Encyclopedia

Benson's algorithm

Wikipedia contributorsAdapted for Fathomly · View authors

1 min read
Source updated January 31, 2019

Benson's algorithm, named after Harold Benson, is a method for solving multi-objective linear programming problems and vector linear programs. This works by finding the "efficient extreme points in the outcome set". The primary concept in Benson's algorithm is to evaluate the upper image of the vector optimization problem by cutting planes.

01Idea of algorithm

Consider a vector linear program

$\min _{C}Px\;{\text{ subject to }}Ax\geq b$

for $P\in \mathbb {R} ^{q\times n}$ , $A\in \mathbb {R} ^{m\times n}$ , $b\in \mathbb {R} ^{m}$ and a polyhedral convex ordering cone $C$ having nonempty interior and containing no lines. The feasible set is $S=\{x\in \mathbb {R} ^{n}:\;Ax\geq b\}$ . In particular, Benson's algorithm finds the extreme points of the set $P[S]+C$ , which is called upper image.

In case of $C=\mathbb {R} _{+}^{q}:=\{y\in \mathbb {R} ^{q}:y_{1}\geq 0,\dots ,y_{q}\geq 0\}$ , one obtains the special case of a multi-objective linear program (multiobjective optimization).

02Dual algorithm

There is a dual variant of Benson's algorithm, which is based on geometric duality for multi-objective linear programs.

03Implementations

Bensolve - a free VLP solver

www.bensolve.org

Inner

Link to github

Watch videos about Benson's algorithmExplainers and documentaries on YouTube (opens in a new tab)

Sources and credits

This article is adapted from the Wikipedia article “Benson's algorithm”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

Fathomly is not affiliated with or endorsed by the Wikimedia Foundation. Spotted a problem? Tell us.