Characterizing Efficiency on Infinite-dimensional Commodity Spaces with Ordering Cones Having Possibly Empty Interior

Abstract

Some production models in finance require infinite-dimensional commodity spaces, where efficiency is defined in terms of an ordering cone having possibly empty interior. Since weak efficiency is more tractable than efficiency from a mathematical point of view, this paper characterizes the equality between efficiency and weak efficiency in infinite-dimensional spaces without further assumptions, like closedness or free disposability. This is obtained as an application of our main result that characterizes the solutions to a unified vector optimization problem in terms of its scalarization. Standard models as efficiency, weak efficiency (defined in terms of quasi-relative interior), weak strict efficiency, strict efficiency, or strong solutions are carefully described. In addition, we exhibit two particular instances and compute the efficient and weak efficient solution set in Lebesgue spaces.