Dually cone-boundedness of a set and applications
Marius Durea, Elena-Cristina Stamate
TL;DR
The paper introduces dually cone-boundedness, a weak cone-relative boundedness notion defined by requiring $x^*(A)$ to be bounded below for all $x^* \in K^+$, where $K$ is a closed convex cone in a normed space. It proves a hierarchy of conditions: (i) $A \subset \ell D_X + K$, (ii) $\inf x^*(A) \ge \rho$ for all $x^* \in K^+ \cap D_{X^*}$, and (iii) $x^*(A)$ bounded below for all $x^* \in K^+$, with (i) ⇔ (ii) ⇒ (iii), while (iii) does not imply (i) in general; equivalence can hold in finite dimensions or when $K^+$ is finitely generated. The authors derive conic cancellation rules under dually $K$-boundedness and present two key applications: Pareto minimality results and a Radström-type embedding for the corresponding set spaces, extending prior results and providing a unified framework for cone optimization under weaker hypotheses. The work broadens the toolkit for cone optimization by enabling weaker assumptions while preserving core procedures, with potential extensions to Pettis integrability and domination-type properties in vector optimization.
Abstract
We introduce and study a generalized concept of boundedness of a subset of a normed vector space with respect to a cone, which is defined as lower boundedness of the images of the underlying set through all the positive functionals of the cone. We show that this is a weaker notion when compared to other similar ones and we explore several links with the existing literature. We subsequently demonstrate that this concept furnishes the properties required to obtain various generalizations of important results and techniques, including conic cancellation rules and the Rådström embedding procedure.