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

Andreas Löhne, Benjamin Weißing

TL;DR

The result implies that an arbitrary vector linear program (with arbitrary polyhedral ordering cone) can be solved by solving a multiple objective linear program with one additional objective space dimension.

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.

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

TL;DR

The result implies that an arbitrary vector linear program (with arbitrary polyhedral ordering cone) can be solved by solving a multiple objective linear program with one additional objective space dimension.

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.

Paper Structure

This paper contains 3 sections, 5 theorems, 84 equations, 2 figures.

Table of Contents

  1. Problem formulations and solution concepts
  2. Equivalence between (PP), (MOLP) and (VLP)
  3. Examples and remarks

Key Result

Theorem 3

Let a polyhedral projection problem P be given. If P is feasible, then a solution $(S^{\rm poi},S^{\rm dir})$ (compare Definition def.sltnVLP) of the associated multiple objective linear program associated_molp exists. Every solution $(S^{\rm poi},S^{\rm dir})$ of associated_molp is also a solution

Figures (2)

  • Figure 1: The feasible set $S$ and the upper image $P[S]+C$ of Example \ref{['ex:8']}. A solution is given by the feasible points $x^1$,$x^2$ and the feasible direction $x^3$. Their respective image-vectors $y^1$,$y^2$ and $y^3$ generate the upper image. It can be seen that $x^4$ is not part of a solution, as the image-vector $y^4$ belongs to the ordering cone $C$ and is therefore not a minimal direction.
  • Figure 2: The polytope $P[W]$ of Example \ref{['ex1']} computed by BensolvebensolveLoeWei16 via MOLP reformulation. The resulting polytope has $43680$ vertices and $26186$ facets. The upper image $\mathcal{P}\subseteq \mathbb{R}^4$ of the corresponding MOLP has $43680$ vertices and $26187$ facets. The displayed polytope is one of these facets, the only one that is bounded.

Theorems & Definitions (14)

  • Definition 1
  • Definition 2
  • Theorem 3
  • proof
  • Proposition 4
  • proof
  • Theorem 5
  • proof
  • Corollary 6
  • Theorem 7
  • ...and 4 more