Functions with Uniform Sublevel Sets on Cones

Abstract

Extended real-valued functions on a real vector space with uniform sublevel sets are important in optimization theory. Weidner studied these functions in [1]. In the present paper, we study the class of these functions, which coincides with the class of Gerstewitz functionals, on cones. These cone are not necessarily embeddable in vector spaces. Almost any Weidner's results are not true on cones without extra conditions. We show that the mentioned conditions are necessary, by nontrivial examples. Specially for element k from the cone $\mathcal{P}$, we define $k$-directional closed subsets of the cone and prove some properties of them. For a subset $A$ of the cone $\mathcal{P}$, we characterize domain of the $\varphi_{A,k}$ (function with uniform sublevel set) and show that this function is $k$-transitive. One of the important conditions for satisfying the results, is that $k$ has the symmetric element in the cone. Also, we prove that, under some conditions, the class of Gerstewitz functionals coincides with the class of $k$-translative functions on $\mathcal{P}$.