# Facially Dual Complete (Nice) cones and lexicographic tangents

## TL;DR

This paper investigates the structure of facially dual complete cones, introducing lexicographic tangent cones and new exposure notions to characterize their duality properties in convex optimization.

## Contribution

It introduces lexicographic tangent cones and new exposure notions, providing necessary and sufficient conditions for cones to be facially dual complete.

## Key findings

- Established a necessary condition for facially dual complete cones.
- Provided a sufficient condition using lexicographic tangent cones.
- Connected lexicographic tangent cones to Nesterov's derivatives and subtransversality.

## Abstract

We study the boundary structure of closed convex cones, with a focus on facially dual complete (nice) cones. These cones form a proper subset of facially exposed convex cones, and they behave well in the context of duality theory for convex optimization. Using the well-known and commonly used concept of tangent cones in nonlinear optimization, we introduce some new notions for exposure of faces of convex sets. Based on these new notions, we obtain a necessary condition and a sufficient condition for a cone to be facially dual complete. In our sufficient condition, we utilize a new notion called lexicographic tangent cones (these are a family of cones obtained from a recursive application of the tangent cone concept). Lexicographic tangent cones are related to Nesterov's lexicographic derivatives and to the notion of subtransversality in the context of variational analysis.