Equivalence between polyhedral projection, multiple objective linear programming and vector linear programming

TL;DR

This paper establishes an equivalence between polyhedral projection, multiple objective linear programming, and vector linear programming, showing how solutions to one can be transformed into solutions for the others, simplifying complex optimization problems.

Contribution

It demonstrates that polyhedral projection problems are equivalent to multiple objective linear programs, enabling solutions via standard vector linear programming methods with an added objective.

Findings

Polyhedral projection can be solved via multiple objective linear programming.

Vector linear programs with arbitrary cones can be reduced to standard form with one extra objective.

The equivalence simplifies solving complex polyhedral and vector optimization problems.

Abstract

Let a polyhedral convex set be given by a finite number of linear inequalities and consider the problem to project this set onto a subspace. This problem, called polyhedral projection problem, is shown to be equivalent to multiple objective linear programming. The number of objectives of the multiple objective linear program is by one higher than the dimension of the projected polyhedron. The result implies that an arbitrary vector linear program (with arbitrary polyhedral ordering cone) can be solved by solving a multiple objective linear program (i.e. a vector linear program with the standard ordering cone) with one additional objective space dimension.