Calculus of convex polyhedra and polyhedral convex functions by utilizing a multiple objective linear programming solver

TL;DR

This paper introduces a method for performing operations on convex polyhedra and functions using multiple objective linear programming, enabling explicit calculations via P-representations and a new computational toolbox.

Contribution

It develops a polyhedral calculus toolbox based on MOLP for efficient computation of polyhedral operations and introduces the P-representation concept for convex polyhedra.

Findings

Polyhedral calculus operations can be expressed through P-representations.

MOLP is equivalent to polyhedral projection, enabling computational solutions.

A MATLAB/Octave toolbox for polyhedral calculus has been implemented and tested.

Abstract

The article deals with operations defined on convex polyhedra or polyhedral convex functions. Given two convex polyhedra, operations like Minkowski sum, intersection and closed convex hull of the union are considered. Basic operations for one convex polyhedron are, for example, the polar, the conical hull and the image under affine transformation. The concept of a P-representation of a convex polyhedron is introduced. It is shown that many polyhedral calculus operations can be expressed explicitly in terms of P-representations. We point out that all the relevant computational effort for polyhedral calculus consists in computing projections of convex polyhedra. In order to compute projections we use a recent result saying that multiple objective linear programming (MOLP) is equivalent to the polyhedral projection problem. Based on the MOLP-solver bensolve a polyhedral calculus toolbox for Matlab and GNU Octave is developed. Some numerical experiments are discussed.