On the approximation error for approximating convex bodies using multiobjective optimization

TL;DR

This paper analyzes the approximation error when representing convex bodies with polyhedral shapes derived from multiobjective convex programming, providing bounds based on algorithm stopping criteria.

Contribution

It introduces error bounds for polyhedral convex body approximations using Benson type algorithms, linking approximation quality to algorithm stopping criteria.

Findings

Error bounds in Hausdorff distance for convex body approximations

Dependence of approximation accuracy on primal and dual algorithm stopping criteria

Theoretical analysis of polyhedral approximation quality

Abstract

A polyhedral approximation of a convex body can be calculated by solving approximately an associated multiobjective convex program (MOCP). An MOCP can be solved approximately by Benson type algorithms, which compute outer and inner polyhedral approximations of the problem's upper image. Polyhedral approximations of a convex body can be obtained from polyhedral approximations of the upper image of the associated MOCP. We provide error bounds in terms of the Hausdorff distance for the polyhedral approximations of a convex body in dependence of the stopping criterion of the primal and dual Benson type algorithms which are applied to the associated MOCP.