Projection of polyhedral cones and linear vector optimization

TL;DR

This paper explores how projecting polyhedral cones relates to solving linear vector optimization problems, introducing methods to find extreme directions and dual solutions, and providing an alternative proof of geometric duality.

Contribution

It establishes a connection between cone projection problems and linear vector optimization, offering a new algorithmic approach and an alternative proof of duality theorems.

Findings

Projection problems help solve linear vector optimization tasks.

The approach yields algorithms for arbitrary linear vector optimization problems.

An alternative proof of the geometric duality theorem is provided.

Abstract

Consider a polyhedral convex cone which is given by a finite number of linear inequalities. We investigate the problem to project this cone into a subspace and show that this problem is closely related to linear vector optimization: We define a cone projection problem using the data of a given linear vector optimization problem and consider the problem to determine the extreme directions and a basis of the lineality space of the projected cone $K$. The result of this problem yields a solution of the linear vector optimization problem. Analogously, the dual cone projection problem is related to the polar cone of $K$: One obtains a solution of the geometric dual linear vector optimization problem. We sketch the idea of a resulting algorithm for solving arbitrary linear vector optimization problems and provide an alternative proof of the geometric duality theorem based on duality of polytopes.