On the approximation error for approximating convex bodies using multiobjective optimization

TL;DR

This work provides 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.

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.

Figures (2)

• Figure 1: Illustration of Example \ref{['ex:p']} for $q=2$.
• Figure 2: Illustration of Example \ref{['exam:dual']} for $q=2$.

Theorems & Definitions (25)

• Proposition 1
• proof
• Proposition 2
• proof
• Theorem 3: heyde2013
• Lemma 4
• proof
• Proposition 5
• proof
• Proposition 6