Benson's algorithm

Not to be confused with Benson's algorithm (Go), a method to find the unconditionally alive stones in the game Go.

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".^[1] The primary concept in Benson's algorithm is to evaluate the upper image of the vector optimization problem by cutting planes.^[2]

Idea of algorithm

Consider a vector linear program

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

for ${\displaystyle P\in \mathbb {R} ^{q\times n}}$, ${\displaystyle A\in \mathbb {R} ^{m\times n}}$, ${\displaystyle b\in \mathbb {R} ^{m}}$ and a polyhedral convex ordering cone ${\displaystyle C}$ having nonempty interior and containing no lines. The feasible set is ${\displaystyle S=\{x\in \mathbb {R} ^{n}:\;Ax\geq b\}}$. In particular, Benson's algorithm finds the extreme points of the set ${\displaystyle P[S]+C}$, which is called upper image.^[2]

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

Dual algorithm

There is a dual variant of Benson's algorithm,^[3] which is based on geometric duality^[4] for multi-objective linear programs.

Implementations

Bensolve - a free VLP solver

• www.bensolve.org

Inner

• Link to github

References

1. Harold P. Benson (1998). "An Outer Approximation Algorithm for Generating All Efficient Extreme Points in the Outcome Set of a Multiple Objective Linear Programming Problem". Journal of Global Optimization. 13 (1): 1–24. doi:10.1023/A:1008215702611.
2. Andreas Löhne (2011). Vector Optimization with Infimum and Supremum. Springer. pp. 162–169. ISBN 9783642183508.
3. Ehrgott, Matthias; Löhne, Andreas; Shao, Lizhen (2011). "A dual variant of Benson's "outer approximation algorithm" for multiple objective linear programming". Journal of Global Optimization. 52 (4): 757–778. doi:10.1007/s10898-011-9709-y. ISSN 0925-5001.
4. Heyde, Frank; Löhne, Andreas (2008). "Geometric Duality in Multiple Objective Linear Programming" (PDF). SIAM Journal on Optimization. 19 (2): 836–845. doi:10.1137/060674831. ISSN 1052-6234.