A Benson-Type Algorithm for Bounded Convex Vector Optimization Problems with Vertex Selection

TL;DR

An algorithm for approximately solving bounded convex vector optimization problems that provides both an outer and an inner polyhedral approximation of the upper image and proposes a new and efficient selection rule for deciding which vertex to cutoff.

Abstract

We present an algorithm for approximately solving bounded convex vector optimization problems. The algorithm provides both an outer and an inner polyhedral approximation of the upper image. It is a modification of the primal algorithm presented by Löhne, Rudloff, and Ulus in 2014. There, vertices of an already known outer approximation are successively cut off to improve the approximation error. We propose a new and efficient selection rule for deciding which vertex to cut off. Numerical examples are provided which illustrate that this method may solve fewer scalar problems overall and therefore may be faster while achieving the same approximation quality.

Figures (4)

• Figure 1: Illustration of an $\varepsilon$-solution ${\mathcal{X}}\subseteq S$ (see Definition \ref{['def:3']}). The four points of $F[\mathcal{X}]$ are $C$-minimal in $\mathcal{P}$, hence ${\mathcal{X}}$ is a set of minimizers. The Hausdorff distance between ${\mathop{\mathrm{conv}}\nolimits F[\mathcal{X}] +C}$ and $\mathcal{P}$ is $\varepsilon$.
• Figure 2: Left: Outer (red) and inner (blue) approximations of $\mathcal{P}$ after iteration $k$. Center: The vertex $v$ and direction $c$ are obtained by the vertex selection. The point $F(x)$ is obtained by solving \ref{['P2']}. Right: The updated outer and inner approximations after cutting off $v$ and adding $F(x)$ as a vertex to $\mathcal{I}^k$.
• Figure 3: Inner approximations of the upper image for Example \ref{['ex:1']} with $a = 7$; with vertex selection \ref{['fig:ex1vs']} and without \ref{['fig:ex1wo']}. Each vertex corresponds to a weak minimizer. One can see that without vertex selection there are many vertices in close proximity to each other, especially in regions that exhibit a large curvature (bottom right). With vertex selection, the vertices are "spread more evenly" across the surface.
• Figure 4: The planar 10-member truss from Example \ref{['ex:3']} with two fixed supports and four free nodes. The colored arrows illustrate different loads corresponding to weak minimizers.

Theorems & Definitions (15)

• Definition 3.1
• Definition 3.2
• Definition 3.3
• Remark 3.4
• Proposition 3.5
• Proposition 3.6
• Proposition 3.7
• Proposition 3.8
• Proposition 4.1
• Corollary 4.2