Roughly d-Convex Functions on Undirected Tree Networks

Daniela Marian

In this paper we establish some properties of roughly d-convex functions on undirected tree networks. It is pointed out that these roughly d-convex functions have the following properties concerning the property of minimum: each local minimum of a midpoint $\delta$-d-convex or lightly $\gamma$-d-convex functions is a global minimum, where a local minimizer has to yield the minimal function value in its neighborhood with radius equal to the roughness degree. Since every $\rho$-d-convex or $\delta$-d-convex function is midpoint $\delta$-d-convex and every $\gamma$-d-convex function is lightly $\gamma$-d convex, this conclusion holds for them, too. We also state weaker but sufficient conditions for roughly d-convex functions. We adopt the definition of network as metric space introduced by Dearing P.M. and Francis R.L. in 1974.