Benson's algorithm

Benson's algorithm, named after Harold Benson, is a method for solving linear multi-objective optimization problems. 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.

Idea 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).

Implementations

Bensolve - a free VLP solver (C programming language)

• www.bensolve.org