# A Representation of Generalized Convex Polyhedra and Applications

## TL;DR

This paper extends the representation formulas for convex polyhedra to generalized convex polyhedra in locally convex spaces, enabling new solution existence results for generalized linear programming and vector optimization problems.

## Contribution

It develops new representation formulas for generalized convex polyhedra in locally convex spaces, generalizing prior Banach space results.

## Key findings

- Representation formulas for generalized convex polyhedra in locally convex spaces.
- Application of formulas to prove solution existence in generalized linear programming.
- Application to generalized linear vector optimization problems.

## Abstract

It is well known that finite-dimensional polyhedral convex sets can be generated by finitely many points and finitely many directions. Representation formulas in this spirit are obtained for convex polyhedra and generalized convex polyhedra in locally convex Hausdorff topological vector spaces. Our results develop those of X. Y. Zheng (Set-Valued Anal., Vol. 17, 2009, 389-408), which were established in a Banach space setting. Applications of the representation formulas to proving solution existence theorems for generalized linear programming problems and generalized linear vector optimization problems are shown.