Monomial tropical cones for multicriteria optimization

TL;DR

This paper introduces a novel algorithm for multicriteria optimization that leverages tropical convex hull computations and a unique duality, improving the efficiency of finding all nondominated points.

Contribution

It presents a new algorithm utilizing tropical convexity and duality for multicriteria optimization, extending existing methods with a different mathematical approach.

Findings

Algorithm computes all nondominated points with complexity similar to previous methods.

Uses tropical convex hulls and a special duality related to monomial ideals.

Achieves efficient enumeration of solutions in multicriteria problems.

Abstract

We present an algorithm to compute all $n$ nondominated points of a multicriteria discrete optimization problem with $d$ objectives using at most $\mathcal{O}(n^{\lfloor d/2 \rfloor})$ scalarizations. The method is similar to algorithms by Przybylski et al. (2010) and by Klamroth et al. (2015) with the same complexity. As a difference, our method employs a tropical convex hull computation, and it exploits a particular kind of duality which is special for the tropical cones arising. This duality can be seen as a generalization of the Alexander duality of monomial ideals.