce the notion of a self-concordant convex-concave function, establish basic properties of these functions and develop a path-following interior point method for approximating saddle points of “good enough” convex-concave functions – those which admit natural self-concordant convex-concave regularizations. The approach is illustrated by its applications to developing an ...

Convex and concave relaxations of implicit functions